The trissotetras: or, a most exquisite table for resolving all manner of triangles, whether plaine or sphericall, rectangular or obliquangular, with greater facility, then ever hitherto hath been practised: most necessary for all such as would attaine to the exact knowledge of fortification, dyaling, navigation, surveying, architecture, the art of shadowing, taking of heights, and distances, the use of both the globes, perspective, the skill of making the maps, the theory of the planets, the calculating of their motions, and of all other astronomicall computations whatsoever. Now lately invented, and perfected, explained, commented on, and with all possible brevity, and perspicuity, in the hiddest, and most re-searched mysteries, from the very first grounds of the science it selfe, proved, and convincingly demonstrated. / By Sir Thomas Urquhart of Cromartie Knight. Published for the benefit of those that are mathematically affected.
         Urquhart, Thomas, Sir, 1611-1660.
      
       
         
           1645
        
      
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             The trissotetras: or, a most exquisite table for resolving all manner of triangles, whether plaine or sphericall, rectangular or obliquangular, with greater facility, then ever hitherto hath been practised: most necessary for all such as would attaine to the exact knowledge of fortification, dyaling, navigation, surveying, architecture, the art of shadowing, taking of heights, and distances, the use of both the globes, perspective, the skill of making the maps, the theory of the planets, the calculating of their motions, and of all other astronomicall computations whatsoever. Now lately invented, and perfected, explained, commented on, and with all possible brevity, and perspicuity, in the hiddest, and most re-searched mysteries, from the very first grounds of the science it selfe, proved, and convincingly demonstrated. / By Sir Thomas Urquhart of Cromartie Knight. Published for the benefit of those that are mathematically affected.
             Urquhart, Thomas, Sir, 1611-1660.
          
           [14], 96, [18] p. : ill.
           
             Printed by Iames Young.,
             London, :
             1645.
          
           
             Reproduction of the original in the British Library.
          
        
      
    
     
       
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           Mathematics -- Early works to 1800.
        
      
    
     
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           THE
           TRISSOTETRAS
           :
           OR
           A
           MOST
           EXQUISITE
           TABLE
           FOR
           Resolving
           all
           manner
           of
           Triangles
           ,
           whether
           Plaine
           or
           Sphericall
           ,
           Rectangular
           or
           Obliquangular
           ,
           with
           greater
           facility
           ,
           then
           ever
           hitherto
           hath
           been
           practised
           :
           Most
           necessary
           for
           all
           such
           as
           would
           attaine
           to
           the
           exact
           knowledge
           of
           Fortification
           ,
           Dyaling
           ,
           Navigation
           ,
           Surveying
           ,
           Architecture
           ,
           the
           Art
           of
           Shadowing
           ,
           taking
           of
           Heights
           ,
           and
           Distances
           ,
           the
           use
           of
           both
           the
           Globes
           ,
           Perspective
           ,
           the
           skill
           of
           making
           of
           Maps
           ,
           the
           Theory
           of
           the
           Planets
           ,
           the
           calculating
           of
           their
           motions
           ,
           and
           of
           all
           other
           Astronomicall
           computations
           whatsoever
           .
        
         
           Now
           lately
           invented
           ,
           and
           perfected
           ,
           explained
           ,
           commented
           on
           ,
           and
           ,
           with
           all
           possible
           brevity
           ,
           and
           perspicuity
           ,
           in
           the
           hiddest
           ,
           and
           most
           re-searched
           mysteries
           ,
           from
           the
           very
           first
           grounds
           of
           the
           Science
           it selfe
           ,
           proved
           ,
           and
           convincingly
           demonstrated
           .
        
         
           By
           Sir
           THOMAS
           URQUHART
           of
           Cromartie
           Knight
           .
        
         
           Published
           for
           the
           benefit
           of
           those
           that
           are
           Mathematically
           affected
           .
        
         
           LONDON
           ,
           Printed
           by
           
             Iames
             Young.
          
           1645.
           
        
      
       
         
         
         
           TO
           THE
           RIGHT
           HONOVRABLE
           ,
           And
           most
           noble
           LADY
           ,
           My
           deare
           and
           loving
           Mother
           ,
           the
           Lady
           DOWAGER
           of
           Cromartie
           .
        
         
           
             MADAM
             ,
          
        
         
           FILIALL
           duty
           being
           the
           more
           binding
           in
           me
           ,
           that
           I
           doe
           owe
           it
           to
           the
           best
           of
           Mothers
           ;
           if
           in
           the
           discharge
           thereof
           I
           observe
           not
           the
           usuall
           manner
           of
           other
           sonnes
           ,
           I
           am
           the
           lesse
           to
           blame
           ,
           that
           their
           obligation
           is
           not
           so
           great
           as
           mine
           :
           Therefore
           in
           that
           doe
           
           presume
           to
           imprint
           your
           Ladiships
           name
           in
           the
           Frontispiece
           of
           this
           Book
           ,
           and
           proffer
           unto
           you
           a
           Dedication
           of
           that
           ,
           which
           is
           beyond
           the
           capacity
           of
           other
           Ladies
           ;
           my
           boldnesse
           therein
           is
           the
           more
           excusable
           ,
           that
           in
           your
           person
           the
           most
           
             vertuous
             Woman
          
           in
           the
           world
           is
           intreated
           to
           Patronize
           that
           ,
           which
           by
           the
           
             learnedest
             men
          
           may
           happily
           be
           perused
           .
        
         
           I
           am
           confident
           
             (
             Madam
          
           )
           that
           your
           gracious
           acceptance
           of
           this
           Present
           is
           the
           more
           easily
           obtainable
           ,
           in
           that
           it
           is
           a
           grand-child
           of
           your
           own
           ,
           whom
           I
           thus
           make
           tender
           of
           ,
           to
           be
           sheltered
           under
           the
           favor
           of
           your
           protection
           ;
           and
           that
           unto
           your
           Ladiship
           it
           will
           not
           be
           the
           more
           unwelcome
           ,
           for
           proceeding
           from
           the
           braines
           of
           him
           ,
           whose
           body
           is
           not
           more
           yours
           by
           generation
           ,
           then
           by
           a
           most
           equitable
           purchase
           are
           the
           faculties
           of
           his
           mind
           ;
           the
           dominion
           which
           over
           my
           better
           halfe
           you
           ,
           by
           your
           goodnesse
           ,
           have
           acquired
           ,
           being
           ,
           in
           regard
           of
           my
           obedience
           ,
           no
           lesse
           voluntary
           ,
           then
           that
           of
           the
           other
           is
           for
           procreation
           naturall
           .
        
         
           Thus
           
             (
             Madam
          
           )
           unto
           you
           doe
           I
           totally
           belong
           ,
           but
           so
           ,
           as
           that
           those
           exteriour
           parts
           of
           mine
           ,
           which
           by
           birth
           are
           from
           your
           Ladiship
           derived
           ,
           cannot
           be
           more
           fortunate
           in
           this
           their
           subjection
           (
           notwithstanding
           the
           egregious
           advantages
           of
           bloud
           ,
           and
           
           consanguinity
           thereby
           to
           them
           accruing
           )
           then
           my selfe
           am
           happy
           (
           as
           from
           my
           heart
           I
           doe
           acknowledge
           it
           )
           in
           the
           just
           right
           ,
           your
           Ladiship
           hath
           to
           the
           eternall
           possession
           of
           the
           never-dying
           powers
           of
           my
           soule
           .
           For
           ,
           though
           (
           Soveraignty
           excepted
           )
           there
           be
           none
           in
           this
           Island
           more
           honourably
           descended
           then
           is
           your
           Ladiship
           ,
           nor
           whose
           progenitors
           ,
           these
           many
           ages
           past
           ,
           have
           been
           (
           on
           either
           side
           )
           of
           a
           more
           Noble
           extraction
           :
           Yet
           ,
           laying
           apart
           
             Nobility
             ,
             beauty
             ,
             wealth
             ,
             parentage
             ,
          
           and
           friends
           ,
           which
           (
           together
           with
           many
           other
           gifts
           of
           fortune
           )
           have
           hitherto
           served
           to
           adorn
           your
           Ladiship
           beyond
           others
           of
           your
           sex
           ,
           who
           for
           all
           these
           have
           been
           deservedly
           renowned
           ;
           and
           (
           in
           some
           measure
           )
           not
           esteeming
           that
           properly
           to
           be
           yours
           ,
           the
           receiving
           whereof
           did
           not
           altogether
           depend
           upon
           your
           owne
           election
           :
           it
           is
           the
           treasure
           of
           those
           excellent
           graces
           ,
           wherewith
           inwardly
           you
           are
           enriched
           ,
           that
           ,
           in
           praising
           of
           your
           Ladiship
           is
           most
           to
           be
           pitch'd
           upon
           ,
           and
           for
           the
           which
           you
           are
           most
           highly
           to
           be
           commended
           ;
           seeing
           by
           the
           means
           of
           them
           ,
           you
           ,
           from
           your
           tenderest
           yeeres
           upwards
           ,
           untill
           this
           time
           ,
           in
           the
           state
           of
           both
           Virginity
           and
           Matrimony
           ,
           have
           so
           constantly
           ,
           and
           indefatigably
           proceeded
           in
           the
           course
           of
           vertue
           ,
           with
           such
           alacrity
           fixed
           your
           gallant
           thoughts
           on
           the
           sweetnesse
           
           thereof
           ,
           and
           thereunto
           so
           firmly
           and
           cheerefully
           devoted
           all
           your
           words
           and
           actions
           ,
           as
           if
           righteousnesse
           in
           your
           Ladiship
           had
           been
           an
           inbred
           quality
           ,
           and
           that
           in
           your
           Will
           there
           had
           beene
           no
           aptitude
           of
           declining
           from
           the
           way
           of
           reason
           .
        
         
           This
           much
           is
           sufficiently
           well
           known
           to
           those
           ,
           that
           have
           at
           any
           time
           enjoyed
           the
           honour
           of
           your
           Ladiships
           conversation
           ,
           by
           whose
           most
           unpartiall
           reports
           ,
           the
           Splendour
           of
           your
           reputation
           is
           both
           in
           this
           ,
           and
           foraine
           Nations
           accounted
           precious
           ,
           in
           the
           minds
           even
           of
           those
           ,
           that
           have
           never
           seen
           you
           .
           But
           in
           so
           much
           more
           especially
           ,
           doe
           the
           most
           judicious
           of
           either
           sex
           admire
           the
           rare
           and
           sublime
           endowments
           ,
           wherewith
           your
           Ladiship
           is
           qualified
           ,
           that
           (
           as
           a
           patterne
           of
           perfection
           ,
           worthy
           to
           be
           universally
           followed
           )
           other
           Ladies
           (
           of
           what
           dignity
           soever
           )
           are
           truly
           by
           them
           esteemed
           of
           the
           choiser
           merit
           ,
           the
           nearer
           they
           draw
           to
           the
           Paragon
           proposed
           ,
           and
           resemble
           your
           Ladiship
           ;
           for
           that
           ,
           by
           vertue
           of
           your
           beloved
           society
           ,
           your
           neighbouring
           Countesses
           ,
           and
           other
           greater
           Dames
           of
           your
           kindred
           and
           acquaintance
           ,
           become
           the
           more
           illustrious
           in
           your
           imitation
           ;
           amidst
           whom
           ,
           as
           Cynthia
           amongst
           the
           obscurer
           Planets
           ,
           your
           Ladiship
           shines
           ,
           and
           darteth
           the
           Angelick
           rayes
           of
           your
           matchlesse
           example
           
           on
           the
           spirits
           of
           those
           ,
           who
           by
           their
           good
           Genius
           have
           been
           brought
           into
           your
           favourable
           presence
           to
           be
           enlightned
           by
           them
           .
        
         
           Now
           
             (
             Madam
          
           )
           lest
           ,
           by
           insisting
           any
           longer
           upon
           this
           straine
           ,
           I
           should
           seeme
           to
           offend
           that
           modesty
           ,
           and
           humility
           ,
           which
           (
           without
           derogating
           from
           your
           heroick
           vertues
           )
           are
           seated
           in
           a
           considerable
           place
           of
           your
           soule
           ,
           I
           will
           here
           ,
           in
           all
           submission
           ,
           most
           humbly
           take
           my
           leave
           of
           your
           Ladiship
           ,
           and
           beseech
           Almighty
           God
           ,
           that
           it
           may
           please
           his
           Divine
           Majesty
           so
           to
           blesse
           your
           Ladiship
           with
           continuance
           of
           dayes
           ,
           that
           the
           sonnes
           of
           those
           whom
           I
           have
           not
           as
           yet
           begot
           ,
           may
           attaine
           to
           the
           happinesse
           of
           presenting
           unto
           your
           Ladiship
           a
           brain-babe
           of
           more
           sufficiencie
           and
           consequence
           ;
           and
           that
           your
           Ladiship
           may
           live
           with
           as
           much
           health
           ,
           and
           prosperity
           ,
           to
           accept
           thereof
           ,
           and
           cherish
           it
           then
           ,
           as
           (
           I
           hope
           )
           you
           doe
           now
           ,
           at
           your
           vouchsafing
           to
           receive
           this
           ,
           which
           (
           though
           disproportionable
           ,
           both
           to
           your
           Ladiships
           high
           deserts
           ,
           and
           to
           that
           fervencie
           of
           willingnesse
           in
           me
           ,
           sometime
           to
           make
           offer
           of
           what
           is
           of
           better
           worth
           ,
           and
           more
           sutable
           to
           the
           grandour
           of
           your
           acceptance
           )
           in
           all
           sincerity
           of
           heart
           (
           confiding
           in
           that
           candour
           and
           ingenuity
           ,
           wherby
           your
           Ladiship
           is
           accustomed
           to
           value
           gifts
           ,
           
           according
           to
           the
           intention
           of
           the
           giver
           )
           and
           in
           all
           duty
           ,
           and
           lowlinesse
           of
           mind
           ,
           together
           with
           my selfe
           in
           whole
           ,
           and
           all
           my
           best
           endeavours
           ,
           I
           tender
           unto
           your
           Ladiship
           ,
           as
           becometh
           ,
           
             (
             Madam
          
           )
        
         
           
             Your
             Ladiships
             most
             affectionate
             Sonne
             and
             humble
             servant
             ,
             THOMAS
             URQUHART
             .
          
        
      
       
         
         
           To
           the
           Reader
           .
        
         
           TO
           write
           of
           Trigonometry
           ,
           and
           not
           make
           mention
           of
           the
           illustrious
           Lord
           Neper
           of
           Marchiston
           ,
           the
           inventer
           of
           Logarithms
           ,
           were
           to
           be
           unmindfull
           of
           him
           that
           is
           our
           daily
           Benefactor
           ;
           these
           artificiall
           numbers
           by
           him
           first
           excogitated
           and
           perfected
           ,
           being
           of
           such
           incomparable
           use
           ,
           that
           ,
           by
           them
           ,
           we
           may
           operate
           more
           in
           one
           day
           ,
           and
           with
           lesse
           danger
           of
           errour
           ,
           then
           can
           be
           done
           without
           them
           in
           the
           space
           of
           a
           whole
           week
           .
           A
           secret
           which
           would
           have
           beene
           so
           precious
           to
           Antiquity
           ,
           that
           Pythagoras
           ,
           all
           the
           seven
           wise
           men
           of
           
             Greece
             ,
             Archimedes
             ,
             Socrates
             ,
             Plato
             ,
             Euclid
             ,
          
           and
           Aristotle
           ,
           had
           (
           if
           coaevals
           )
           joyntly
           adored
           him
           ,
           and
           unanimously
           concurred
           to
           the
           deifying
           of
           the
           revealer
           of
           so
           great
           a
           Mystery
           :
           and
           truly
           (
           besides
           them
           )
           a
           great
           many
           other
           learned
           men
           ,
           who
           for
           the
           laboriousnesse
           of
           long
           and
           various
           Multiplications
           ,
           Divisions
           ,
           and
           Radicall
           extractions
           of
           severall
           sorts
           ,
           were
           deterred
           from
           the
           prosecuting
           ,
           and
           divulging
           of
           their
           knowledge
           in
           the
           chiefest
           ,
           and
           most
           noble
           parts
           of
           the
           Mathematicks
           ,
           would
           have
           left
           behind
           them
           diverse
           exquisite
           Volumes
           ,
           of
           an
           incomprehensible
           value
           ,
           if
           the
           Arithmetical
           equality
           of
           difference
           ,
           agreeable
           to
           every
           continued
           Geometricall
           proportion
           ,
           had
           been
           made
           known
           unto
           them
           .
        
         
           Wherefore
           ,
           I
           am
           infallibly
           perswaded
           ,
           that
           ,
           in
           the
           estimation
           of
           scientifically
           disposed
           spirits
           ,
           the
           Philosophers
           stone
           is
           but
           trash
           to
           this
           invention
           ,
           which
           will
           alwayes
           (
           in
           their
           judicious
           opinions
           )
           be
           accounted
           of
           more
           worth
           to
           the
           Mathematicall
           world
           ,
           then
           was
           the
           finding
           out
           of
           America
           ,
           to
           the
           King
           of
           Spaine
           ;
           or
           the
           discovery
           of
           the
           nearest
           
           way
           to
           the
           East-indies
           ,
           would
           be
           to
           the
           Northerly
           occidentall
           Merchants
           .
        
         
           What
           the
           merit
           then
           of
           the
           Author
           is
           ,
           let
           the
           most
           envious
           judge
           :
           for
           my
           owne
           part
           ,
           I
           doe
           not
           praise
           him
           so
           much
           ,
           for
           that
           he
           is
           my
           Compatriot
           ,
           as
           I
           must
           extoll
           the
           happinesse
           of
           my
           Countrey
           ,
           for
           having
           produced
           so
           brave
           a
           spark
           ,
           in
           whom
           alone
           (
           I
           may
           with
           confidence
           averre
           ,
           it
           is
           more
           glorious
           ,
           then
           if
           it
           had
           beene
           the
           conquering
           Kingdome
           of
           five
           hundred
           potent
           Nations
           :
           for
           ,
           by
           how
           much
           the
           gifts
           of
           the
           mind
           ,
           are
           more
           excellent
           then
           those
           of
           either
           body
           or
           fortune
           ;
           by
           so
           much
           the
           divine
           effects
           of
           the
           faculties
           thereof
           ,
           are
           of
           greater
           consequence
           ,
           then
           what
           is
           performed
           by
           meer
           force
           of
           Armes
           ,
           or
           chance
           of
           Warre
           .
        
         
           I
           might
           say
           more
           in
           commendation
           of
           this
           gallant
           man
           ,
           but
           that
           my
           discourse
           being
           directed
           to
           the
           Reader
           ,
           he
           will
           possibly
           expect
           to
           be
           entertained
           with
           some
           other
           purpose
           then
           Encomiasticks
           ;
           and
           therefore
           ,
           to
           undeceive
           him
           of
           those
           hopes
           (
           if
           any
           such
           there
           be
           )
           I
           will
           assure
           him
           ,
           that
           to
           no
           other
           end
           I
           did
           require
           his
           observance
           here
           ,
           but
           to
           be
           informed
           by
           me
           of
           the
           laudable
           endowments
           of
           that
           honourable
           Baron
           ,
           whose
           eminencie
           above
           others
           (
           wher-ever
           he
           be
           spoke
           of
           )
           deserveth
           such
           an
           ample
           Elogie
           by
           it selfe
           ,
           that
           the
           paper
           ,
           graced
           with
           the
           receiving
           of
           his
           name
           and
           character
           ,
           should
           not
           be
           blurred
           with
           the
           course
           impression
           of
           any
           other
           stuffe
           .
        
         
           However
           the
           Reader
           ought
           not
           to
           conceive
           amisse
           for
           his
           being
           detained
           so
           long
           upon
           this
           Eulogistick
           subject
           ,
           without
           the
           variety
           of
           any
           peculiar
           instruction
           bestowed
           on
           him
           ;
           seeing
           I
           am
           certaine
           there
           is
           nothing
           more
           advantagious
           to
           him
           ,
           or
           that
           more
           efficaciously
           can
           tend
           to
           his
           improvement
           ,
           then
           the
           imitation
           of
           that
           admirable
           Gentleman
           ,
           whose
           immortall
           fame
           ,
           in
           spite
           of
           time
           ,
           will
           out-last
           all
           ages
           ,
           and
           look
           eternity
           in
           the
           face
           .
        
         
           
             The
             Readers
             well-wisher
             .
             T.
             U.
             
          
        
      
       
         
         
           An
           Epaenetick
           and
           Doxologetick
           Expresse
           ,
           in
           commendation
           of
           this
           Book
           ,
           and
           the
           Author
           thereof
           .
        
         
           
             To
             all
             Philomathets
             .
          
        
         
           SEeing
           Trigonometry
           ,
           which
           handsomly
           unlocketh
           the
           choycest
           ,
           and
           most
           intime
           mysteries
           of
           the
           Mathematicks
           ,
           hath
           beene
           hitherto
           exposed
           to
           the
           world
           in
           a
           method
           ,
           whose
           intricacy
           deterreth
           many
           from
           adventuring
           on
           it
           ;
           We
           are
           all
           ,
           and
           every
           one
           of
           us
           ,
           by
           duty
           bound
           to
           acknowledge
           our selves
           beholding
           to
           the
           Author
           of
           this
           Treatise
           ;
           who
           ,
           by
           reducing
           all
           the
           secrets
           of
           that
           noble
           Science
           into
           a
           most
           exquisite
           order
           ,
           hath
           so
           facilitated
           the
           way
           to
           the
           Learner
           ,
           that
           in
           seven
           weeks
           ,
           at
           most
           ,
           he
           may
           attaine
           to
           more
           knowledge
           therein
           ,
           then
           otherwise
           he
           could
           doe
           for
           his
           heart
           in
           the
           space
           of
           a
           twelve-moneth
           :
           And
           who
           ,
           for
           the
           better
           encouragement
           of
           the
           studious
           ,
           hath
           so
           gentilely
           expatiated
           his
           spirits
           upon
           all
           its
           Actioms
           ,
           Principles
           ,
           Analogies
           ,
           precepts
           ,
           and
           whole
           subjected
           matter
           ;
           that
           this
           Mathematicall
           Tractate
           doth
           no
           lesse
           bespeak
           him
           a
           good
           Poet
           ,
           and
           good
           Orator
           ,
           then
           by
           his
           elaboured
           Poems
           he
           hath
           showne
           himselfe
           already
           a
           good
           Philosopher
           ,
           and
           Mathematician
           .
        
         
           Thus
           doth
           the
           various
           mixture
           of
           most
           excellent
           qualities
           
           in
           him
           ,
           give
           such
           evidence
           of
           the
           transcendent
           faculties
           of
           his
           mind
           ,
           that
           ,
           as
           the
           Muses
           never
           yet
           inspired
           sublimer
           conceptions
           in
           a
           more
           refined
           stile
           ,
           then
           is
           to
           be
           found
           in
           the
           accurate
           strain
           of
           his
           most
           ingenious
           Epigrams
           :
           so
           ,
           on
           the
           other
           part
           ,
           are
           the
           abstrusest
           difficulties
           of
           this
           Science
           by
           him
           so
           neatly
           unfolded
           ,
           and
           with
           such
           exactnesse
           hath
           he
           resolved
           the
           hardest
           ,
           and
           most
           intangled
           doubts
           thereof
           ;
           that
           ,
           I
           may
           justly
           say
           ,
           what
           praise
           (
           in
           his
           Epistle
           ,
           or
           rather
           Preface
           ,
           to
           the
           Reader
           )
           he
           hath
           beene
           pleased
           (
           out
           of
           his
           ingenuity
           ,
           to
           confer
           on
           the
           learned
           ,
           and
           honourable
           Neper
           ,
           doth
           ,
           without
           any
           diminution
           ,
           in
           every
           jot
           ,
           as
           duly
           belong
           unto
           himselfe
           .
        
         
           For
           ,
           I
           am
           certainly
           perswaded
           ,
           he
           that
           useth
           Logarithms
           ,
           shall
           not
           gaine
           so
           much
           time
           on
           the
           Worker
           by
           the
           naturall
           Sines
           and
           Tangents
           ,
           as
           ,
           by
           vertue
           of
           this
           succinct
           manner
           of
           calculation
           ,
           shall
           be
           got
           on
           him
           that
           knoweth
           it
           not
           ,
           how
           compendiously
           soever
           else
           ,
           with
           Addition
           and
           Subtraction
           ,
           or
           Addition
           alone
           ,
           he
           frame
           his
           Computations
           .
           However
           ,
           he
           who
           ,
           together
           with
           that
           of
           the
           Logarithms
           ,
           maketh
           use
           of
           this
           invention
           ,
           is
           in
           a
           way
           which
           will
           bring
           him
           so
           straight
           and
           readily
           to
           the
           perfect
           practice
           of
           Trigonometry
           ,
           that
           ,
           compared
           with
           the
           old
           beaten
           path
           ,
           trod
           upon
           by
           
             Regiomontanus
             ,
             Ptolomy
          
           ,
           and
           other
           ancient
           Mathematicians
           ,
           it
           is
           like
           the
           Sea
           voyage
           ,
           in
           regard
           of
           that
           by
           Land
           ,
           betwixt
           the
           two
           Pillars
           of
           
             (
             Hercules
          
           commonly
           called
           ,
           the
           Straits
           of
           Gibraltar
           )
           whereof
           the
           one
           is
           but
           of
           six
           houres
           sailing
           at
           most
           ,
           and
           the
           other
           a
           journey
           of
           seven
           thousand
           long
           miles
           .
        
         
         
           If
           we
           then
           consider
           how
           a
           great
           many
           ,
           despairing
           ever
           to
           get
           out
           (
           if
           once
           entred
           )
           of
           the
           confused
           obscurity
           wherein
           the
           doctrine
           of
           Triangles
           hath
           beene
           from
           time
           to
           time
           involved
           ,
           have
           rather
           contented
           themselves
           barely
           with
           Scale
           and
           Compasse
           ,
           and
           other
           mechanick
           tooles
           and
           instruments
           ,
           to
           prosecute
           their
           operations
           ,
           and
           in
           any
           reasonable
           measure
           to
           glance
           somewhat
           neare
           the
           truth
           ,
           then
           ,
           thorough
           so
           many
           pesterments
           ,
           and
           harsh
           incumbrances
           ,
           to
           touch
           it
           to
           a
           point
           ,
           in
           its
           most
           indivisible
           and
           infallible
           reality
           .
           And
           how
           others
           ,
           for
           all
           their
           being
           more
           industrious
           ,
           in
           proving
           their
           Conclusions
           by
           the
           Mediums
           from
           which
           they
           are
           necessarily
           inferr'd
           ,
           are
           neverthelesse
           (
           even
           when
           they
           have
           bestowed
           half
           an
           age
           in
           the
           Trigonometricall
           practice
           )
           oftentimes
           so
           farre
           to
           seek
           ,
           that
           ,
           without
           a
           great
           deale
           of
           premeditation
           advisement
           ,
           and
           recollecting
           of
           themselves
           ,
           they
           know
           not
           how
           to
           discusse
           some
           queries
           ,
           corollaries
           ,
           problems
           ,
           consectaries
           ,
           proportions
           ,
           wayes
           of
           perpendicular
           falling
           ,
           and
           other
           such
           like
           occurring
           debatable
           matters
           ,
           incident
           to
           the
           scientifick
           measuring
           of
           Triangles
           ;
           We
           cannot
           choose
           (
           these
           things
           being
           maturely
           perpended
           )
           but
           be
           much
           taken
           with
           the
           pregnancy
           of
           this
           device
           ,
           whereby
           we
           shall
           sooner
           hit
           to
           a
           minut
           upon
           the
           verity
           of
           an
           Angle
           or
           Side
           demanded
           ,
           and
           trace
           it
           to
           the
           very
           source
           and
           originall
           ,
           from
           whence
           it
           flowes
           ,
           then
           another
           mechanically
           shall
           be
           able
           to
           come
           within
           three
           degrees
           thereof
           ,
           although
           he
           cannot
           ,
           for
           the
           same
           little
           he
           doth
           ,
           afford
           any
           reason
           at
           all
           ;
           And
           so
           suddenly
           resolve
           any
           Trigonometricall
           question
           (
           without
           
           paines
           or
           labour
           ,
           how
           perplexed
           soever
           it
           be
           )
           with
           all
           the
           dependances
           thereto
           belonging
           ,
           as
           if
           it
           were
           a
           knowledge
           meerly
           infused
           from
           above
           ,
           and
           revealed
           by
           the
           peculiar
           inspiration
           of
           some
           favourable
           Angel.
           
        
         
           Besides
           these
           advantages
           ,
           administred
           unto
           us
           by
           the
           meanes
           of
           this
           exquisite
           Book
           ,
           this
           maine
           commodity
           accreweth
           to
           the
           diligent
           Perusers
           of
           it
           ,
           that
           ,
           instead
           of
           three
           quarters
           of
           a
           yeere
           ,
           usually
           by
           Professors
           allowed
           to
           their
           Schollers
           for
           the
           right
           conceiving
           of
           this
           Science
           ,
           which
           (
           notwithstanding
           )
           through
           any
           little
           discontinuance
           ,
           is
           by
           them
           so
           apt
           to
           be
           forgotten
           ,
           that
           the
           expence
           of
           a
           week
           or
           two
           will
           hardly
           suffice
           to
           reseat
           it
           in
           their
           memories
           ;
           they
           shall
           not
           need
           ,
           by
           this
           method
           ,
           to
           bestow
           above
           a
           moneth
           ,
           and
           with
           such
           ease
           and
           facility
           for
           retention
           ,
           when
           they
           have
           learned
           and
           acquired
           it
           ,
           that
           ,
           if
           multiplicity
           of
           businesses
           ,
           or
           serious
           plodding
           upon
           other
           studies
           happen
           to
           blot
           it
           out
           of
           their
           minds
           ,
           they
           may
           as
           firmly
           recover
           in
           one
           quarter
           of
           an
           hour
           the
           whole
           knowledge
           and
           remembrance
           thereof
           ,
           as
           when
           they
           had
           it
           best
           ,
           and
           were
           most
           punctually
           versed
           in
           it
           .
        
         
           A
           secret
           (
           in
           my
           opinion
           )
           so
           precious
           ,
           that
           (
           as
           the
           Author
           spoke
           of
           Marchiston
           )
           I
           may
           with
           the
           like
           pertinencie
           avouch
           of
           him
           ,
           that
           his
           Countrey
           and
           kindred
           would
           not
           have
           been
           more
           honoured
           by
           him
           ,
           had
           he
           purchased
           millions
           of
           gold
           ,
           and
           severall
           rich
           territories
           of
           a
           great
           and
           vast
           extent
           ;
           then
           for
           this
           subtile
           and
           divine
           invention
           ,
           which
           will
           out-last
           the
           continuance
           of
           any
           inheritance
           ,
           and
           remaine
           fresh
           in
           the
           understandings
        
         
         
           of
           men
           of
           profound
           Literature
           ,
           when
           houses
           and
           possessions
           will
           change
           their
           owners
           ,
           the
           wealthy
           become
           poor
           ,
           and
           the
           children
           of
           the
           needy
           enjoy
           the
           treasures
           of
           those
           ,
           whose
           heires
           are
           impoverished
           .
        
         
           Therefore
           ,
           seeing
           for
           the
           many-fold
           uses
           thereof
           in
           divers
           Arts
           and
           Sciences
           ,
           in
           speculation
           and
           practice
           ,
           peace
           and
           war
           ,
           sport
           and
           earnest
           ,
           with
           the
           admirable
           furtherances
           we
           reape
           by
           it
           ,
           in
           the
           knowledge
           of
           Sea
           and
           Land
           ,
           and
           Heaven
           and
           Earth
           ,
           it
           cannot
           be
           otherwise
           then
           permanent
           ,
           together
           with
           the
           Authors
           fame
           ,
           so
           long
           as
           any
           of
           those
           endure
           ;
           I
           will
           (
           God
           willing
           )
           in
           the
           ruines
           of
           all
           these
           ,
           and
           when
           time
           it selfe
           is
           expired
           ,
           in
           testimony
           of
           my
           thankfulnesse
           in
           particular
           for
           so
           great
           a
           benefit
           ,
           (
           if
           after
           the
           Resurrection
           ,
           there
           be
           any
           complementall
           affability
           )
           expresse
           my selfe
           then
           ,
           as
           I
           doe
           now
           ,
        
         
           
             The
             Authors
             most
             affectionate
             ,
             and
             most
             humbly
             devoted
             servant
             J.
             A.
             
          
        
      
       
         
         
           The
           Diatyposis
           of
           the
           whole
           Doctrine
           of
           Triangles
           .
        
         
           
             The
             plane
             Triangles
             have
             13.
             
             Moods
             .
             
               
                 Planorectangulars
                 7.
                 
                 
                   
                     1
                     Upalem
                     .
                     
                  
                   
                     2
                     Uberman
                     .
                     
                  
                   
                     3
                     Uphener
                     .
                     
                  
                   
                     4
                     Ekarul
                     .
                     
                  
                   
                     5
                     Egalem
                     .
                     
                  
                   
                     6
                     Echemun
                     .
                     
                  
                   
                     7
                     Etena●
                     .
                     
                  
                
              
               
                 Planobliquangulars
                 .
                 6.
                 
                 
                   
                     1
                     Danarele
                     .
                     
                  
                   
                     2
                     There●abmo
                     .
                     
                  
                   
                     3
                     Zelemabne
                     .
                     
                  
                   
                     4
                     Xemenoro
                     .
                     
                  
                   
                     5
                     Shenerolem
                     .
                     
                  
                   
                     6
                     Pserelema
                     .
                     
                  
                
              
               
                 The
                 Sphericals
                 have
                 28
                 Moods
                 .
                 
                   
                     Orthogonosphericals
                     .
                     16.
                     
                     
                       
                         1.
                         
                         Upalam
                         .
                      
                       
                         2.
                         
                         Ubamen
                         .
                      
                       
                         3.
                         
                         Uphanep
                         .
                      
                       
                         4.
                         
                         Ukelamb
                         .
                      
                       
                         5.
                         
                         Ugemon
                         .
                      
                       
                         6.
                         Uchener
                      
                       
                         7.
                         E●alum
                      
                       
                         8.
                         
                         Edamon
                         .
                      
                       
                         9.
                         
                         Ethaner
                         .
                      
                       
                         10.
                         
                         Ezolum
                         .
                      
                       
                         11.
                         
                         Exoman
                         .
                      
                       
                         12.
                         
                         Epsoner
                         .
                      
                       
                         13.
                         
                         Alamun
                         .
                      
                       
                         14.
                         
                         Amaner
                         .
                      
                       
                         15.
                         
                         Enerul
                         .
                      
                       
                         16.
                         
                         E●elam
                         .
                      
                    
                     
                       
                         
                           Of
                           these
                           16.
                           
                           Mood
                           〈◊〉
                           Moods
                           of
                           V●●●gen
                           ●re
                           prounded
                           upon
                           the
                           Axi●●re
                           of
                           Supro●●●
                           .
                        
                         
                           The
                           8.
                           of
                           Pubkutetkepsaler
                           on
                           Sbaprotea
                           :
                           and
                           the
                           5.
                           of
                           Uchedezexam
                           on
                           Proso
                        
                      
                    
                     
                       By
                       these
                       16.
                       
                       Representatives
                       ,
                       1.
                       
                       Le●
                       .
                       2.
                       
                       Yet
                       .
                       3.
                       
                       R●c
                       4.
                       
                       Cle.
                       5.
                       
                       Lu.
                       6.
                       
                       Tul.
                       7.
                       
                       Tere.
                       8.
                       
                       Tol.
                       9.
                       
                       Le●
                       10.
                       
                       At.
                       11.
                       
                       Tul.
                       12.
                       
                       Clet
                       .
                       13.
                       
                       Cret
                       .
                       14.
                       
                       Tur.
                       15.
                       
                       Tur.
                       16.
                       
                       Le
                       (
                       A.
                       signifying
                       an
                       oblique
                       Angle
                       .
                       E.
                       the
                       Perpendicular
                       U.
                       the
                       subtendent
                       C.
                       Initiall
                       ,
                       the
                       complemēt
                       of
                       a
                       side
                       to
                       a
                       quadrant
                       ●
                       .
                       finall
                       ,
                       the
                       side
                       continued
                       to
                       the
                       Radius
                       or
                       a
                       Quadrant
                       .
                       I.
                       left
                       .
                       R.
                       right
                       .
                       and
                       T.
                       one
                       of
                       the
                       top
                       Triangles
                       of
                       the
                       Scheme
                       )
                       it
                       is
                       evidenced
                       in
                       what
                       part
                       of
                       the
                       Diagram
                       the
                       Analogy
                       of
                       my
                       of
                       the
                       16.
                       
                       Moods
                       begins
                       ,
                       which
                       being
                       once
                       knowne
                       ,
                       the
                       progressive
                       sequence
                       of
                       the
                       proportionable
                       Sides
                       &
                       Angles
                       is
                       easily
                       discerned
                       out
                       of
                       the
                       orderly
                       in
                       volutions
                       of
                       the
                       Figure
                       it selfe
                       .
                       Here
                       it
                       is
                       to
                       be
                       observed
                       ,
                       that
                       as
                       the
                       Book
                       explaineth
                       the
                       Trissotetral
                       Table
                       :
                       so
                       this
                       Trigonodiatyposis
                       unfoldeth
                       ▪
                       all
                       the
                       intricate
                       difficulties
                       of
                       the
                       Book
                       .
                    
                  
                   
                     Loxogonosphericals
                     .
                     12.
                     
                     
                       That
                       the
                       Schemes
                       and
                       Types
                       of
                       Triangulary
                       Analogies
                       are
                       not
                       seated
                       in
                       the
                       roomes
                       ,
                       where
                       they
                       are
                       treated
                       of
                       ,
                       I
                       purposely
                       have
                       done
                       it
                       ;
                       to
                       the
                       end
                       ,
                       that
                       being
                       all
                       perceived
                       at
                       one
                       view
                       ,
                       their
                       multiplicity
                       (
                       which
                       would
                       appeare
                       confused
                       in
                       their
                       ●ispersed
                       method
                       )
                       might
                       ●●ot
                       any
                       way
                       discourage
                       〈…〉
                       der
                       :
                       besides
                       that
                       ,
                       this
                       their
                       ●eing
                       together
                       in
                       their
                       ●u●
                       order
                       ,
                       and
                       rancked
                       ●ecording
                       to
                       the
                       exigence
                       of
                       the
                       Sides
                       or
                       Angles
                       ,
                       is
                       such
                       a
                       furtherance
                       to
                       the
                       memory
                       ,
                       and
                       illustration
                       to
                       the
                       judgement
                       ,
                       that
                       it
                       maketh
                       Trigonometry
                       ,
                       which
                       of
                       all
                       Sciences
                       was
                       accounted
                       the
                       abstrusest
                       ,
                       to
                       be
                       in
                       effect
                       the
                       most
                       〈◊〉
                       and
                       〈◊〉
                       .
                    
                     
                       
                         Monurgeticks
                         .
                         4
                         
                           
                             1
                             Lamaneprep
                             .
                             
                          
                           
                             2
                             Menerolo
                             .
                             
                          
                           
                             3
                             Nerelema
                             .
                             
                          
                           
                             4
                             Ralam●●●
                             
                          
                        
                      
                       
                         
                         Of
                         the
                         Disergeticks
                         there
                         be
                         8.
                         
                         Moods
                         ,
                         each
                         whereof
                         is
                         divided
                         into
                         foure
                         Cases
                         .
                         
                           
                             Ahalebmane
                             .
                             
                               
                                 1.
                                 Alamebna
                                 
                                   
                                     Dasimforaug
                                     .
                                     
                                  
                                   
                                     Dadisfo●●ug
                                     .
                                     
                                  
                                   
                                     Dadisgatin
                                     .
                                     
                                  
                                   
                                     Simomatin
                                     .
                                     
                                  
                                
                              
                               
                                 2.
                                 
                                 Alamebne
                                 .
                                 
                                   
                                     Dasimforauxy
                                     .
                                     
                                  
                                   
                                     Dadiscracforeug
                                     .
                                     
                                  
                                   
                                     Dadiscramgatin
                                     .
                                     
                                  
                                   
                                     Simomatin
                                     .
                                     
                                  
                                
                              
                            
                          
                           
                             Ahamepnare
                             .
                             
                               
                                 〈…〉
                                 
                                   
                                     Dadissepamforaur
                                     .
                                     
                                  
                                   
                                     Dadissexamforeur
                                     .
                                     
                                  
                                   
                                     Dasimatin
                                     .
                                     
                                  
                                   
                                     Simomatin
                                     .
                                     
                                  
                                
                              
                               
                                 4.
                                 Ammaneprela
                                 
                                   
                                     Dadissepamfor
                                     .
                                     
                                  
                                   
                                     Dadisse●amfor
                                     .
                                     
                                  
                                   
                                     Dasimin
                                     .
                                     
                                  
                                   
                                     Simomatin
                                     .
                                     
                                  
                                
                              
                            
                          
                           
                             Ehenabrole
                             .
                             
                               
                                 〈…〉
                                 
                                   
                                     Dacramfor
                                     .
                                     
                                  
                                   
                                     Damracfor
                                     .
                                     
                                  
                                   
                                     Dasimquzin
                                     .
                                     
                                  
                                   
                                     Simomatin
                                     .
                                     
                                  
                                
                              
                               
                                 6.
                                 
                                 E●neral●la
                                 .
                                 
                                   
                                     Dacforamb
                                     .
                                     
                                  
                                   
                                     Damforac
                                     .
                                     
                                  
                                   
                                     Da●imat●m
                                     .
                                     
                                  
                                   
                                     Simomatin
                                     .
                                     
                                  
                                
                              
                            
                          
                           
                             Eherolabme
                             
                               
                                 〈…〉
                                 
                                   
                                     Dacracforeur
                                     .
                                     
                                  
                                   
                                     Dambracforeur
                                     .
                                     
                                  
                                   
                                     Dacrambatin
                                     .
                                     
                                  
                                   
                                     Simomatin
                                     .
                                     
                                  
                                
                              
                               
                                 8.
                                 E●relome
                                 
                                   
                                     Dakyxamfor
                                     .
                                     
                                  
                                   
                                     Dambyxamfor
                                     .
                                     
                                  
                                   
                                     Dakypambin
                                     .
                                     
                                  
                                   
                                     Simomatin
                                     .
                                     
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
       
         
         
           In
           Eruditum
           D.
           
             Thomae
             Vrquharti
          
           equitis
           Trissotetrados
           librum
           .
        
         
           SI
           cupis
           aetherios
           tutò
           peragrare
           meatus
           ,
        
         
           Et
           sulcare
           audes
           si
           vada
           salsa
           maris
           :
        
         
           Vel
           tibi
           si
           cordi
           est
           terrae
           spatia
           ampla
           metiri
           ,
        
         
           Huc
           ades
           ,
           hunc
           doctum
           percipe
           mente
           librum
           .
        
         
           Hoc
           ,
           sine
           Daedaleis
           pennis
           volitare
           per
           auras
           ,
        
         
           Et
           sine
           Neptuno
           nare
           per
           alta
           vales
           .
        
         
           Hoc
           duce
           ,
           jam
           Lybicos
           poteris
           superare
           calores
           ,
        
         
           Atque
           pati
           Scythici
           frigora
           saeva
           poli
           .
        
         
           Perge
           Thoma
           ;
           tali
           tandem
           gaudebit
           alumno
        
         
           SCOTIA
           ,
           quam
           scriptis
           tollis
           in
           astra
           tuis
           .
        
         
           
             Al.
             Ross
             .
          
        
      
    
     
       
         
         
           POSITIONS
           .
        
         
           EVery
           Circle
           is
           divided
           into
           three
           hundred
           and
           sixty
           parts
           ,
           called
           Degrees
           ,
           whereof
           each
           one
           is
           Sexagesimated
           ,
           Subsexagesimated
           ,
           Resubsexagesimated
           ,
           and
           Biresubsexagesimated
           ,
           in
           Minutes
           ,
           Seconds
           ,
           Thirds
           ,
           Fourths
           ,
           and
           so
           far
           forth
           as
           any
           Computist
           is
           pleased
           to
           proceed
           for
           the
           exactnesse
           of
           a
           Research
           ,
           in
           the
           calculation
           of
           any
           Orbiculary
           Dimension
           .
        
         
           2.
           
           As
           Degrees
           are
           the
           measure
           of
           Arches
           ,
           so
           are
           they
           of
           Angles
           ;
           but
           that
           those
           are
           called
           Circumferentiall
           ,
           these
           Angulary
           Degrees
           ,
           each
           whereof
           is
           the
           three
           hundred
           and
           sixtieth
           part
           of
           four
           right
           Angles
           ,
           which
           are
           nothing
           else
           but
           the
           surface
           of
           a
           Plain
           to
           any
           point
           circumjacent
           ;
           for
           any
           space
           whatsoever
           about
           a
           point
           ,
           is
           divided
           in
           360.
           parts
           :
           And
           the
           better
           to
           conceive
           the
           Analogie
           that
           is
           betwixt
           these
           two
           sorts
           of
           graduall
           Measures
           ,
           we
           must
           know
           ,
           that
           there
           is
           the
           same
           proportion
           of
           any
           Angle
           to
           4.
           right
           Angles
           ,
           as
           of
           an
           arch
           of
           so
           many
           circumferential
           degrees
           to
           the
           whole
           circumference
           .
        
         
           3.
           
           Hence
           is
           it
           ,
           that
           the
           same
           number
           serves
           the
           Angle
           ,
           and
           the
           Arch
           that
           vaults
           it
           ,
           and
           that
           divers
           quantities
           are
           measured
           (
           as
           it
           were
           )
           with
           the
           same
           graduall
           measure
           .
           Angles
           and
           Arches
           then
           are
           Analogicall
           ,
           and
           the
           same
           reason
           is
           of
           both
           .
        
         
           4.
           
           Seeing
           any
           given
           proportion
           may
           be
           found
           in
           numbers
           ,
           and
           that
           any
           two
           quantities
           have
           the
           same
           proportion
           that
           the
           two
           numbers
           have
           ,
           according
           to
           the
           which
           they
           are
           measured
           :
           if
           for
           the
           measuring
           of
           Triangles
           there
           must
           be
           certain
           proportions
           of
           all
           the
           parts
           of
           a
           Triangle
           ,
           to
           one
           another
           known
           ,
           and
           those
           proportions
           explained
           in
           numbers
           it
           is
           most
           certain
           ,
           all
           Magnitudes
           ,
           being
           Figures
           at
           least
           in
           power
           ,
           and
           all
           Figures
           either
           Triangles
           ,
           or
           Triangled
           ,
           that
           the
           Arithmeticall
           Solution
           of
           any
           Geometricall
           question
           ,
           dependeth
           on
           the
           Doctrine
           of
           Triangles
           .
        
         
           5.
           
           And
           though
           the
           proportion
           betwixt
           the
           parts
           of
           a
           Triangle
           cannot
           be
           without
           some
           errour
           ;
           because
           of
           crooked
           lines
           to
           right
           lines
           ,
           and
           of
           crooked
           lines
           amongst
           themselves
           ,
           the
           reason
           is
           inscrutable
           ,
           no
           man
           being
           able
           to
           finde
           out
           the
           exact
           proportion
           of
           the
           Diameter
           to
           the
           Circumference
           :
           yet
           both
           in
           plain
           Triangles
           ,
           where
           the
           measure
           of
           the
           Angles
           is
           of
           a
           different
           species
           from
           the
           sides
           ,
           and
           in
           Sphericalls
           ,
           wherein
           both
           the
           Angles
           and
           sides
           are
           of
           a
           circular
           nature
           ,
           crooked
           lines
           are
           in
           some
           measure
           reduced
           to
           right
           lines
           by
           the
           definition
           of
           quantity
           which
           right
           lines
           ,
           viz.
           Sines
           ,
           Tangents
           ,
           
           and
           Secants
           ,
           applyed
           to
           a
           Circle
           have
           in
           respect
           of
           the
           Radius
           ,
           o●
           half-Diameter
           .
        
         
           6.
           
           And
           therefore
           ,
           though
           the
           Circles
           Quadrature
           be
           not
           found
           out
           ,
           it
           being
           in
           our
           power
           to
           make
           the
           Diameter
           ,
           or
           the
           semi-Diameter
           ,
           which
           is
           the
           Radius
           of
           as
           many
           parts
           as
           we
           please
           ,
           and
           being
           sure
           so
           much
           the
           more
           that
           the
           Radius
           be
           taken
           ,
           the
           error
           will
           be
           the
           lesser
           ;
           for
           albeit
           the
           Sines
           ,
           Tangents
           and
           Secants
           ,
           be
           irrationall
           thereto
           for
           the
           most
           part
           ,
           and
           their
           proportion
           inexplicable
           by
           any
           number
           whatsoever
           ,
           whither
           whole
           or
           broken
           :
           yet
           if
           they
           be
           rightly
           made
           ,
           they
           will
           be
           such
           ,
           as
           that
           in
           them
           all
           no
           number
           will
           be
           different
           from
           the
           truth
           by
           an
           integer
           ,
           or
           unity
           of
           those
           parts
           ,
           whereof
           the
           Radius
           is
           taken
           :
           which
           is
           so
           exactly
           done
           by
           some
           ,
           especially
           by
           Petiscus
           ,
           who
           assumed
           a
           Radius
           of
           twenty
           six
           places
           ,
           that
           according
           to
           his
           supputation
           (
           the
           Diameter
           of
           the
           Earth
           being
           known
           ,
           and
           the
           Globe
           thereof
           supposed
           to
           be
           perfectly
           round
           )
           one
           should
           not
           fail
           in
           the
           dimension
           of
           its
           whole
           Circuit
           ,
           the
           nine
           hundreth
           thousand
           scantling
           of
           the
           Million
           part
           of
           an
           Inch
           ,
           and
           yet
           not
           be
           able
           ,
           for
           all
           that
           ,
           to
           measure
           it
           without
           amisse
           ;
           for
           so
           indivisible
           the
           truth
           of
           a
           thing
           is
           ,
           that
           come
           you
           never
           so
           neer
           it
           ,
           unlesse
           you
           hit
           upon
           it
           just
           to
           a
           point
           ,
           there
           is
           an
           errour
           still
           .
        
      
       
         
           DEFINITIONS
           .
        
         
           A
           Cord
           ,
           or
           Subtense
           ,
           is
           a
           right
           line
           ,
           drawn
           from
           the
           one
           extremity
           to
           the
           other
           of
           an
           Arch.
           
        
         
           2.
           
           A
           right
           Sine
           is
           the
           half
           Cord
           of
           the
           double
           Arch
           proposed
           ,
           and
           from
           one
           extremity
           of
           the
           Arch
           falleth
           perpendicularly
           on
           the
           Radius
           ,
           passing
           by
           the
           other
           end
           thereof
           .
        
         
           3.
           
           A
           Tangent
           is
           a
           right
           line
           ,
           drawn
           from
           the
           Secant
           by
           one
           end
           of
           the
           Arch
           ,
           perpendicularly
           on
           the
           extremity
           of
           the
           Diameter
           ,
           passing
           by
           the
           other
           end
           of
           the
           said
           Arch.
           
        
         
           4.
           
           A
           Secant
           is
           the
           prolonged
           Radius
           ,
           which
           passeth
           by
           the
           upper
           extremity
           of
           the
           Arch
           ,
           till
           it
           meet
           with
           the
           sine
           Tangent
           of
           the
           said
           Arch.
           
        
         
           5.
           
           Complement
           is
           the
           difference
           betwixt
           the
           lesser
           Arch
           ,
           and
           a
           Quadrant
           ,
           or
           betwixt
           a
           right
           Angle
           and
           an
           Acute
           .
        
         
           6.
           
           The
           complement
           to
           a
           semi-Circle
           ,
           is
           the
           difference
           betwixt
           the
           half-Circumference
           and
           any
           Arch
           lesser
           ,
           or
           betwixt
           two
           right
           Angles
           ,
           and
           an
           Oblique
           Angle
           ,
           whither
           blunt
           or
           sharp
           .
        
         
           7.
           
           The
           versed
           sine
           is
           the
           remainder
           of
           the
           Radius
           ,
           the
           sine
           
           Complement
           being
           subtracted
           from
           it
           ,
           and
           though
           great
           use
           may
           be
           made
           of
           the
           versed
           sines
           ,
           for
           finding
           out
           of
           the
           Angles
           by
           the
           sides
           ,
           and
           sides
           by
           the
           Angles
           :
           yet
           in
           Logarithmicall
           calculations
           they
           are
           altogether
           uselesse
           ,
           and
           therefore
           in
           my
           Trissotetras
           there
           is
           no
           mention
           made
           of
           them
           .
        
         
           8.
           
           In
           Amblygonosphericall●
           ,
           which
           admit
           both
           of
           an
           Extrinsecall
           ,
           and
           Intrinsecall
           demission
           of
           the
           perpendicular
           ,
           nineteen
           severall
           parts
           are
           to
           be
           considered
           :
           viz.
           The
           Perpendicular
           ,
           the
           Subtendentall
           ,
           the
           Subtendentine
           ,
           two
           Cosubtendents
           ,
           the
           Basall
           ,
           the
           Basidion
           ,
           the
           chief
           Segment
           of
           the
           Base
           ,
           two
           Cobases
           ,
           the
           double
           Verticall
           ,
           the
           Verticall
           ,
           the
           Verticaline
           ,
           two
           Coverticalls
           ,
           the
           next
           Cathetopposite
           ,
           the
           prime
           Cathetopposite
           ,
           and
           the
           two
           Cocathetopposites
           :
           fourteen
           whereof
           ,
           (
           to
           wit
           )
           the
           Subtendentall
           ,
           the
           Subtendentine
           ,
           the
           Cosubtendents
           ,
           the
           Basall
           ,
           the
           Basidion
           ,
           the
           Cobases
           ,
           the
           Verticall
           ,
           the
           Verticaline
           ,
           the
           Coverticalls
           ,
           and
           Cocathetopposites
           ,
           are
           called
           the
           first
           ,
           either
           Subtendent
           ,
           Base
           ,
           Topangle
           ,
           or
           Cocathetopposite
           ,
           whither
           in
           the
           great
           Triangle
           or
           the
           little
           ,
           or
           in
           the
           Correctangle
           ,
           if
           they
           be
           ingredients
           of
           that
           Rectangular
           ,
           whereof
           most
           parts
           are
           known
           ,
           which
           parts
           are
           alwayes
           a
           Subtendent
           and
           a
           Cathetopposite
           :
           but
           if
           they
           be
           in
           the
           other
           Triangle
           ,
           they
           are
           called
           the
           second
           Subtendents
           ,
           Bases
           ,
           and
           so
           forth
           .
        
         
           9.
           
           The
           externall
           double
           Verticall
           is
           included
           by
           the
           Perpendicular
           ,
           and
           Subtendentall
           ,
           and
           divided
           by
           the
           Subtendentine
           :
           the
           internall
           is
           included
           by
           cosubtendents
           ,
           and
           divided
           by
           the
           Perpendicular
           .
        
      
       
         
           APODICTICKS
           .
        
         
           THe
           Angles
           made
           by
           a
           right
           Line
           ,
           falling
           on
           another
           right
           Line
           ,
           are
           equall
           to
           two
           right
           Angles
           ;
           because
           every
           Angle
           being
           measured
           by
           an
           Arch
           ,
           or
           part
           of
           a
           Circumference
           ,
           and
           a
           right
           Angle
           by
           ninety
           Degrees
           ,
           if
           upon
           the
           middle
           of
           the
           ground
           line
           ,
           as
           Center
           ,
           be
           described
           a
           semi-Circle
           ,
           it
           will
           be
           the
           measure
           of
           the
           Angles
           ,
           comprehended
           betwixt
           the
           falling
           ,
           and
           sustaining
           lines
           .
        
         
           2.
           
           Hence
           it
           is
           ,
           that
           the
           four
           opposite
           Angles
           made
           by
           one
           line
           ,
           crossing
           another
           ,
           are
           always
           each
           to
           its
           own
           opposite
           equall
           ;
           for
           if
           upon
           the
           point
           of
           Intersection
           ,
           as
           Center
           ,
           be
           described
           a
           Circle
           ,
           every
           two
           of
           those
           Angles
           will
           fill
           up
           the
           semi-Circle
           ;
           therefore
           the
           first
           and
           second
           will
           be
           equall
           to
           the
           second
           and
           third
           ,
           and
           consequently
           the
           second
           ,
           which
           is
           the
           common
           Angle
           to
           both
           these
           couples
           being
           removed
           ,
           the
           first
           will
           remain
           equall
           to
           the
           
           third
           ,
           and
           by
           the
           same
           reason
           ,
           the
           second
           to
           the
           fourth
           ,
           which
           was
           to
           be
           demonstrated
           .
        
         
           3.
           
           If
           a
           right
           line
           falling
           upon
           two
           other
           right
           lines
           ,
           make
           the
           alternat
           Angles
           equall
           ,
           these
           lines
           must
           needs
           be
           Paralell
           ;
           for
           if
           they
           did
           meet
           ,
           the
           alternat
           Angles
           would
           not
           be
           equall
           ;
           because
           in
           all
           plain
           Triangles
           ,
           the
           outward
           Angle
           is
           greater
           ,
           then
           any
           of
           the
           remote
           inward
           Angles
           ,
           which
           is
           proved
           by
           the
           first
           .
        
         
           4.
           
           If
           one
           of
           the
           sides
           of
           a
           Triangle
           be
           produced
           ,
           the
           outward
           Angle
           is
           equall
           to
           both
           the
           inner
           ,
           and
           opposite
           Angles
           together
           ;
           because
           according
           to
           the
           acclining
           or
           declining
           of
           the
           conterminall
           side
           ,
           is
           left
           an
           Angulary
           space
           ,
           for
           the
           receiving
           of
           a
           paralell
           to
           the
           opposite
           side
           ,
           in
           the
           point
           of
           whose
           occourse
           at
           the
           base
           ,
           the
           Exterior
           Angle
           is
           divided
           into
           two
           ,
           which
           for
           their
           like
           ,
           and
           alternat
           situation
           with
           the
           two
           Interior
           Angles
           ,
           are
           equall
           each
           to
           its
           own
           conform
           to
           the
           nature
           of
           Angles
           ,
           made
           by
           a
           right
           line
           crossing
           divers
           paralells
           .
        
         
           5.
           
           From
           hence
           we
           gather
           ,
           that
           the
           three
           Angles
           of
           a
           plain
           Triangle
           ,
           are
           equall
           to
           two
           rights
           ;
           for
           the
           two
           inward
           ,
           being
           equall
           to
           the
           Externall
           one
           ,
           and
           there
           remaining
           of
           the
           three
           ,
           but
           one
           ,
           which
           was
           proved
           in
           the
           first
           Apodictick
           ,
           to
           be
           the
           Externall
           Angles
           complement
           to
           two
           rights
           ;
           it
           must
           needs
           fall
           forth
           (
           what
           are
           equall
           to
           a
           third
           ,
           being
           equall
           amongst
           themselves
           )
           that
           the
           three
           Angles
           of
           a
           plain
           Triangle
           ,
           are
           equall
           to
           two
           right
           Angles
           ,
           the
           which
           we
           undertook
           to
           prove
           .
        
         
           6.
           
           By
           the
           same
           reason
           ,
           the
           two
           acute
           of
           a
           Rectangled
           plain
           Triangle
           ,
           are
           equall
           to
           one
           right
           Angle
           ,
           and
           any
           one
           of
           them
           ,
           the
           others
           complement
           thereto
           .
        
         
           7.
           
           In
           every
           Circle
           ,
           an
           Angle
           from
           the
           Center
           ,
           is
           two
           in
           the
           Limb
           ,
           both
           of
           them
           having
           one
           part
           of
           the
           Circumference
           for
           base
           ;
           for
           being
           an
           Externall
           Angle
           ,
           and
           consequently
           equall
           to
           both
           the
           Intrinsecall
           Angles
           ,
           and
           therefore
           equall
           to
           one
           another
           ;
           because
           of
           their
           being
           subtended
           by
           equall
           bases
           ,
           viz.
           the
           semi-Diameters
           ,
           it
           must
           needs
           be
           the
           double
           of
           the
           foresaid
           Angle
           in
           the
           limb
           .
        
         
           8.
           
           Triangles
           standing
           between
           two
           paralells
           ,
           upon
           one
           and
           the
           fame
           base
           ,
           are
           equall
           ;
           for
           the
           Identity
           of
           the
           base
           ,
           whereon
           they
           are
           seated
           ,
           together
           with
           the
           Equidistance
           of
           the
           Lines
           ,
           within
           the
           which
           they
           are
           confined
           ,
           maketh
           them
           of
           such
           a
           nature
           ,
           that
           how
           long
           so
           ever
           the
           line
           paralell
           to
           the
           base
           be
           protracted
           ,
           the
           Diagonall
           cutting
           of
           in
           one
           off
           the
           Triangles
           ,
           as
           much
           of
           bredth
           ,
           as
           it
           
           gains
           of
           length
           ,
           (
           the
           ones
           losse
           accruing
           to
           the
           profit
           of
           the
           other
           )
           Quantifies
           them
           both
           to
           an
           equality
           ,
           the
           thing
           we
           did
           intend
           to
           prove
           .
        
         
           9.
           
           Hence
           do
           we
           inferre
           ,
           that
           Triangles
           betwixt
           two
           paralells
           ,
           are
           in
           the
           same
           proportion
           with
           their
           bases
           .
        
         
           10.
           
           Therefore
           if
           in
           a
           Triangle
           ,
           be
           drawn
           a
           paralell
           to
           any
           of
           the
           sides
           ,
           it
           divideth
           the
           other
           sides
           ,
           through
           which
           it
           passeth
           proportionally
           ;
           for
           besides
           that
           it
           maketh
           the
           four
           segments
           ,
           to
           be
           four
           bases
           ,
           it
           becomes
           (
           if
           two
           Diagonall
           lines
           be
           extended
           from
           the
           ends
           thereof
           ,
           to
           the
           ends
           of
           its
           paralell
           )
           a
           common
           base
           to
           two
           equall
           Triangles
           ,
           to
           which
           two
           ,
           the
           Triangle
           of
           the
           first
           two
           segments
           ,
           having
           reference
           according
           to
           the
           difference
           of
           their
           bases
           ,
           and
           these
           two
           being
           equall
           ,
           as
           it
           is
           to
           the
           one
           ,
           so
           must
           it
           be
           to
           the
           other
           ,
           and
           therefore
           the
           first
           base
           ,
           must
           be
           to
           the
           second
           ,
           (
           which
           are
           the
           Segments
           of
           one
           side
           of
           the
           Triangle
           )
           as
           the
           third
           to
           the
           fourth
           ,
           (
           which
           are
           the
           Segments
           of
           the
           second
           )
           all
           which
           was
           to
           be
           demonstrated
           .
        
         
           11.
           
           From
           hence
           do
           we
           collect
           ,
           that
           Equiangled
           Triangles
           have
           their
           sides
           about
           the
           equall
           Angles
           proportionall
           to
           one
           another
           .
           
             This
             sayes
          
           Petiscus
           ,
           
             is
             the
             golden
             Foundation
             ,
             and
             chief
             ground
             of
             Trigonometry
             .
          
        
         
           12.
           
           An
           Angle
           in
           a
           semi-Circle
           is
           right
           ;
           because
           it
           is
           equall
           to
           both
           the
           Angles
           at
           the
           base
           ,
           which
           (
           by
           cutting
           the
           Diameter
           in
           two
           )
           is
           perceivable
           to
           any
           .
        
         
           13.
           
           Of
           four
           proportionall
           lines
           ,
           the
           Rectangled
           figure
           ,
           made
           of
           the
           two
           extreames
           ,
           is
           equall
           to
           the
           Rectangular
           ,
           composed
           of
           the
           means
           ;
           for
           as
           four
           and
           one
           ,
           are
           equall
           to
           two
           and
           three
           ,
           by
           an
           Arithmeticall
           proportion
           :
           and
           the
           fourth
           term
           Geometrically
           exceeding
           ,
           or
           being
           lesse
           then
           the
           third
           ,
           as
           the
           second
           is
           more
           ,
           or
           lesse
           then
           the
           first
           ;
           what
           the
           fourth
           hath
           ,
           or
           wanteth
           ,
           from
           and
           above
           the
           third
           ,
           is
           supplyed
           ,
           or
           impaired
           by
           the
           Surplusage
           ,
           or
           deficiency
           of
           the
           first
           from
           and
           above
           the
           second
           :
           These
           Analogies
           being
           still
           taken
           in
           a
           Geometricall
           way
           ,
           make
           the
           oblong
           of
           the
           two
           middle
           ,
           equall
           to
           that
           of
           the
           extreams
           ,
           which
           was
           to
           be
           proved
           .
        
         
           14.
           
           In
           all
           plain
           Rectangled
           Triangles
           ,
           the
           Ambients
           are
           equall
           in
           power
           to
           the
           Subtendent
           ;
           for
           by
           demitting
           from
           the
           right
           Angle
           a
           Perpendicular
           ,
           there
           will
           arise
           two
           Correctangles
           ,
           from
           whose
           Equiangularity
           with
           the
           great
           Rectangle
           ,
           will
           proceed
           such
           a
           proportion
           amongst
           the
           Homologall
           sides
           ,
           of
           all
           the
           three
           ,
           that
           if
           you
           set
           them
           right
           in
           the
           rule
           ,
           beginning
           your
           Analogy
           at
           the
           main
           Subtendent
           ,
           
           (
           seeing
           the
           including
           sides
           of
           the
           totall
           Rectangle
           ,
           prove
           Subtendents
           in
           the
           partiall
           Correctangles
           ,
           and
           the
           bases
           of
           those
           Rectanglets
           ,
           the
           Segments
           of
           the
           great
           Subtendent
           )
           it
           will
           fall
           out
           ,
           that
           as
           the
           main
           Subtendent
           is
           to
           his
           base
           ,
           on
           either
           side
           (
           for
           either
           of
           the
           legs
           of
           a
           Rectangled
           Triangle
           ,
           in
           reference
           to
           one
           another
           ,
           is
           both
           base
           and
           Perpendicular
           )
           so
           the
           same
           bases
           ,
           which
           are
           Subtendents
           in
           the
           lesser
           Rectangles
           ,
           are
           to
           their
           bases
           ,
           the
           Segment
           ,
           of
           the
           prime
           Subtendent
           :
           Then
           by
           the
           Golden
           rule
           we
           find
           ,
           that
           the
           multiplying
           of
           the
           middle
           termes
           (
           which
           is
           nothing
           else
           ,
           but
           the
           squaring
           of
           the
           comprehending
           sides
           of
           the
           prime
           Rectangular
           )
           affords
           two
           products
           ,
           equall
           to
           the
           oblongs
           made
           of
           the
           great
           Subtendent
           ,
           and
           his
           respective
           Segments
           ,
           the
           aggregat
           whereof
           by
           equation
           is
           the
           same
           with
           the
           square
           of
           the
           chief
           Subtendent
           ,
           or
           Hypotenusa
           ,
           which
           was
           to
           be
           demonstrated
           .
        
         
           15.
           
           In
           every
           totall
           square
           ,
           the
           supplements
           about
           the
           partiall
           ,
           and
           Interior
           squares
           ,
           are
           equall
           the
           one
           to
           the
           other
           ;
           for
           by
           drawing
           a
           Diagonall
           line
           ,
           the
           great
           square
           being
           divided
           into
           two
           equall
           Triangles
           ,
           because
           of
           their
           standing
           on
           equall
           bases
           betwixt
           two
           paralells
           ,
           by
           the
           ninth
           Apodictick
           ,
           it
           is
           evident
           ,
           that
           in
           either
           of
           these
           great
           Triangles
           ,
           there
           being
           two
           partiall
           ones
           ,
           equall
           to
           the
           two
           of
           the
           other
           ,
           each
           to
           his
           own
           ,
           by
           the
           same
           Reason
           of
           the
           ninth
           :
           If
           from
           equall
           things
           
             (
             viz.
          
           the
           totall
           Triangles
           )
           be
           taken
           equall
           things
           (
           to
           wit
           ,
           the
           two
           pairs
           of
           partiall
           Triangles
           )
           equall
           things
           must
           needs
           remain
           ,
           which
           are
           the
           foresaid
           supplements
           ,
           whose
           equality
           I
           undertook
           to
           prove
           ,
        
         
           16.
           
           If
           a
           right
           line
           cut
           into
           two
           equall
           parts
           be
           increased
           ,
           the
           square
           made
           of
           the
           additonall
           line
           ,
           and
           one
           of
           the
           Bisegments
           ,
           joyned
           in
           one
           ,
           lesse
           by
           the
           Square
           of
           the
           half
           of
           the
           line
           Bisected
           ,
           is
           equall
           to
           the
           oblong
           contained
           under
           the
           prolonged
           line
           ,
           and
           the
           line
           of
           Continuation
           ;
           for
           if
           annexedly
           to
           the
           longest
           side
           of
           the
           proposed
           oblong
           ,
           be
           described
           the
           foresaid
           Square
           ,
           there
           will
           jet
           out
           beyond
           the
           Quadrat
           Figure
           ,
           a
           space
           or
           Rectangle
           ,
           which
           for
           being
           powered
           by
           the
           Bisegment
           and
           Additionall
           line
           ,
           will
           be
           equall
           to
           the
           neerest
           supplement
           ,
           and
           consequently
           to
           the
           other
           (
           the
           equality
           of
           supplements
           being
           proved
           by
           the
           last
           Apodictick
           )
           by
           vertue
           whereof
           ,
           a
           Gnomon
           in
           the
           great
           Square
           ,
           lacking
           nothing
           of
           its
           whole
           Area
           ,
           but
           the
           space
           of
           the
           square
           of
           the
           Bisected
           line
           ,
           is
           apparent
           to
           equalize
           the
           Parallelogram
           proposed
           ,
           which
           was
           to
           be
           demonstrated
           .
        
         
         
           17.
           
           From
           hence
           proceedeth
           this
           Sequell
           ,
           that
           if
           from
           any
           point
           without
           a
           circle
           ,
           two
           lines
           cutting
           it
           be
           protracted
           to
           the
           other
           extremity
           thereof
           ,
           making
           two
           cords
           ,
           the
           oblongs
           contained
           under
           the
           totall
           lines
           ,
           and
           the
           excesse
           of
           the
           Subtenses
           ,
           are
           equall
           one
           to
           another
           ;
           for
           whether
           any
           of
           the
           lines
           passe
           through
           the
           Center
           ,
           or
           not
           ,
           if
           the
           Subtenses
           be
           Bisected
           ,
           seeing
           all
           lines
           from
           the
           Center
           fall
           Perpendicularly
           upon
           the
           Chordall
           point
           of
           Bisection
           (
           because
           the
           two
           semi-Diameters
           ,
           and
           Bisegments
           substerned
           under
           equall
           Angles
           ,
           in
           two
           Triangles
           evince
           the
           equality
           of
           the
           third
           Angle
           ,
           to
           the
           third
           ,
           by
           the
           fift
           Apodictick
           ,
           which
           two
           Angles
           being
           made
           by
           the
           falling
           of
           one
           right
           line
           upon
           another
           ,
           must
           needs
           be
           right
           by
           the
           tenth
           definition
           of
           the
           first
           of
           Euchilde
           )
           the
           Bucarnon
           of
           Pythagoras
           ,
           demonstrated
           in
           my
           fourteenth
           Apodictick
           ,
           will
           by
           Quadrosubductions
           of
           Ambients
           ,
           from
           one
           another
           ,
           and
           their
           Quadrobiquadrequation●
           with
           the
           Hypotenusa
           ,
           together
           with
           other
           Analogies
           of
           equation
           with
           the
           powers
           of
           like
           Rectangular
           Triangles
           ,
           comprehended
           within
           the
           same
           circle
           ,
           manifest
           the
           equality
           of
           long
           Squares
           ,
           or
           oblongs
           Radically
           meeting
           in
           an
           Exterior
           point
           ,
           and
           made
           of
           the
           prolonged
           Subtenses
           ,
           and
           the
           lines
           of
           interception
           ,
           betwixt
           the
           limb
           of
           the
           circle
           ,
           and
           the
           point
           of
           concourse
           ,
           
             quod
             probandum
             fuit
          
           .
        
         
           18.
           
           Now
           to
           look
           back
           on
           the
           eleaventh
           Apodictick
           ,
           where
           according
           to
           Petiscus
           ,
           I
           said
           that
           upon
           the
           mutuall
           proportion
           of
           the
           sides
           of
           Equiangled
           Triangles
           ,
           is
           founded
           the
           whole
           Science
           of
           Trigonometry
           ,
           I
           do
           here
           respeak
           it
           ,
           and
           with
           confidence
           maintain
           the
           truth
           thereof
           ;
           because
           ,
           besides
           many
           others
           ,
           it
           is
           the
           ground
           of
           these
           Subsequent
           Theorems
           :
           1.
           
           The
           right
           sine
           of
           an
           Arch
           ,
           is
           to
           its
           co-sine
           ,
           as
           the
           Radius
           to
           the
           co-tangent
           of
           the
           said
           Arch.
           2.
           
           The
           co-sine
           of
           an
           Arch
           ,
           is
           to
           its
           sine
           ,
           as
           the
           Radius
           to
           the
           Tangent
           of
           the
           said
           Arch.
           3.
           
           The
           Sines
           ,
           and
           co-Secants
           :
           the
           Secants
           ,
           and
           co-Sines
           :
           and
           the
           Tangents
           ,
           and
           co-Tangents
           ,
           are
           reciprocally
           proportionall
           .
           4.
           
           The
           Radius
           is
           a
           mean
           proportionall
           ,
           betwixt
           the
           Sine
           ,
           and
           co-Secant
           :
           the
           Secant
           ,
           and
           co-Sine
           :
           and
           the
           Tangent
           ,
           and
           co-Tangent
           :
           The
           verity
           of
           all
           these
           ▪
           (
           If
           a
           Quadrant
           be
           described
           ,
           and
           upon
           the
           two
           Radiuses
           two
           Tangents
           ,
           and
           two
           or
           three
           Sines
           be
           erected
           (
           which
           in
           respect
           of
           other
           Arches
           will
           be
           co-Sines
           and
           co-Tangents
           )
           and
           two
           Secants
           drawn
           (
           which
           are
           likewise
           co-Secants
           )
           from
           the
           Center
           to
           the
           top
           of
           the
           Tangents
           )
           will
           appear
           by
           the
           foresaid
           reasons
           ,
           out
           of
           my
           eleaventh
           Apodictick
           .
        
      
       
         
         
           The
           Trissotetras
           .
        
         
           
             
               Plain
               .
            
             
               Sphericall
               .
            
          
           
             
               Plain
               Trissotetras
               .
               Axiomes
               four
               .
            
             
               
                 1.
                 Rulerst
              
               
                 Vradesso
                 :
              
               
                 Directory
                 :
              
               
                 Enodandas
                 .
              
            
             
               
                 Eradetul
                 :
              
               
                 Vphechet
                 :
              
               
                 3.
                 
                 Orth.
                 1.
                 
                 Obl.
                 
              
            
             
               
                 2.
                 Eproso
              
               
                 Directorie
                 :
              
               
                 Enodandas
              
               
                 3.
                 
                 
                   Ax.
                   Grediftal
                   :
                   Dir.
                
                 ●
                 .
              
            
             
               
                 Pubkegdaxesh
                 :
              
               
                 4.
                 
                 Orth.
                 
              
               
                 4.
                 
                 
                   Ax.
                   Bagrediffiu
                   :
                   Dir.
                
                 ●
                 .
              
            
             
               
                 3.
                 
                 Obl.
                 
              
            
          
           
             
               The
               Planorectangular
               Table
               :
               Figures
               four
               ▪
            
             
               
                 1.
                 
                 
                   Va*
                   le
                
              
               
                 
                   Datas
                   .
                   Quaesitas
                
                 .
              
               
                 Resolvers
                 .
              
            
             
               
                 
                   Vp*
                   Al§em
                
                 .
              
               
                 
                   Rad
                   —
                   V
                   —
                   Sapy
                
                 ☞
                 Yr.
                 
              
            
             
               
                 2.
                 
                 
                   Ve*
                   mane
                
              
               
                 
                   Vb*
                   em
                   §an
                
                 .
              
               
                 
                   V
                   —
                   Rad
                   —
                   Eg
                
                 ☞
                 So.
                 
              
            
             
               
                  
              
               
                 Praesubserv
                 .
              
               
                 Possubserv
                 .
              
            
             
               
                 
                   Vph*
                   en
                   §er
                
                 .
              
               
                 
                   Vb*
                   em
                   §an
                
                 .
              
               
                 
                   Vp*
                   al§em
                   ,
                   or
                   ,
                   Eg*
                   al§em
                   .
                
              
            
             
               
                 3.
                 
                 
                   Ena*
                   ve
                
              
               
                 
                   Ek*
                   ar
                   §ul
                
                 .
              
               
                 
                   Sapeg
                   —
                   Eg
                   —
                   Rad
                
                 ☞
                 Vr
                 .
              
            
             
               
                 
                   Eg*
                   al
                   §em
                
                 .
              
               
                 
                   Rad
                   —
                   Taxeg
                   —
                   Eg
                
                 ☞
                 Yr.
                 
              
            
             
               
                  
              
               
                  
              
               
                 Praesubserv
                 .
              
               
                 Possubserv
                 .
              
            
             
               
                 4.
                 
                 
                   Ere*
                   va
                
              
               
                 
                   Ech*
                   em
                   §un
                
                 .
              
               
                 
                   Et*
                   en
                   §ar
                
                 .
              
               
                 
                   Ek*
                   ar
                   §ul
                
                 .
              
            
             
               
                 
                   Et*
                   en
                   §ar
                
                 .
              
               
                 
                   E
                   —
                   Ge
                   —
                   Rad
                
                 ☞
                 Toge
                 .
              
            
          
           
             
               The
               Planobliquangular
               Table
               :
               Figures
               four
               .
            
             
               
                 1.
                 
                 Alahe
                 *
                 me
              
               
                 
                   Da*na*re
                   §le
                
                 .
              
               
                 
                   Sapeg
                   —
                   Eg
                   —
                   Sapyr
                
                 ☞
                 Yr.
                 
              
            
             
               
                 2.
                 Emena*role
              
               
                 
                   The*re*
                   lab
                   §mo
                
                 .
              
               
                 
                   Aggres
                   —
                   Zes
                   —
                   Talfagros
                
                 ☞
                 Talzo
                 .
              
            
             
               
                  
              
               
                 Praesubserv
                 .
              
               
                 Possubserv
                 .
              
            
             
               
                 
                   Ze*le*mab
                   §ne
                
                 .
              
               
                 
                   The*re*
                   lab
                   §mo
                
                 .
              
               
                 
                   Da*na*re
                   §le
                
                 .
              
            
             
               
                 3.
                 Enero*lome
              
               
                 
                   Xe*
                   me*
                   no
                   §ro
                
                 .
              
               
                 
                   E
                   —
                   So
                   —
                   Ge
                
                 ☞
                 So.
                 
              
               
                  
              
            
             
               
                  
              
               
                 Praesubserv
                 .
              
               
                 Possubserv
                 .
              
            
             
               
                 
                   She*ne*
                   ro
                   §lem
                
                 .
              
               
                 
                   Xe*
                   me*
                   no
                   §ro
                
                 .
              
               
                 
                   Da*na*re
                   §le
                
              
            
             
               
                  
              
               
                  
              
               
                 Praesubserv
                 .
              
               
                 Possubserv
                 .
              
            
             
               
                 4.
                 
                 
                   Erele*
                   a
                
              
               
                 
                   Pse*
                   re*
                   le
                   §ma
                
                 .
              
               
                 Bagreziu
                 .
              
               
                 
                   Vb*em
                   §an
                
                 .
              
            
             
               
                 Finall
                 Resolver
                 .
              
               
                 
                   Vxi●q
                   —
                   Rad
                   —
                   〈◊〉
                
                 —
                 ☞
                 Sor.
                 
              
            
          
           
             
             
               The
               Sphericall
               Trissotetras
               .
               Axiomes
               three
               .
            
             
               1.
               
               
                 Suprosca
                 .
                 Dir.
                 uphugen
              
               .
            
             
               2.
               
               
                 Sbaprotca
                 .
                 pubkutethepsaler
              
               .
            
             
               3.
               
               
                 Seproso
                 .
                 uchedezexam
              
               .
            
          
           
             
               The
               Orthogonospherical
               Table
               .
               Figures
               6.
               
               Datoquaeres
               16.
               
            
             
               
                  
              
               
                 Dat.
                 Quaes
                 .
              
               
                 Resolvers
                 .
              
            
             
               
                 1.
                 Valam*menep
              
               
                 Vp*al§am
                 .
              
               
                 
                   Torb
                   —
                   Tag
                   —
                   Nu
                
                 ☞
                 Mir.
                 
              
            
             
               
                 Vb*am§en
                 .
              
               
                 
                   Nag
                   —
                   Mu
                   —
                   Torp
                
                 ☞
                 Myr.
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   Torp
                   —
                   Mu
                   —
                   Lag
                
                 ☞
                 Myr.
                 
              
            
             
               
                 
                   Vph*an
                   §ep
                
                 .
              
               
                 
                   Tol
                   —
                   Sag
                   —
                   Su
                
                 ☞
                 Syr.
                 
              
            
             
               
                 2.
                 Veman*nore
              
               
                 Vk*el§amb
                 .
              
               
                 
                   Meg
                   —
                   Torp
                   —
                   Mu
                
                 ☞
                 Nir.
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   Torp
                   —
                   Teg
                   —
                   Mu
                
                 ☞
                 Nir.
                 
              
            
             
               
                 
                   Ug*em
                   §on
                
                 .
              
               
                 
                   Su
                   —
                   Seg
                   —
                   Tom
                
                 ☞
                 Sir.
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   Tom
                   —
                   Seg
                   —
                   Ru
                
                 ☞
                 Sir.
                 
              
            
             
               
                 
                   Uch*en
                   §er
                
                 .
              
               
                 
                   Neg
                   —
                   To
                   —
                   Nu
                
                 ☞
                 Nyr
                 .
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   To
                   —
                   Le
                   —
                   Nu
                
                 ☞
                 Nyr
                 .
              
            
             
               
                 3.
                 Enar*rulome
              
               
                 Et*al§um
                 .
              
               
                 
                   Torp
                   —
                   Me
                   —
                   Nag
                
                 ☞
                 Mur.
                 
              
            
             
               
                 Ed*am§on
                 .
              
               
                 
                   To
                   —
                   Neg
                   —
                   Sa
                
                 ☞
                 Nir.
                 
              
            
             
               
                 Eth*an§er
                 .
              
               
                 
                   Torb
                   —
                   Tag
                   —
                   Se
                
                 ☞
                 Tyr.
                 
              
            
             
               
                 4.
                 Erol*lumane
              
               
                 Ez*●l§um
                 .
              
               
                 
                   Sag
                   —
                   Sep
                   —
                   Rad
                
                 ☞
                 Sur.
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   Rad
                   —
                   Seg
                   —
                   Rag
                
                 ☞
                 Sur.
                 
              
            
             
               
                 
                   Ex*●●
                   §an
                
                 .
              
               
                 
                   Ne
                   —
                   To
                   —
                   Nag
                
                 ☞
                 Sir.
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   To
                   —
                   Le
                   —
                   Nag
                
                 ☞
                 Sir.
                 
              
            
             
               
                 
                   Eps*
                   on
                   §er
                
                 .
              
               
                 
                   Tag
                   —
                   Tolb
                   —
                   Te
                
                 ☞
                 Syr.
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   Tolb
                   —
                   Mag
                   —
                   Te
                
                 ☞
                 Syr.
                 
              
            
             
               
                 5.
                 
                 
                   Acha*
                   ve
                
              
               
                 
                   Al*
                   am
                   §un
                
                 .
              
               
                 
                   Tag
                   —
                   Torb
                   —
                   Ma
                
                 ☞
                 Nur.
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   Torb
                   —
                   Mag
                   —
                   Ma
                
                 ☞
                 Nur.
                 
              
            
             
               
                 
                   Am*
                   an
                   §er
                
                 .
              
               
                 
                   Say
                   —
                   Nag
                   —
                   T●
                
                 ☞
                 Nyr
                 .
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   Tω
                   —
                   Noy
                   —
                   Ray
                
                 ☞
                 Nyr
                 .
              
            
             
               
                 6.
                 Eshe*va
              
               
                 
                   En*er
                   §ul
                
                 .
              
               
                 
                   Ton
                   —
                   Neg
                   —
                   Ne
                
                 ☞
                 Nur.
                 
              
            
             
               
                 
                   Er*
                   el
                   §am
                
                 .
              
               
                 
                   Sei
                   —
                   Teg
                   —
                   Torb
                
                 ☞
                 Tir.
                 or
                 ,
              
            
             
               
                  
              
               
                 
                   Torb
                   —
                   Tepi
                   —
                   Rexi
                
                 ☞
                 Tir.
                 
              
            
          
        
         
         
           
             
               The
               Loxogonospherical
               Trissotetras
               .
            
             
               Monurgetick
            
             
               Disergetick
               .
            
          
           
             
               The
               Monurgetick
               Loxogonospherical
               Table
               .
               Axiomes
               two
               .
            
             
               1.
               
               
                 Seproso
                 .
                 Dir.
                 Lame
              
               .
               Figures
               two
               .
            
             
               2.
               
               
                 Parses
                 .
                 Dir.
                 Nera
              
               .
               Moods
               four
               .
            
          
           
             
               
                 Figures
                 .
              
               
                 Datas
                 .
              
               
                 Quaes
                 .
              
               
                 Resolvers
                 .
              
            
             
               
                 1.
                 Datamista
              
               
                 Lam*an*ep
              
               
                 §
                 rep
                 .
              
               
                 
                   Sapeg
                   —
                   Se
                   —
                   Sapy
                   ☞
                   Syr.
                   
                
              
            
             
               
                 Me*ne*ro
              
               
                 §
                 lo.
                 
              
               
                 
                   Sepag
                   —
                   Sa
                   —
                   Sepi
                   ☞
                   Sir.
                   
                
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                 ad
              
            
             
               
                 2.
                 Datapura
              
               
                 Ne*re*le
              
               
                 §
                 ma.
                 
              
               
                 
                   Hal
                   Basaldileg
                   Sad
                   Sab
                   Re
                   Regals
                   Bis*ir
                   .
                
              
            
             
               
                  
              
               
                  
              
               
                 ab
              
            
             
               
                  
              
               
                  
              
               
                 
                   Parses
                   —
                   Powto
                   —
                   Parsadsab
                   ☞
                   PowsalvertiR
                
              
            
             
               
                 Ra*la*ma
              
               
                 §
                 ne
                 .
              
               
                 
                   Kour
                   Bfasines
                   (
                   ereled
                   )
                   Kouf
                   Br*axypopyx
                   .
                
              
            
          
           
             
               The
               Loxogonospherical
               Disergeticks
               Axiomes
               foure
               .
            
             
               1.
               
               
                 Na
                 Bad
                 prosver
                 .
                 Dir.
                 Alama
                 .
              
            
             
               2.
               
               
                 Naverpr
                 or
                 Tes.
                 Allera
              
               .
            
             
               3.
               
               
                 Siubpror
                 Tab.
                 Ammena
              
               .
            
             
               4.
               
               
                 Niub
                 prodesver
                 .
                 Errenna
              
               .
            
          
           
             
               Figures
               4.
               
               Moods
               8.
               
            
             
               
                  
              
               
                 Fig.
                 
              
               
                 M.
                 
              
               
                 Sub
              
               
                 Res
                 .
              
               
                 Dat.
                 
              
               
                 Praen
                 .
              
               
                 Cathetothesis
                 .
              
               
                 Final
                 Resolvers
                 .
              
            
             
               
                 1.
                 
              
               
                 Ab
              
               
                 A
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Cafregpiq
                 .
              
               
                  
              
            
             
               
                  
              
               
                 La
              
               
                 Vp
              
               
                 Tag
              
               
                 ut
                 *
                 Op
              
               
                 §
                 At
              
               
                 Dasimforaug
              
               
                 Sat-nop-Seud
                 †
                 
                   nob
                   .
                   Kir
                
                 .
              
            
             
               
                 A
              
               
                 Meb
              
               
                 Al
              
               
                 Nu
              
               
                 ud
                 *
                 Ob
              
               
                 §
                 Aud
              
               
                 Dadisforeug
              
               
                 Saud-nob-Sat
                 †
                 
                   nop
                   .
                   Ir.
                
                 
              
            
             
               
                  
              
               
                 Na.
                 
              
               
                 Am
              
               
                 Mir
              
               
                 
                   uth*
                   Oph
                
              
               
                 §
                 Auth
              
               
                 Dadisgatin
              
               
                 Sauth-noph-Seuth
                 †
                 
                   nops
                   Ir.
                
                 
              
            
             
               
                 Leb
              
               
                 2.
                 
              
               
                 Sub.
                 
              
               
                 Res
                 .
              
               
                 Dat.
                 
              
               
                 P●ae●
                 .
              
               
                 Cathetothesis
                 .
              
               
                 Final
                 Resolvers
                 .
              
            
             
               
                  
              
               
                 Al
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Cafyxegeq
                 .
              
               
                  
              
            
             
               
                 Ma
              
               
                 La
              
               
                 Vp
              
               
                 Tag
              
               
                 ut
                 *
                 op
              
               
                 §
                 at
              
               
                 dasimforauxy
              
               
                 nat-mut-naud
                 †
                 mwd
              
            
             
               
                  
              
               
                 Meb
              
               
                 Al
              
               
                 Nu
              
               
                 ud
                 *
                 ob
              
               
                 §
                 aud
              
               
                 dadiscracforeug
              
               
                 naud-mud-nat
                 †
                 mwt
              
            
             
               
                 Ne
              
               
                 Ne
              
               
                 Am
              
               
                 Mir
              
               
                 uth
                 *
                 oph
              
               
                 §
                 auth
              
               
                 dadiscramgatin
              
               
                 nauth-muth-neuth
                 †
                 mwth
              
            
          
        
         
         
           
             
               
                 Fig.
                 M.
                 
              
               
                  
              
               
                  
              
               
                  
              
               
                 Cathetothesis
                 .
              
               
                 
                   Plus
                   minus
                
                 .
              
            
             
               
                 A
              
               
                 A
              
               
                 Sub.
                 
              
               
                 Re.
                 
              
               
                 Dat.
                 
              
               
                 Pr.
                 
              
               
                 Cafriq
                 .
              
               
                 Final
                 Resolvers
                 .
              
               
                 Sindifora
                 .
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 At
              
            
             
               
                  
              
               
                 Ma
              
               
                 up
              
               
                 Tag
              
               
                 ut*Op
              
               
                 §
                 At
              
               
                 Dadissepamforaur
              
               
                 
                   Nop-Sat-Nob
                   ☞
                
              
               
                 
                   Seudfr
                   Autir
                
                 .
              
            
             
               
                 Ha
              
               
                 Nep
              
               
                 Al
              
               
                 Nu
              
               
                 ud*Ob
              
               
                 §
                 Aud
              
               
                 Dadissexamforeur
              
               
                 
                   Nob-Saud-Nop
                   ☞
                
              
               
                 
                   Satfr
                   Eutir
                
                 .
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Aud
              
            
             
               
                  
              
               
                 Ra
              
               
                 Am
              
               
                 Mir
              
               
                 
                   uth*
                   Oph
                
              
               
                 §
                 Auth
              
               
                 Dasimatin
              
               
                 
                   Noph-Seuth-Nops
                   ☞
                
              
               
                 
                   Soethj
                   Authir
                
                 .
              
            
             
               
                 Mep
              
               
                 4.
                 
              
               
                  
              
               
                  
              
               
                  
              
               
                 Cathetothesis
                 .
              
               
                 
                   Plus
                   minus
                
                 .
              
            
             
               
                  
              
               
                 Am
              
               
                 Sub.
                 
              
               
                 Re.
                 
              
               
                 Dat.
                 
              
               
                 .
                 Pr.
                 
              
               
                 Cafregpagiq
                 .
              
               
                 Final
                 Resolvers
                 .
              
               
                 Sindiforiu
                 .
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Aet
              
            
             
               
                 Na
              
               
                 Ma
              
               
                 ub
              
               
                 Mu
              
               
                 
                   Ut*
                   Op
                
              
               
                 §
                 Aet
              
               
                 Dadissepamfor
              
               
                 
                   Tob-Top-Saet
                   ☞
                
              
               
                 
                   Soedfr
                   Dyr
                
                 .
              
            
             
               
                  
              
               
                 Nep
              
               
                 Am
              
               
                 Lag
              
               
                 
                   Ud*
                   Ob
                
              
               
                 §
                 Aed
              
               
                 Dadissexamfor
              
               
                 
                   Top-Tob-Saed
                   ☞
                
              
               
                 
                   Soetfr
                   Dyr
                
                 .
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Aed
              
            
             
               
                 Re
              
               
                 Reb
              
               
                 En
              
               
                 Myr
              
               
                 
                   Uth*
                   Oph
                
              
               
                 §
                 aeth
              
               
                 Dasimin
              
               
                 
                   Tops-Toph-SAEth
                   ☞
                
              
               
                 
                   Soethj
                   aeth
                   Syr.
                
                 
              
            
          
           
             
               
                 Fig.
                 M.
                 
              
               
                 Cathetothesis
                 .
              
            
             
               
                 Eb
              
               
                 En
              
               
                 Sub.
                 
              
               
                 Re.
                 
              
               
                 Dat.
                 
              
               
                 Pr.
                 
              
               
                 Cafregpigeq
                 .
              
               
                 Final
                 Resolvers
                 .
              
            
             
               
                  
              
               
                 Er
              
               
                 Ub
              
               
                 Mu
              
               
                 
                   Ut*
                   Op
                
              
               
                 §
                 aet
              
               
                 Dacramfor
              
               
                 Soed-Top-Saet
                 ☞
                 
                   Tob.
                   Kir
                
                 .
              
            
             
               
                 En
              
               
                 Ab
              
               
                 Am
              
               
                 Lag
              
               
                 
                   Ud*
                   Ob
                
              
               
                 §
                 ad
              
               
                 Damracfor
              
               
                 
                   Soet-Tob-Saed
                   ☞
                   Top.
                   Ir.
                
                 
              
            
             
               
                  
              
               
                 Lo
              
               
                 En
              
               
                 Myr
              
               
                 
                   Uth*
                   Oph
                
              
               
                 §
                 aeth
              
               
                 Dasimquaein
              
               
                 
                   Soeth-Toph-Saeth
                   ☞
                   Tops
                   .
                   Ir.
                
                 
              
            
             
               
                 Ab
              
               
                 6.
                 
              
               
                 Cathetothesis
                 .
              
            
             
               
                  
              
               
                 En
              
               
                 Sub.
                 
              
               
                 Re.
                 
              
               
                 Dat.
                 Q.
                 
              
               
                 Pr.
                 
              
               
                 Cafregpiq
                 .
              
               
                 Final
                 Resolvers
                 .
              
            
             
               
                 Ro
              
               
                 Ne
              
               
                 Ub
              
               
                 Mu
              
               
                 
                   ut*
                   Op
                
              
               
                 §
                 aet
              
               
                 Dacforamb
              
               
                 
                   Naet-Nut-Noed
                   ☞
                   Nwd
                   .
                   Yr.
                
                 
              
            
             
               
                  
              
               
                 Rab
              
               
                 Am
              
               
                 Lag
              
               
                 
                   ud*
                   Ob
                
              
               
                 §
                 aed
              
               
                 Damforac
              
               
                 
                   Naed-Nud-Noet
                   ☞
                   Nwt
                   .
                   Yr.
                
                 
              
            
             
               
                 Le
              
               
                 Le
              
               
                 En
              
               
                 Myr
              
               
                 
                   uth*
                   Oph
                
              
               
                 §
                 aeth
              
               
                 Dakinatam
              
               
                 
                   Naeth-Nuth-Noeth
                   ☞
                   Nwth
                   .
                   Yr.
                
                 
              
            
          
        
         
           
             
               
                 Fig.
                 M.
                 
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Cathteothesis
                 .
              
               
                 
                   Plus
                   minus
                
                 .
              
            
             
               
                 Eb
              
               
                 E
              
               
                 Sub.
                 
              
               
                 Re.
                 
              
               
                 Dat.
                 
              
               
                 Pr.
                 
              
               
                 Cafriq
                 .
              
               
                 Final
                 Resolvers
                 .
              
               
                 Sindifora
                 .
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 At
              
            
             
               
                  
              
               
                 Re
              
               
                 Up
              
               
                 Tag
              
               
                 
                   Ut*
                   Op
                
              
               
                 §
                 at
              
               
                 Dacracforaur
              
               
                 
                   Mut-Nat-Mwd
                   ☞
                
              
               
                 
                   Neudfr
                   Autir
                
                 .
              
            
             
               
                 Er
              
               
                 Lo
              
               
                 Al
              
               
                 Nu
              
               
                 
                   Ud*
                   Ob
                
              
               
                 §
                 aud
              
               
                 Dambracforeur
              
               
                 
                   Mud-Naud-Mwt
                   ☞
                
              
               
                 
                   Natfr
                   Autir
                
                 .
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Aud
              
            
             
               
                  
              
               
                 Mab
              
               
                 Am
              
               
                 Mir
              
               
                 
                   Uth*
                   Oph
                
              
               
                 §
                 auth
              
               
                 Dacrambatin
              
               
                 
                   Muth-Nauth-Mwth
                   ☞
                
              
               
                 
                   Neuthj
                   Authir
                
              
            
             
               
                 Om
              
               
                 8.
                 
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Cathetothesis
                 .
              
               
                 
                   Plus
                   minus
                
                 .
              
            
             
               
                  
              
               
                 Er
              
               
                 Sub.
                 
              
               
                 Re.
                 
              
               
                 Dat.
                 
              
               
                 .
                 Pr.
                 
              
               
                 Cacurgyq
                 .
              
               
                 Final
                 Resolvers
                 .
              
               
                 Sindiforiu
                 .
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Aet
              
            
             
               
                 Ab
              
               
                 Re
              
               
                 Ub
              
               
                 Mu
              
               
                 
                   Ut*
                   Op
                
              
               
                 §
                 aet
              
               
                 Dakyxamfor
              
               
                 
                   Nut-Nat-Nwd
                   ☞
                
              
               
                 
                   Noedfr
                   Dyr
                
                 .
              
            
             
               
                  
              
               
                 Lo
              
               
                 Am
              
               
                 Lag
              
               
                 
                   Ud*
                   Ob
                
              
               
                 §
                 aed
              
               
                 Dambyxamfor
              
               
                 
                   Nud-Nad-Nwt
                   ☞
                
              
               
                 
                   Noetfr
                   Dyr
                
                 .
              
            
             
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
               
                 Aed
              
            
             
               
                 Me
              
               
                 Me
              
               
                 En
              
               
                 Myr
              
               
                 
                   Uth*
                   Oph
                
              
               
                 §ath
              
               
                 Dakypambin
              
               
                 
                   Nuth-Nath-Nwth
                   ☞
                
              
               
                 
                   Noethj
                   Aeth
                   Syr.
                
                 
              
            
          
        
         
         
           THe
           novelty
           of
           these
           words
           I
           know
           will
           seeme
           strange
           to
           some
           ,
           and
           to
           the
           eares
           of
           illiterate
           hearers
           sound
           like
           termes
           of
           Conjuration
           :
           yet
           seeing
           that
           since
           the
           very
           infancie
           of
           learning
           ,
           such
           inventions
           have
           beene
           made
           use
           of
           ,
           and
           new
           words
           coyned
           ,
           that
           the
           knowledge
           of
           severall
           things
           representatively
           confined
           within
           a
           narrow
           compasse
           ,
           might
           the
           more
           easily
           be
           retained
           in
           a
           memory
           susceptible
           of
           their
           impression
           (
           as
           is
           apparent
           by
           the
           names
           of
           
             Barbara
             ,
             Celarent
             ,
             Darii
             ,
             Ferio
             ,
          
           and
           fifteen
           more
           Syllogistick
           Moods
           ,
           and
           by
           those
           likewise
           of
           
             Gammuth
             ,
             A-re
             ,
             B-mi
             ,
             C-fa-uth
             ,
          
           and
           seventeen
           other
           steps
           of
           
             Guidos
             Scale
          
           ,
           which
           are
           universally
           received
           by
           men
           of
           understanding
           ,
           and
           that
           have
           their
           spirits
           tuned
           to
           the
           harmony
           of
           reason
           )
           I
           know
           not
           why
           Logick
           and
           Musick
           should
           be
           rather
           fitted
           with
           such
           helps
           then
           Trigonometrie
           ,
           which
           ,
           for
           certitude
           of
           demonstration
           ,
           hath
           been
           held
           inferior
           to
           no
           science
           ,
           and
           for
           sublimity
           and
           variety
           of
           object
           ,
           is
           the
           primest
           of
           the
           Mathematicks
           .
           This
           is
           the
           cause
           why
           I
           framed
           the
           Trissotetras
           ,
           wherein
           the
           termes
           by
           me
           invented
           ,
           without
           regard
           of
           the
           initiall
           letters
           of
           the
           words
           by
           them
           expressed
           ,
           are
           composed
           of
           such
           as
           ,
           joyned
           together
           ,
           are
           of
           most
           easie
           pronunciation
           ;
           as
           the
           Tangent
           complement
           of
           a
           Subtendent
           is
           sooner
           uttered
           by
           Mu
           then
           by
           T
           C
           S
           ;
           and
           the
           Secant
           complement
           of
           the
           side
           required
           ,
           by
           Ry
           ,
           then
           (
           in
           the
           usuall
           apocopating
           way
           )
           by
           the
           first
           syllables
           or
           letters
           of
           Secant
           complement
           ,
           side
           ,
           and
           required
           ;
           and
           considering
           that
           without
           opening
           of
           the
           mouth
           no
           word
           can
           be
           spoken
           ,
           which
           overture
           is
           performed
           by
           the
           vowel
           ,
           to
           all
           the
           sides
           and
           Angles
           I
           designed
           vowels
           ,
           that
           in
           the
           coalescencie
           of
           syllables
           ,
           Sines
           ,
           Tangents
           ,
           and
           Secants
           might
           the
           better
           consound
           therewith
           .
        
      
       
         
           The
           explanation
           of
           the
           Trissotetras
           .
        
         
           A.
           signifieth
           an
           Angle
           :
           Ab.
           in
           the
           Resolvers
           signifieth
           abstraction
           ,
           but
           in
           the
           Figures
           and
           Datoquaeres
           the
           
             Angle
             between
             :
             Ac.
          
           or
           Ak.
           the
           
             acute
             Angle
             .
             Ad.
          
           Addition
           .
           AE
           .
           the
           first
           base
           :
           Amb.
           or
           Am.
           an
           obtuse
           Angle
           :
           As
           Angles
           in
           the
           plurall
           number
           .
           At.
           the
           double
           verticall
           ,
           
             whether
             externall
             or
             internall
             .
             Au.
          
           the
           first
           verticall
           Angle
           :
           Ay
           ,
           the
           Angle
           adjoyning
           to
           the
           side
           required
           .
        
         
         
           B.
           or
           Ba.
           the
           true
           base
           :
           Bis
           the
           double
           of
           a
           thing
           .
        
         
           Ca.
           the
           perpendicular
           :
           Cra.
           the
           concurse
           of
           a
           given
           and
           required
           side
           :
           Cur.
           the
           concurse
           of
           two
           given
           sides
           .
        
         
           D.
           the
           partiall
           or
           little
           rectangle
           or
           rectanglet
           .
           Da.
           the
           datas
           .
           Di.
           or
           Dif
           .
           the
           difference
           :
           Dir.
           the
           directories
           .
           
             D.
             q.
          
           Datoquaeres
           .
           Diss
           .
           of
           unlike
           natures
           .
        
         
           E.
           a
           side
           :
           Eb.
           the
           side
           between
           :
           Enod
           ,
           enodandas
           :
           Ereled
           .
           turned
           into
           sides
           :
           Es
           ,
           sides
           in
           the
           plurall
           number
           :
           Ei
           ,
           the
           side
           conterminall
           with
           the
           Angle
           required
           :
           Eu
           ,
           the
           second
           verticall
           Angle
           .
        
         
           F.
           the
           new
           base
           ,
           or
           angularie
           base
           ,
           it
           being
           an
           Angle
           converted
           into
           a
           side
           :
           Fig.
           figures
           :
           
             Fin.
             Res
          
           .
           finall
           resolvers
           :
           For
           ,
           or
           Fo
           ,
           outwardly
           ,
           often
           made
           use
           of
           in
           the
           
             Cathetothesis
             :
             Fr.
          
           a
           subducting
           of
           a
           lesser
           from
           a
           greater
           ,
           whether
           it
           be
           Side
           or
           Angle
           .
        
         
           G.
           An
           Angle
           or
           Side
           given
           :
           Gre
           ,
           or
           aggre
           ,
           the
           summe
           or
           aggregat
           .
        
         
           Hal
           ,
           or
           Al
           ,
           the
           halfe
           .
        
         
           
             I.
             Vowel
          
           ,
           an
           Angle
           required
           :
           
             I
             Consonant
          
           ,
           the
           addition
           of
           one
           thing
           to
           another
           used
           in
           the
           clausuls
           of
           some
           of
           the
           finall
           Resolvers
           .
           In
           ,
           intus
           or
           inwardly
           ,
           and
           sometimes
           turned
           into
           .
           Iu
           ,
           the
           segments
           of
           the
           base
           ,
           or
           the
           segmented
           base
           .
        
         
           K.
           The
           complement
           of
           an
           Angle
           to
           a
           Semicircle
           .
        
         
           L.
           The
           Secant
           :
           Leg
           ,
           one
           of
           the
           comprehending
           sides
           of
           an
           Angle
           .
           This
           representative
           is
           once
           only
           mentioned
           .
        
         
           M.
           A
           Tangent
           complement
           .
        
         
           N.
           A
           Sine
           complement
           .
        
         
           O.
           An
           opposite
           Angle
           ,
           or
           rather
           Cathetopposite
           :
           Ob.
           the
           next
           cathetopposite
           Angle
           ,
           by
           some
           called
           the
           first
           opposite
           :
           Op.
           the
           prime
           cathetopposite
           Angle
           ,
           by
           some
           called
           the
           second
           opposite
           Oph
           ,
           the
           first
           of
           the
           coopposite
           Angles
           :
           Orth
           ,
           an
           acute
           Angle
           :
           Ops
           ,
           the
           second
           of
           the
           coopposits
           :
           Os
           ,
           opposite
           Angles
           in
           the
           plurall
           number
           .
           Oe
           ,
           the
           second
           base
           ;
           Ou
           ,
           the
           Angle
           opposite
           to
           the
           base
           .
        
         
           P.
           Opposite
           ,
           whether
           Angle
           or
           side
           :
           Par.
           a
           parallelogram
           or
           oblong
           .
           Praes
           .
           praesubservient
           :
           Possub
           .
           possubservient
           :
           Pro.
           proportionall
           :
           Prod.
           directly
           proportionall
           :
           Pror
           ,
           reciprocally
           proportionall
           :
           Pow.
           the
           Square
           of
           a
           Line
           :
           Pran
           .
           praenoscendas
           .
        
         
         
           Q.
           Continued
           if
           need
           be
           .
           
             Quaes
             .
             Quaesitas
             .
             Quae.
             Quaere
             ,
          
           or
           Required
           .
        
         
           R.
           The
           Secant
           complement
           ,
           and
           sometimes
           in
           the
           middle
           of
           the
           Cathetothesis
           signifies
           required
           ,
           as
           alwayes
           in
           the
           latter
           end
           of
           a
           
             finall
             resolver
          
           it
           doth
           by
           way
           of
           emphasis
           ,
           when
           it
           followes
           I.
           or
           
             Y.
             R.
          
           likewise
           in
           the
           Axiom
           of
           Rulerst
           stands
           for
           Radius
           .
        
         
           Ra.
           the
           Radius
           ,
           and
           in
           the
           Scheme
           the
           middle
           angularie
           Radius
           .
        
         
           S.
           The
           Sine
           ,
           and
           in
           the
           close
           of
           some
           Resolvers
           ,
           the
           
             Summe
             .
             Sim.
          
           of
           like
           affection
           or
           nature
           :
           Subs
           .
           Subservient
           .
        
         
           T.
           The
           Tangent
           .
           To.
           the
           Radius
           or
           total
           Sine
           ,
           but
           in
           the
           Diagram
           it
           is
           taken
           for
           the
           left
           angularie
           Radius
           :
           
           Tω
           .
           the
           right
           angularie
           Radius
           in
           the
           Scheme
           proposed
           :
           Tol.
           the
           first
           hypotenusal
           Radius
           thereof
           .
           Tom.
           the
           second
           hyp
           .
           Radius
           .
           Ton.
           the
           third
           hyp
           .
           Rad.
           Tor.
           the
           fourth
           hyp
           .
           Rad.
           Tolb
           .
           the
           basiradius
           on
           the
           left
           hand
           .
           Torb
           .
           the
           basiradius
           on
           the
           right
           .
           Tolp.
           the
           Cathetorabdos
           ,
           or
           Radius
           on
           the
           left
           .
           Torp
           .
           the
           Cathetoradius
           on
           the
           right
           .
           Th.
           the
           correctangle
           .
        
         
           U.
           The
           Subtendent
           side
           .
           V.
           consonant
           ,
           to
           avoid
           vastnesse
           of
           gaping
           ,
           expresseth
           the
           same
           in
           severall
           figures
           .
           Ur.
           the
           Subtendent
           required
           .
        
         
           W.
           The
           second
           Subtendent
           .
        
         
           X.
           Adjacent
           or
           Conterminal
           .
        
         
           Y.
           The
           side
           required
           .
        
         
           Z.
           The
           difference
           of
           Segments
           ,
           and
           is
           the
           same
           with
           di
           ,
           or
           
             dif
             .
             Neverthelesse
             the
          
           Reader
           
             may
             be
             pleased
             to
             observe
             ,
             that
             no
             Consonants
             in
             the
          
           Figures
           or
           Moods
           
             are
             representative
             save
          
           P.
           and
           B.
           
             and
             that
             only
             in
             a
             few
             ;
             both
             these
             two
             and
             all
             the
             other
             Consonants
             meerly
             serving
             to
             expresse
             the
             order
             and
             series
             of
             the
          
           Moods
           and
           Figures
           
             respectively
             amongst
             themselves
             ,
             and
             of
             their
             constitutive
             parts
             in
             regard
             of
             one
             another
             .
          
        
      
       
         
           ANIMADVERSIONS
           .
        
         
           IN
           the
           letter
           T.
           I
           have
           been
           something
           large
           in
           the
           enumeration
           of
           severall
           Radiuses
           ;
           for
           there
           being
           eleven
           made
           use
           of
           in
           the
           grand
           Scheme
           ,
           whereof
           eight
           are
           Circumferentiall
           ,
           and
           three
           Angularie
           ,
           that
           they
           might
           be
           the
           better
           distinguished
           
           from
           one
           another
           ,
           when
           falling
           in
           proportion
           we
           should
           have
           occasion
           to
           expresse
           them
           ;
           I
           thought
           good
           to
           allot
           to
           every
           one
           of
           them
           its
           owne
           peculiar
           Character
           :
           all
           which
           I
           have
           done
           with
           the
           more
           exactnesse
           ,
           that
           by
           the
           variety
           of
           the
           Radiuses
           amongst
           themselves
           ,
           when
           any
           one
           of
           them
           in
           particular
           is
           pitched
           upon
           ,
           we
           may
           the
           sooner
           know
           what
           part
           of
           the
           Diagram
           ,
           by
           meanes
           thereof
           ,
           is
           fittest
           for
           the
           resolving
           of
           any
           Orthogonosphericall
           problem
           :
           though
           indeed
           ,
           I
           must
           confesse
           ,
           when
           sometimes
           to
           a
           question
           propounded
           ,
           I
           adapt
           a
           figure
           apart
           ,
           I
           doe
           indifferently
           (
           excluding
           all
           other
           characters
           )
           make
           use
           of
           To
           ,
           or
           Rad
           ,
           or
           R
           onely
           for
           the
           totall
           Sine
           ,
           which
           ,
           without
           any
           obscurity
           or
           confusion
           at
           all
           ,
           I
           have
           practised
           for
           brevities
           sake
           .
        
         
           Likewise
           ,
           it
           being
           my
           maine
           designe
           in
           the
           framing
           of
           this
           Table
           ,
           to
           make
           alcapable
           trigonometrically-affected
           Students
           with
           much
           facility
           and
           litle
           labour
           attaine
           to
           the
           whole
           knowledge
           of
           the
           noble
           Science
           of
           the
           doctrine
           of
           Triangles
           ,
           I
           deemed
           it
           expedient
           ,
           the
           more
           firmely
           and
           readily
           to
           imprint
           the
           severall
           Datoquaeres
           or
           praescinded
           Problems
           thereof
           in
           their
           memories
           ,
           to
           accommodate
           them
           accordingly
           with
           letters
           proper
           for
           the
           purpose
           ;
           which
           ,
           if
           the
           ingenious
           Reader
           will
           be
           pleased
           to
           consider
           ,
           he
           will
           find
           ,
           by
           the
           very
           letters
           themselves
           ,
           the
           place
           and
           number
           of
           each
           Datoquaere
           :
           This
           is
           the
           reason
           why
           my
           Trissotetras
           (
           conforme
           to
           the
           Etymologie
           of
           its
           name
           )
           is
           in
           so
           many
           divers
           Ternaries
           ,
           and
           Quaternaries
           divided
           ;
           and
           that
           the
           sharp
           ,
           meane
           ,
           blunt
           ,
           double
           ,
           and
           Liquid
           Consonants
           of
           the
           Greek
           Alphabet
           ,
           are
           so
           orderly
           bestowed
           in
           their
           severall
           roomes
           ,
           being
           all
           and
           every
           one
           of
           them
           seated
           according
           to
           the
           nature
           of
           the
           Moods
           and
           Figures
           ,
           whose
           characteristicks
           they
           are
           .
        
         
           Thirdly
           ,
           the
           Moods
           of
           the
           Planotriangular
           Table
           ,
           being
           in
           all
           thirteene
           ,
           whereof
           there
           be
           seven
           Rectangular
           ,
           and
           six
           Obliquangular
           ,
           are
           fitly
           comprehended
           by
           the
           three
           blunt
           ,
           three
           meane
           ,
           three
           sharp
           ,
           and
           soure
           double
           Consonants
           ,
           the
           Hebrew
           Shin
           being
           accounted
           for
           one
           of
           them
           .
        
         
           Fourthly
           ,
           the
           sixteen
           Moods
           of
           the
           Orthogonosphericall
           Trissotetras
           are
           contained
           under
           three
           sharp
           ,
           three
           mean
           ,
           three
           blunt
           ,
           three
           double
           ,
           and
           foure
           Liquids
           ,
           which
           foure
           doe
           orderly
           particularise
           the
           Binaries
           of
           the
           last
           two
           Figures
           .
        
         
           Fifthly
           ,
           the
           foure
           Monurgetick
           Loxogonosphericals
           are
           deciphred
           
           by
           each
           its
           owne
           Liquid
           in
           front
           ,
           according
           to
           their
           literall
           order
           .
        
         
           Sixthly
           ,
           the
           eight
           Loxogonosphericall
           Disergeticks
           are
           also
           distinguished
           by
           the
           foure
           Liquids
           ,
           but
           with
           this
           difference
           from
           the
           Monurgeticks
           ,
           that
           the
           Vowels
           of
           A
           and
           E
           precede
           them
           in
           the
           first
           syllable
           ,
           importing
           thereby
           the
           Datas
           of
           an
           Angle
           or
           a
           Side
           .
           Now
           because
           these
           Disergeticks
           are
           eight
           in
           all
           ,
           there
           being
           allotted
           to
           every
           Liquid
           that
           characteriseth
           the
           Figures
           ,
           the
           better
           to
           diversifie
           the
           first
           and
           second
           Datas
           of
           each
           respective
           binarie
           from
           one
           another
           ,
           (
           in
           so
           farre
           as
           they
           have
           reference
           to
           each
           its
           own
           Quaesitum
           )
           the
           Figurative
           Liquid
           is
           doubled
           when
           a
           Side
           is
           required
           ,
           and
           remaineth
           single
           when
           an
           Angle
           .
        
         
           Furthermore
           ,
           in
           the
           Oblique
           Sphericodisergeticks
           ,
           so
           farre
           as
           the
           sense
           of
           the
           Resolvers
           could
           beare
           it
           ,
           I
           did
           trinifie
           them
           with
           letters
           convenient
           for
           the
           purpose
           ,
           according
           to
           the
           severall
           cases
           of
           their
           Datoquaeres
           ,
           whose
           diversity
           reacheth
           not
           above
           the
           extent
           of
           π.
           β.
           φ.
           and
           τ.
           δ.
           θ.
           
        
         
           I
           had
           almost
           omitted
           to
           tell
           you
           ,
           that
           for
           the
           more
           variety
           in
           the
           last
           two
           Figures
           of
           the
           Orthogonosphericals
           are
           set
           downe
           the
           two
           letters
           of
           Ch.
           and
           Shin
           ,
           the
           first
           a
           Spanish
           ,
           and
           the
           second
           an
           Hebrew
           letter
           .
           Now
           if
           to
           those
           helps
           for
           the
           memorie
           which
           in
           this
           Table
           I
           have
           afforded
           the
           Reader
           ,
           both
           by
           the
           Alphabetical
           order
           of
           some
           Consonants
           ,
           and
           homogeneity
           of
           others
           in
           their
           affections
           of
           sharpnesse
           ,
           meannesse
           ,
           obtusity
           ,
           and
           duplicity
           ,
           he
           joyne
           that
           artificiall
           aid
           in
           having
           every
           part
           of
           th●●
           Chem●locally
           in
           his
           mind
           (
           of
           all
           wayes
           both
           for
           facility
           in
           remembring
           ,
           and
           stedfastnesse
           of
           retention
           ,
           without
           doubt
           ,
           the
           most
           expedite
           )
           or
           otherwise
           place
           the
           representatives
           of
           words
           ,
           according
           to
           the
           method
           of
           the
           Art
           of
           memory
           ,
           in
           the
           severall
           corners
           of
           a
           house
           (
           which
           ,
           in
           regard
           of
           their
           paucity
           are
           containable
           within
           a
           Parlour
           or
           dining
           roome
           at
           most
           )
           he
           may
           with
           ease
           get
           them
           all
           by
           heart
           in
           lesse
           then
           the
           space
           of
           an
           houre
           :
           
             which
             is
             no
             great
             expence
             of
             time
             ,
             though
             bestowed
             on
             matters
             of
             meaner
             consequence
             .
          
        
      
       
         
         
           The
           Commentary
           .
        
         
           THe
           Axiomes
           of
           plain
           Triangles
           are
           foure
           ,
           
             viz.
             Rulerst
             ,
             Eproso
             ,
             Grediftal
             ,
          
           and
           Bagrediffus
           .
        
         
           Rulerst
           ,
           that
           is
           to
           say
           ,
           the
           Subtendent
           in
           plain
           Triangles
           may
           be
           either
           Radius
           or
           Secant
           ,
           and
           the
           Ambients
           either
           Radius
           ,
           Sines
           ,
           or
           Tangents
           ;
           for
           it
           is
           a
           maxime
           in
           Planangular
           Triangles
           ,
           that
           any
           side
           may
           be
           put
           for
           Radius
           ,
           grounded
           on
           this
           ,
           that
           from
           any
           point
           at
           any
           distance
           a
           Circle
           may
           be
           described
           :
           therefore
           if
           any
           of
           the
           sides
           of
           a
           plain
           Triangle
           be
           given
           together
           with
           the
           Angler
           ,
           each
           of
           the
           other
           two
           sides
           is
           given
           by
           a
           threefold
           proportion
           ,
           that
           is
           ,
           whether
           you
           put
           that
           ,
           or
           this
           ,
           or
           the
           third
           side
           for
           the
           Radius
           ;
           which
           difference
           occasioneth
           both
           in
           plaine
           and
           Sphericall
           Triangles
           great
           variety
           in
           their
           calculations
           .
        
         
           The
           Branches
           of
           this
           Axiom
           are
           Vradesso
           and
           Eradetul
           .
        
         
           Vradesso
           ,
           that
           is
           when
           the
           Hypotenusa
           is
           Radius
           ,
           the
           sides
           are
           Sines
           of
           their
           opposit
           Angles
           ;
           so
           that
           there
           be
           two
           Arches
           described
           with
           that
           Hypotenusal
           identity
           of
           distance
           ,
           whose
           Centers
           are
           in
           the
           two
           extremities
           of
           the
           Subtendent
           ;
           for
           so
           the
           case
           will
           be
           made
           plaine
           in
           both
           the
           Legs
           ,
           which
           otherwise
           would
           not
           appeare
           but
           in
           one
           .
        
         
           Eradetul
           ,
           when
           any
           of
           the
           sides
           is
           Radius
           ,
           the
           other
           of
           them
           is
           a
           Tangent
           ,
           and
           the
           Subtendent
           a
           Secant
           .
           The
           reason
           of
           this
           is
           found
           in
           the
           very
           definitions
           of
           the
           Sines
           ,
           Tangents
           ,
           and
           Secants
           ,
           to
           the
           which
           ,
           if
           the
           Reader
           please
           ,
           he
           may
           have
           recourse
           ;
           for
           I
           have
           set
           them
           downe
           amongst
           my
           Definitions
           .
           Hence
           it
           is
           (
           according
           to
           Mr.
           Speidels
           observation
           in
           his
           book
           of
           Sphericals
           )
           that
           the
           Sine
           of
           any
           Arch
           being
           Radius
           ,
           that
           which
           was
           the
           totall
           Sine
           becomes
           the
           Secant
           complement
           of
           the
           said
           Arch
           ,
           and
           that
           the
           Tangent
           of
           any
           Arch
           being
           Radius
           ,
           what
           was
           Radius
           becomes
           Tangent
           complement
           of
           that
           Arch.
           
        
         
           The
           Directory
           of
           this
           Axiome
           is
           Vphech●t
           .
           which
           sheweth
           us
           ,
           that
           there
           be
           three
           Planorectangular
           Enodandas
           belonging
           thereto
           ,
           
             viz.
             Vphener
             ,
             Echemun
          
           ,
           and
           Etenar
           ;
           as
           for
           Pserelema
           ,
           which
           is
           the
           Loxogonian
           one
           pointed
           at
           in
           my
           Trissotetras
           ,
           because
           it
           is
           but
           a
           partiall
           Enodandum
           ,
           I
           have
           purposely
           omitted
           to
           mention
           it
           in
           the
           Directory
           of
           Eradetul
           .
        
         
           The
           second
           Axiome
           is
           Epros●
           ,
           that
           is
           ,
           the
           sides
           are
           proportionall
           to
           one
           another
           as
           the
           Sines
           of
           their
           opposite
           Angles
           ;
           for
           seeing
           
           about
           any
           Triangle
           a
           Circle
           may
           be
           circumscribed
           ,
           in
           which
           case
           each
           side
           is
           a
           cord
           or
           Subtense
           ,
           the
           halfe
           whereof
           is
           the
           Sine
           of
           its
           opposite
           Angle
           ,
           and
           there
           being
           alwayes
           the
           same
           reason
           of
           the
           whole
           to
           the
           whole
           ,
           as
           of
           the
           halfe
           to
           the
           halfe
           ,
           the
           sides
           must
           needs
           be
           proportionall
           to
           one
           another
           ,
           as
           the
           Sines
           of
           their
           opposite
           Angles
           ,
           
             quod
             probandum
             erat
          
           .
        
         
           The
           Directory
           of
           this
           second
           Axiome
           is
           Pubkegdaxesh
           ,
           which
           declareth
           that
           there
           are
           seven
           Enodandas
           grounded
           on
           it
           ,
           to
           wit
           ,
           foure
           Rectangular
           ,
           
             Upalem
             ,
             Ubeman
             ,
             Ekarul
             ,
             Egalem
             ,
          
           and
           three
           Obliquangular
           ,
           
             Danarele
             ,
             Xemenoro
          
           ,
           and
           Shenerolem
           .
        
         
           The
           third
           Axiom
           is
           Grediftal
           ,
           that
           is
           ,
           in
           all
           plain
           Triangles
           ,
           As
           the
           summe
           of
           the
           two
           sides
           is
           to
           their
           difference
           ,
           so
           is
           the
           Tangent
           of
           the
           halfe
           sum
           of
           the
           opposite
           Angles
           to
           the
           Tangent
           of
           half
           their
           difference
           ;
           for
           if
           a
           Line
           be
           drawne
           equall
           to
           the
           summe
           of
           the
           two
           sides
           ,
           and
           if
           on
           the
           point
           of
           Extension
           with
           the
           distance
           of
           the
           shorter
           side
           a
           Semicircle
           be
           described
           ,
           and
           that
           from
           the
           extremity
           of
           the
           protracted
           Line
           a
           Diameter
           be
           drawne
           thorough
           the
           Circle
           where
           it
           toucheth
           the
           top
           of
           the
           Triangle
           in
           question
           ,
           till
           it
           occurre
           with
           a
           parallel
           to
           the
           third
           side
           ,
           there
           will
           arise
           two
           Equicrurall
           Triangles
           ,
           one
           whereof
           having
           one
           Angle
           common
           with
           the
           Triangle
           proposed
           ,
           and
           the
           three
           of
           the
           one
           being
           equall
           to
           the
           three
           of
           the
           other
           ,
           any
           one
           of
           the
           equall
           Angles
           in
           the
           foresaid
           Isosceles
           must
           needs
           be
           the
           one
           halfe
           of
           the
           two
           unknowne
           Angles
           .
           This
           is
           the
           first
           step
           to
           the
           obtaining
           of
           what
           we
           demand
           .
           Then
           do
           we
           find
           that
           the
           third
           side
           cutteth
           the
           sides
           of
           the
           greatest
           Triangle
           according
           to
           the
           Analogie
           required
           ,
           which
           is
           perceivable
           enough
           ,
           if
           with
           the
           distance
           of
           the
           outmost
           Parallel
           from
           the
           lower
           end
           thereof
           as
           Center
           ,
           be
           described
           a
           new
           Circle
           ;
           for
           then
           will
           the
           Tangents
           be
           perspicuous
           and
           so
           much
           the
           more
           for
           their
           Rectangularity
           ,
           the
           one
           with
           the
           Radius
           ,
           and
           the
           other
           with
           its
           Parallel
           ,
           which
           ,
           being
           touched
           at
           an
           Angle
           described
           in
           a
           Semicircle
           ,
           confirmeth
           the
           Rectangularity
           of
           both
           .
           By
           the
           Parallels
           likewise
           is
           inferred
           the
           equality
           of
           the
           alternate
           Angles
           ,
           whose
           addition
           and
           subduction
           to
           and
           from
           halfe
           the
           sum
           of
           the
           two
           unknown
           Angles
           make
           up
           both
           the
           greater
           and
           lesser
           Angle
           .
           
             Hereby
             it
             is
             evident
             how
             the
             sum
             of
             the
             two
             sides
             ,
             &c.
             which
             was
             to
             be
             proved
             .
          
        
         
           The
           Directory
           of
           this
           third
           Axiom
           is
           θ.
           onely
           ;
           for
           it
           hath
           no
           Enodandum
           but
           Therelabmo
           .
        
         
         
           The
           fourth
           Axiom
           is
           Bagrediffiu
           ,
           that
           is
           ,
           As
           the
           Base
           or
           greatest
           side
           is
           to
           the
           summe
           of
           the
           other
           sides
           ,
           so
           the
           difference
           of
           the
           other
           sides
           to
           the
           difference
           of
           the
           Segments
           of
           the
           Base
           ;
           for
           if
           upon
           the
           Center
           of
           the
           verticall
           Angle
           with
           the
           distance
           of
           the
           shortest
           side
           be
           described
           a
           Circle
           ,
           it
           will
           so
           cut
           the
           two
           greater
           sides
           of
           the
           given
           Triangle
           ,
           that
           ,
           finding
           thereby
           two
           Oblongs
           of
           the
           nature
           of
           those
           whose
           equality
           is
           demonstrated
           in
           my
           Apodicticks
           ,
           we
           may
           inferre
           (
           the
           Oblong
           made
           of
           the
           summe
           of
           the
           sides
           ,
           and
           difference
           of
           the
           sides
           being
           equall
           to
           the
           Oblong
           made
           of
           the
           Base
           ,
           and
           the
           difference
           of
           its
           Segments
           )
           that
           their
           sides
           are
           reciprocally
           proportionall
           ;
           that
           is
           ,
           As
           the
           greatest
           side
           is
           to
           the
           sum
           of
           the
           other
           sides
           :
           so
           the
           difference
           of
           the
           other
           sides
           ,
           to
           the
           difference
           of
           the
           Segments
           of
           the
           Base
           ,
           or
           greater
           side
           .
        
         
           The
           Directory
           of
           this
           Axiom
           is
           θ.
           and
           its
           onely
           Enodandum
           ,
           (
           though
           but
           a
           partiall
           one
           )
           Pserelema
           .
        
      
       
         
           The
           Planorectangular
           Table
           hath
           foure
           Figures
           .
        
         
           IT
           is
           to
           be
           observed
           ,
           that
           Figure
           here
           is
           not
           taken
           Geometrically
           ,
           but
           in
           the
           sense
           that
           it
           is
           used
           in
           the
           Logicks
           ,
           when
           a
           Syllogism
           is
           said
           to
           be
           in
           the
           first
           ,
           second
           ,
           or
           third
           Figure
           ;
           for
           ,
           as
           there
           by
           the
           various
           application
           of
           the
           Medium
           or
           mean
           terme
           the
           Figures
           are
           constituted
           diverse
           :
           so
           doth
           the
           difference
           of
           the
           Datas
           in
           a
           Triangle
           distinguish
           these
           Trissotetrall
           Figures
           from
           one
           another
           ,
           and
           (
           to
           continue
           yet
           further
           in
           the
           Syllogisticall
           Analogy
           )
           are
           according
           to
           the
           severall
           demands
           (
           when
           the
           Datas
           are
           the
           same
           )
           subdivided
           into
           Moods
           .
        
         
           The
           first
           two
           vowels
           give
           notice
           of
           the
           Data's
           ,
           and
           the
           third
           of
           what
           is
           demanded
           ,
           so
           that
           Uale
           (
           and
           euphonetically
           pronounced
           Vale
           )
           which
           is
           the
           first
           Figure
           ,
           shewes
           that
           the
           Subtendent
           ,
           and
           one
           Angle
           are
           given
           ,
           and
           that
           one
           of
           the
           containing
           sides
           is
           required
           .
           Vemane
           is
           the
           second
           Figure
           ,
           which
           pointeth
           out
           all
           those
           problems
           wherein
           the
           Hypotenusa
           ,
           and
           one
           Leg
           are
           given
           ,
           and
           an
           Angle
           ,
           or
           the
           other
           Leg
           is
           required
           .
        
         
           The
           third
           Figure
           is
           Enave
           ,
           which
           comprehendeth
           all
           the
           Problems
           ,
           
           wherein
           one
           of
           the
           Ambients
           is
           given
           with
           an
           Oblique
           Angle
           ,
           and
           the
           Subtendent
           ,
           or
           other
           Ambient
           required
           .
        
         
           The
           fourth
           and
           last
           of
           the
           Rectangular
           Figures
           is
           Ereva
           ,
           which
           standeth
           for
           those
           Datoquaeres
           ,
           wherein
           the
           including
           Sides
           are
           given
           ,
           and
           the
           Subtendent
           or
           an
           Angle
           demanded
           .
        
      
       
         
           Now
           let
           us
           come
           to
           the
           Moods
           of
           those
           Figures
           .
        
         
           THe
           first
           Figure
           Vale
           hath
           but
           one
           Mood
           ,
           and
           therefore
           of
           as
           great
           extent
           as
           it selfe
           ,
           which
           is
           Upalem
           ;
           whose
           nature
           is
           to
           let
           us
           know
           ,
           when
           a
           plane
           right
           angled
           Triangle
           is
           given
           us
           to
           resolve
           ,
           whose
           Subtendent
           and
           one
           of
           the
           Obliques
           is
           proposed
           ,
           and
           one
           of
           the
           Ambients
           required
           ,
           that
           we
           must
           have
           recourse
           unto
           its
           Resolver
           ,
           which
           being
           Rad
           —
           U
           
             —
             Sapy
          
           ☞
           Yr
           sheweth
           ,
           that
           if
           we
           
             joyne
             the
             artificiall
             Sine
             of
             the
             Angle
             opposite
             to
             the
             side
             demanded
             with
             the
             Logarithm
             of
             the
             Subtendent
             ,
             the
             summe
             searched
             in
             the
             Canon
             of
             absolute
             numbers
             will
             afford
             us
             the
             Logarithm
             of
             the
             side
             required
             .
          
           The
           reason
           hereof
           is
           found
           in
           the
           second
           Axiom
           ,
           the
           first
           Consonant
           of
           whose
           Directory
           evidenceth
           that
           Upalem
           is
           Eprosos
           Enodandum
           ;
           for
           it
           is
           ,
           
             As
             the
             totall
             Sine
             ,
             to
             the
             Hypotenusa
             :
             so
             the
             Sine
             of
             the
             Angle
             opposite
             to
             the
             side
             required
             ,
             is
             to
             the
             said
             required
             side
             ,
          
           according
           to
           the
           nature
           of
           the
           foresaid
           Axiom
           ,
           whereupon
           it
           is
           grounded
           .
        
         
           The
           second
           Figure
           Vemane
           hath
           two
           Moods
           ,
           Ubeman
           and
           Uphener
           ;
           the
           first
           whereof
           comprehendeth
           all
           those
           questions
           ,
           wherein
           the
           Subtendent
           and
           an
           Ambient
           being
           given
           ,
           an
           Oblique
           is
           required
           ,
           and
           by
           its
           Resolver
           
             V
             —
             Rad
             —
             Eg
          
           ☞
           So.
           thus
           satisfieth
           our
           demand
           ,
           that
           if
           we
           
             subtract
             the
             Logarithm
             of
             the
             Subtendent
             from
             the
             summe
             of
             the
             Logarithms
             of
             the
             middle
             termes
             ,
             we
             have
             the
             Logarithm
             of
             the
             Sine
             of
             the
             opposite
             Angle
             we
             seek
             for
          
           ;
           for
           it
           is
           ,
           
             As
             the
             Subtendent
             to
             the
             totall
             Sine
             ,
             so
             the
             containing
             side
             given
             to
             the
             Sine
             of
             the
             opposite
             Angle
             required
             .
          
           The
           reason
           likewise
           of
           this
           Analogy
           is
           found
           in
           the
           second
           Axiom
           Eproso
           ,
           upon
           the
           which
           this
           Mood
           is
           grounded
           ,
           as
           the
           second
           Consonant
           of
           its
           Directory
           giveth
           us
           to
           understand
           .
        
         
           The
           second
           Mood
           or
           Datoquaere
           of
           this
           Figure
           is
           Uphener
           ,
           which
           sheweth
           that
           those
           questions
           in
           plaine
           Triangles
           ,
           
           wherein
           the
           Hypotenusa
           and
           a
           Leg
           being
           given
           ,
           the
           other
           Leg
           is
           demanded
           ,
           are
           to
           be
           calculated
           by
           its
           Resolver
           ,
           which
           (
           because
           the
           Canon
           of
           Logarithms
           cannot
           performe
           it
           at
           one
           operation
           ,
           there
           being
           a
           necessity
           to
           find
           one
           of
           the
           oblique
           Angles
           before
           the
           fourth
           terme
           can
           be
           brought
           into
           an
           Analogie
           )
           alloweth
           two
           Subservients
           for
           the
           atchievement
           thereof
           ,
           
             viz.
             Vbeman
          
           ,
           the
           first
           Mood
           of
           the
           second
           Figure
           ,
           for
           the
           finding
           out
           of
           the
           Angle
           ,
           and
           here
           (
           because
           anterior
           in
           the
           work
           )
           called
           Praesubservient
           :
           then
           Vpalem
           ,
           the
           first
           Mood
           of
           all
           ,
           for
           finding
           out
           of
           the
           Leg
           inquired
           ,
           and
           here
           called
           Possubservient
           ,
           because
           of
           its
           posteriority
           in
           the
           operation
           :
           yet
           were
           it
           not
           for
           the
           facility
           which
           addition
           and
           subtraction
           only
           afford
           us
           in
           this
           manner
           of
           calculation
           ,
           we
           might
           doe
           it
           with
           one
           work
           alone
           by
           the
           Bucarnon
           or
           Pythagorases
           Diodot
           ,
           which
           plainly
           sheweth
           us
           ,
           that
           by
           
             subducing
             the
             square
             of
             the
             Leg
             given
             ,
             from
             the
             square
             of
             the
             Subtēdent
             ,
             we
             have
             for
             the
             remainder
             another
             square
             ,
             whose
             root
             is
             the
             side
             required
             .
          
           The
           reason
           of
           this
           is
           in
           my
           Apodicticks
           :
           but
           that
           of
           the
           former
           Resolver
           by
           two
           operations
           ,
           is
           in
           the
           first
           Axiom
           ,
           as
           by
           the
           first
           syllable
           of
           its
           Directory
           is
           manifest
           .
        
         
           The
           third
           Figure
           is
           Enave
           ,
           which
           hath
           two
           Moods
           ,
           Ekarul
           and
           Egalem
           .
           The
           first
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           one
           of
           the
           including
           sides
           ,
           and
           an
           Angle
           being
           given
           ,
           the
           Subtendent
           is
           required
           ,
           and
           by
           its
           Resolver
           
             Sapeg
             —
             Eg
             —
             Rad
          
           ☞
           Vr
           ,
           sheweth
           ,
           that
           
             if
             we
             subtract
             the
             Sine
             of
             the
             Angle
             opposite
             to
             the
             given
             side
             from
             the
             summe
             of
             the
             middle
             termes
          
           (
           I
           meane
           the
           Logarithms
           of
           the
           one
           and
           the
           other
           )
           which
           are
           the
           totall
           Sine
           ,
           and
           the
           Leg
           proposed
           ,
           
             we
             shall
             have
             the
             Hypotenusa
             required
          
           ;
           for
           it
           is
           ,
           
             As
             the
             Sine
             of
             the
             Angle
             opposite
             to
             the
             side
             given
             ,
             to
             the
             foresaid
             given
             side
             :
             so
             the
             totall
             Sine
             ,
             to
             the
             Subtendent
             required
             .
          
           The
           reason
           of
           this
           proportion
           is
           grounded
           on
           the
           second
           Axiom
           Eproso
           ;
           for
           K.
           the
           third
           Consonant
           of
           its
           Directory
           ,
           giveth
           us
           to
           understand
           ,
           that
           it
           is
           one
           of
           the
           Enodandas
           thereof
           .
        
         
           The
           second
           Mood
           of
           Enave
           is
           Egalem
           ,
           which
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           one
           of
           the
           Ambients
           ,
           and
           an
           oblique
           Angle
           being
           given
           ,
           the
           other
           Ambient
           is
           required
           :
           and
           by
           its
           Resolver
           
             Rad
             —
             Taxeg
             —
             Eg
          
           ☞
           Yr
           sheweth
           ,
           that
           if
           we
           
             adde
             the
             Logarithm
             of
             the
             side
             given
             to
             the
             Logarithm
             of
             the
             Tangent
             of
             the
             Angle
             conterminall
             with
             that
             side
             ,
             and
             from
             the
             summe
             if
             we
             cut
             off
             the
             first
             digit
             on
             the
             left
             hand
          
           (
           which
           is
           equivalent
           to
           the
           subtracting
           
           of
           the
           Radius
           whether
           double
           or
           single
           )
           
             The
             remainder
             will
             afford
             us
             a
             Logarithm
          
           (
           so
           neare
           as
           the
           irrationality
           of
           the
           termes
           will
           admit
           )
           
             in
             the
             Table
             of
             equall
             parts
             ,
             expressive
             of
             the
             side
             required
          
           ;
           for
           it
           is
           
             As
             the
             whole
             Sine
             to
             the
             Tangent
             of
             an
             Angle
             insident
             on
             the
             given
             side
             :
             so
             the
             side
             proposed
             ,
             to
             the
             side
             required
             :
          
           The
           reason
           hereof
           is
           grounded
           on
           the
           second
           Axiom
           ,
           for
           the
           fourth
           Consonant
           of
           its
           Directory
           sheweth
           ,
           that
           Egalem
           is
           Eprosos
           enodandum
           .
        
         
           The
           fourth
           Figure
           is
           Ereva
           ,
           whose
           Moods
           are
           Echemun
           and
           Etenar
           .
        
         
           The
           first
           ,
           
             viz.
             Echemun
          
           ,
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           the
           two
           Ambients
           being
           given
           ,
           the
           Subtendent
           is
           required
           ,
           and
           (
           not
           being
           Logarithmically
           resolvable
           in
           lesse
           then
           two
           operations
           )
           hath
           for
           its
           Prae
           and
           Possubservients
           the
           Moods
           of
           Etenar
           and
           Ekarul
           ;
           for
           an
           Oblique
           Angle
           by
           this
           Method
           is
           to
           be
           searched
           before
           the
           Subtendent
           can
           be
           found
           out
           ,
           and
           by
           reason
           of
           these
           severall
           work●
           ,
           this
           Mood
           is
           grounded
           on
           the
           two
           first
           Axioms
           ,
           and
           is
           an
           Enodandum
           partially
           depending
           on
           Eradetul
           ,
           and
           Eproso
           .
           Yet
           ,
           if
           you
           will
           be
           pleased
           to
           be
           at
           the
           paines
           of
           extracting
           the
           
             Square
             root
          
           ,
           you
           may
           have
           the
           Subtendent
           at
           one
           work
           by
           a
           Quadrobiquadraequation
           as
           the
           Bucarnon
           doth
           instruct
           us
           ,
           whose
           demonstration
           you
           have
           plainly
           set
           downe
           in
           the
           fourteenth
           of
           my
           Apodicticks
           .
        
         
           The
           second
           Mood
           of
           this
           Figure
           is
           Etenar
           ,
           which
           includeth
           all
           those
           questions
           wherein
           the
           two
           
             containing
             Sides
          
           being
           given
           ,
           one
           of
           the
           Obliques
           is
           required
           ,
           and
           by
           its
           
             Resolver
             E
             —
             Ge
             —
             Rad
          
           ☞
           Toge
           manifesteth
           ,
           that
           ,
           
             if
             from
             the
             Summe
             of
             the
             Radius
             and
             Logarithm
             of
             the
             side
             given
             ,
             we
             subtract
             the
             Logarithm
             of
             the
             other
             proposed
             side
             ,
             the
             remainder
             will
             afford
             the
             Tangent
             of
             the
             Angle
             opposite
             to
             one
             of
             the
             given
             sides
             ,
          
           the
           Complement
           of
           which
           Angle
           to
           a
           right
           one
           is
           alwayes
           the
           measure
           of
           the
           other
           Angle
           ,
           by
           the
           fifth
           of
           my
           Apodicticks
           ;
           for
           it
           is
           ,
           
             As
             the
             one
             Ambient
             is
             to
             the
             other
             Ambient
             ,
             so
             the
             totall
             Sine
             to
             the
             Tangent
             of
             an
             Angle
          
           ;
           which
           found
           out
           ,
           is
           either
           the
           Angle
           required
           ,
           or
           the
           Complement
           thereof
           to
           a
           right
           Angle
           .
           The
           reason
           of
           this
           Analogie
           is
           grounded
           on
           the
           second
           Branch
           of
           the
           first
           Axiom
           ,
           as
           by
           the
           Characteristick
           of
           the
           Directory
           is
           perceivable
           enough
           to
           any
           industrious
           Reader
           .
        
      
       
         
         
           Of
           the
           Planobliquangular
           Triangles
           there
           be
           foure
           Figures
           :
           
             Alaheme
             ,
             Emenarole
             ,
             Enerolome
          
           ,
           and
           Erelea
           ,
        
         
           THe
           first
           and
           last
           of
           these
           foure
           are
           Monotropall
           Figures
           ,
           and
           have
           but
           each
           one
           Mood
           :
           but
           the
           other
           two
           have
           a
           couple
           a
           piece
           ,
           so
           that
           for
           the
           Planobliquangulars
           ,
           all
           the
           foure
           together
           afford
           us
           six
           Datoquaeres
           .
        
         
           The
           Mood
           of
           Alaheme
           is
           Danarele
           ,
           which
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           two
           Angles
           being
           given
           and
           a
           Side
           ,
           another
           Side
           is
           demanded
           ,
           and
           by
           its
           Resolver
           
             Sapeg
             —
             Eg
             —
             Sapyr
          
           ☞
           Yr
           ,
           sheweth
           ,
           that
           ,
           if
           
             to
             the
             summe
             of
             the
             Logarithm
             of
             the
             side
             given
             ,
             and
             of
             the
             Sine
             of
             the
             Angle
             opposite
             to
             the
             side
             required
             ,
             we
             adde
             the
             difference
             of
             the
             Secant
             complement
             from
             the
             Radius
             ,
          
           (
           by
           some
           called
           the
           Arithmeticall
           complement
           of
           the
           Sine
           ,
           and
           in
           Master
           Speidels
           Logarithmicall
           Canon
           of
           
             Sines
             ,
             Tangents
          
           ,
           and
           Secants
           with
           good
           reason
           termed
           the
           Secant
           ;
           for
           ,
           though
           it
           doe
           not
           cut
           any
           Arch
           ,
           thereby
           more
           Etymologically
           to
           deserve
           the
           name
           of
           Secant
           ,
           yet
           worketh
           it
           the
           same
           effect
           that
           the
           prolonged
           Radius
           doth
           )
           the
           operation
           will
           proceed
           so
           neatly
           ,
           that
           if
           from
           these
           three
           Logarithms
           thus
           summed
           up
           ,
           we
           onely
           cut
           off
           a
           Digit
           at
           the
           left
           hand
           ,
           we
           will
           find
           as
           much
           by
           addition
           alone
           performed
           in
           this
           case
           ,
           as
           if
           from
           the
           proposed
           summe
           the
           Sine
           of
           the
           Angle
           had
           beene
           abstracted
           ;
           for
           the
           totall
           Sine
           thus
           unradiused
           is
           the
           Logarithm
           of
           the
           side
           required
           .
           But
           such
           as
           are
           not
           acquainted
           with
           this
           compendious
           manner
           of
           calculating
           ,
           or
           peradventure
           are
           not
           accommodated
           with
           a
           convenient
           Canon
           for
           the
           purpose
           ,
           may
           ,
           in
           Gods
           name
           ,
           use
           their
           owne
           way
           ,
           the
           Resolver
           being
           of
           such
           amplitude
           ,
           that
           it
           extends
           it selfe
           to
           all
           sorts
           of
           operations
           ,
           whereby
           the
           truth
           of
           the
           fourth
           Ternary
           in
           this
           Mood
           may
           be
           attaind
           unto
           ;
           for
           it
           is
           Analogised
           thus
           ,
           
             As
             the
             Sine
             of
             the
             Angle
             opposite
             to
             the
             side
             given
             is
             to
             the
             same
             given
             side
             ;
             so
             the
             Sine
             of
             the
             Angle
             opposite
             to
             the
             side
             required
             ,
             to
             the
             required
             side
             .
          
           The
           reason
           of
           this
           proportion
           is
           grounded
           on
           the
           second
           Axiom
           ,
           the
           first
           determinater
           of
           whose
           Directory
           sheweth
           ,
           that
           Danarele
           is
           one
           of
           Eprosos
           Enodandas
           .
        
         
           The
           second
           Figure
           of
           the
           Planobliquangulars
           is
           Emenarole
           ,
           whose
           Moods
           are
           Therelabmo
           and
           Zelemabne
           .
        
         
         
           The
           first
           comprehendeth
           all
           those
           Planobliquangular
           Problems
           wherein
           two
           sides
           being
           given
           with
           an
           
             interjacent
             Angle
          
           ,
           an
           
             opposite
             Angle
          
           is
           demanded
           ,
           and
           by
           its
           Resolver
           
             Aggres
             —
             Zes
             —
             Talfagros
          
           ☞
           Talzos
           ,
           sheweth
           ,
           
             that
             if
             from
             the
             summe
             of
             the
             Logarithm
             of
             the
             difference
             of
             the
             sides
             ,
             and
             Tangent
             of
             halfe
             the
             summe
             of
             the
             opposite
             Angles
             ,
             be
             subduced
             the
             aggregat
             or
             summe
             of
             the
             Logarithms
             of
             the
             two
             proposed
             sides
             ,
             the
             remainder
             thereof
             will
             prove
             the
             Logarithm
             of
             the
             Tangent
             of
             halfe
             the
             difference
             of
             the
             opposite
             Angles
          
           ;
           the
           which
           joyned
           to
           the
           one
           ,
           and
           abstracted
           from
           the
           other
           ,
           affords
           us
           the
           measure
           of
           the
           Angle
           we
           require
           ;
           for
           the
           Theoreme
           is
           ,
           
             As
             the
             aggregat
             of
             the
             given
             sides
             ,
             to
             the
             difference
             of
             th●se
             sides
             :
             So
             the
             Tangent
             of
             halfe
             the
             summe
             of
             the
             opposite
             Angles
             ,
             to
             the
             Tangent
             of
             halfe
             the
             difference
             of
             those
             Angles
          
           ;
           which
           ,
           without
           any
           more
           adoe
           ,
           by
           simple
           Addition
           and
           Subtraction
           affordeth
           the
           Angle
           we
           demand
           .
           The
           third
           Axiom
           and
           the
           Theorem
           of
           the
           Resolver
           of
           this
           Mood
           being
           but
           one
           and
           the
           same
           thing
           ,
           I
           must
           make
           bold
           to
           remit
           you
           to
           my
           Apodicticks
           for
           the
           reason
           of
           the
           Analogie
           thereof
           ,
           the
           onely
           determinater
           of
           whose
           Directory
           being
           θ.
           pointeth
           out
           the
           Mood
           of
           Therelabmo
           for
           the
           sole
           enodandum
           appropriated
           thereunto
           .
        
         
           The
           second
           Mood
           of
           this
           Figure
           is
           Zelemabue
           ,
           which
           involveth
           all
           the
           Planobliquangulary
           Problemes
           ,
           wherein
           two
           sides
           being
           given
           with
           the
           
             Angle
             between
          
           ,
           the
           third
           side
           is
           demanded
           :
           and
           not
           being
           calculable
           by
           the
           
             Logarithmicall
             Canon
          
           in
           lesse
           then
           two
           operations
           ,
           because
           it
           requireth
           the
           finding
           out
           of
           another
           Angle
           before
           it
           can
           fix
           upon
           the
           
             side
             ,
             Therelabmo
          
           is
           allowed
           it
           for
           a
           Praesubservient
           ,
           by
           vertue
           whereof
           an
           opposite
           Angle
           is
           obtained
           ,
           and
           Danarele
           for
           its
           Possubservient
           and
           finall
           Resolver
           ,
           by
           whose
           meanes
           we
           get
           the
           side
           required
           .
           The
           reason
           of
           the
           first
           operation
           is
           grounded
           on
           the
           third
           Axiom
           ,
           and
           of
           the
           second
           operation
           on
           the
           second
           :
           but
           because
           this
           Mood
           is
           meerly
           a
           partiall
           Enodandum
           ,
           neither
           of
           the
           foresaid
           Axioms
           affordeth
           any
           Directory
           concerning
           it
           ,
           otherwise
           then
           in
           the
           two
           Subservients
           thereof
           .
        
         
           The
           third
           Figure
           is
           Eneroloms
           ,
           whose
           two
           Moods
           are
           X●monor●
           and
           Shenerolem
           .
        
         
           The
           first
           Mood
           of
           this
           Figure
           includeth
           all
           those
           Planobliquangularie
           Problems
           ,
           wherein
           two
           sides
           being
           given
           ,
           with
           an
           
             opposite
             Angle
          
           ,
           another
           
             opposite
             Angle
          
           is
           demanded
           ,
           and
           by
           its
           Resolver
           
           
             E
             —
             Sog
             —
             Ge
             ☞
             So
             ,
          
           sheweth
           ,
           that
           
             if
             from
             the
             aggregat
             of
             the
             Logarithm
             of
             one
             of
             the
             given
             sides
             ,
             and
             that
             of
             the
             Sine
             of
             the
             opposite
             Angle
             proposed
             ,
             we
             subtract
             the
             Logarithm
             of
             the
             other
             given
             side
             ,
             the
             residue
             will
             afford
             us
             the
             Logarithm
             of
             the
             Sine
             of
             the
             opposite
             Angle
             required
          
           ;
           for
           it
           is
           Analogised
           thus
           ,
           
             As
             one
             of
             the
             sides
             ,
             to
             the
             Sine
             of
             the
             opposite
             Angle
             given
             :
             so
             the
             other
             side
             proposed
             ,
             to
             the
             Sine
             of
             the
             opposite
             Angle
             required
             .
          
           The
           reason
           of
           this
           proportion
           is
           from
           the
           second
           Axiom
           ,
           the
           sixth
           characteristick
           of
           whose
           Directory
           importeth
           ,
           that
           Xemenoro
           is
           one
           of
           Eprosos
           enodandas
           .
        
         
           The
           second
           Mood
           of
           this
           Figure
           is
           Shenerolem
           ,
           which
           containeth
           all
           those
           Planobliquangularie
           Problems
           ,
           wherein
           two
           sides
           being
           given
           with
           an
           
             opposite
             Angle
          
           ,
           the
           third
           side
           is
           demanded
           ,
           which
           not
           being
           findable
           by
           the
           Logarithmicall
           Table
           upon
           the
           foresaid
           Datas
           in
           lesse
           then
           two
           operations
           (
           because
           an
           Angle
           must
           be
           obtained
           first
           before
           the
           side
           can
           be
           had
           )
           Xemenoro
           Praesubserves
           it
           for
           an
           Angle
           ,
           and
           Danarele
           becomes
           its
           Possubservient
           for
           the
           side
           required
           .
           The
           reason
           of
           both
           these
           operations
           is
           founded
           on
           the
           second
           Axiom
           ,
           the
           last
           Characteristick
           of
           whose
           Directory
           inrolleth
           Shenerolem
           for
           one
           of
           Eprosos
           enodandas
           .
        
         
           The
           fourth
           figure
           is
           Erelea
           ,
           which
           ,
           being
           Monotropall
           ,
           hath
           no
           Mood
           but
           Pserelema
           .
        
         
           This
           Pserelema
           encompasseth
           all
           those
           Planobliquangulary
           Problems
           wherein
           the
           three
           sides
           being
           proposed
           ,
           an
           Angle
           is
           required
           .
           This
           Datoquaere
           not
           being
           resolvable
           by
           the
           Logarithms
           in
           lesse
           then
           two
           operations
           ,
           because
           the
           Segments
           of
           the
           Base
           ,
           or
           sustaining
           side
           must
           needs
           be
           found
           out
           ,
           that
           by
           demitting
           of
           a
           Perpendicular
           from
           the
           
             top
             Angle
          
           ,
           we
           may
           hit
           upon
           the
           Angle
           demanded
           :
           the
           Resolver
           for
           the
           Segments
           is
           
             Ba
             —
             Gres
             —
             Zes
             ☞
             Zius
             ,
          
           whereby
           we
           learne
           ,
           that
           
             if
             from
             the
             Logarithm
             of
             the
             summe
             of
             the
             sides
             ,
             joyned
             to
             the
             Logarithm
             of
             the
             difference
             of
             the
             sides
             ,
             we
             subtract
             the
             Logarithm
             of
             the
             Base
             ,
             the
             remainder
             is
             the
             Logarithm
             of
             the
             difference
             of
             the
             Segments
             ,
          
           which
           difference
           being
           taken
           from
           the
           whole
           Base
           ,
           halfe
           the
           difference
           proves
           to
           be
           the
           lesser
           Segment
           .
           This
           Theorem
           being
           thus
           the
           Praesubservient
           of
           this
           Mood
           ,
           its
           Possubservient
           is
           Vbeman
           ,
           whose
           generall
           Resolver
           
             V
             —
             Rad
             —
             Eg
             ☞
             Sor
             ,
          
           is
           particularised
           for
           this
           case
           
             Uxiug
             —
             Rad
             —
             Ing
             ☞
             Sor
             ,
          
           which
           sheweth
           ,
           that
           
             if
             from
             the
             summe
             of
             the
             Logarithms
             of
             the
             totall
             Sine
             ,
             and
             of
             one
             of
             the
             Segments
             given
             ,
             we
             subduce
             the
             Logarithm
          
           
           
             of
             the
             Hypotenusa
             conterminall
             with
             the
             Segment
             proposed
             ,
             the
             remainer
             will
             be
             the
             Logarithm
             of
             the
             Sine
             of
             the
             opposite
             Angle
             required
          
           ;
           for
           the
           demitting
           of
           the
           Perpendicular
           opens
           a
           way
           to
           have
           the
           Theorem
           to
           be
           first
           in
           generall
           propounded
           thus
           ,
           
             As
             the
             Subtendent
             to
             the
             totall
             Sine
             ,
             so
             the
             containing
             side
             given
             to
             the
             Sine
             of
             the
             Angle
             required
             :
          
           or
           in
           particular
           thus
           ,
           
             As
             the
             Sine
             of
             the
             Cosubtendent
             adjoyning
             the
             Segment
             given
             is
             to
             the
             Radius
             ,
             so
             is
             the
             said
             Segment
             proposed
             to
             the
             Sine
             of
             the
             Angle
             required
             .
          
        
      
       
         
           Thus
           farre
           for
           the
           calculating
           of
           plaine
           Triangles
           ,
           both
           right
           and
           oblique
           :
           now
           follow
           the
           Sphericals
           .
        
         
           THere
           be
           three
           principall
           Axioms
           upon
           which
           dependeth
           the
           resolving
           of
           Sphericall
           Triangles
           ,
           to
           wit
           ,
           
             Suprosca
             ,
             Sbaprotca
          
           ,
           and
           Seproso
           .
        
         
           The
           first
           Maxime
           or
           
             Axiom
             ,
             Suprosca
          
           ,
           sheweth
           ,
           that
           of
           severall
           rectangled
           Sphericals
           ,
           which
           have
           one
           and
           the
           same
           
             acute
             Angle
          
           at
           the
           Base
           ,
           the
           
             Sines
             of
             the
             Hypotenusas
             are
             proportionall
             to
             the
             Sines
             of
             their
             Perpendiculars
          
           ;
           for
           ,
           from
           the
           same
           inclination
           every
           where
           of
           the
           one
           plaine
           to
           the
           other
           ,
           there
           ariseth
           an
           equiangularity
           in
           the
           two
           rectangles
           ,
           out
           of
           which
           we
           may
           confidently
           inferre
           the
           homologall
           sides
           (
           which
           are
           the
           Sines
           of
           the
           Subtendents
           ,
           and
           of
           the
           Perpendiculars
           of
           the
           one
           ,
           and
           the
           other
           )
           to
           be
           amongst
           themselves
           proportionall
           .
           It
           s
           Directory
           is
           Uphugen
           ,
           by
           the
           which
           we
           learn
           ,
           that
           
             Uphanep
             ,
             Ugemon
          
           ,
           and
           Enarul
           ,
           are
           its
           three
           enodandas
           .
        
         
           The
           second
           Axiom
           is
           Sbaprotca
           ;
           whereby
           we
           learne
           ,
           that
           in
           all
           rectangled
           Sphericals
           that
           have
           one
           and
           the
           same
           acute
           Angle
           at
           the
           Base
           ,
           
             the
             Sines
             of
             the
             Bases
             are
             proportionall
             to
             the
             Tangents
             of
             their
             Perpendiculars
             :
          
           which
           Analogie
           proceedeth
           from
           the
           equiangularity
           of
           such
           rectangled
           Sphericals
           ,
           by
           the
           semblable
           inclining
           of
           the
           plaine
           towards
           them
           both
           .
           This
           proportion
           neverthelesse
           will
           never
           hold
           betwixt
           the
           Sines
           of
           the
           Bases
           ,
           and
           the
           Sines
           of
           their
           Perpendiculars
           ;
           because
           ,
           if
           the
           Sines
           of
           the
           Bases
           were
           proportionall
           to
           the
           Sines
           of
           the
           Perpendiculars
           (
           the
           Sines
           
           of
           the
           Perpendiculars
           being
           already
           demonstrated
           proportionall
           to
           the
           Sines
           of
           the
           Subtendents
           )
           either
           the
           Sine
           of
           the
           Perpendicular
           ,
           or
           the
           Sine
           of
           the
           Base
           would
           be
           the
           cord
           of
           the
           same
           Arch
           ,
           whereof
           it
           is
           a
           Sine
           ;
           which
           is
           impossible
           ,
           by
           reason
           that
           nothing
           can
           be
           both
           a
           whole
           ,
           and
           a
           part
           ,
           in
           regard
           of
           one
           and
           the
           same
           thing
           ;
           and
           therefore
           doe
           we
           onely
           say
           ,
           that
           
             the
             Sines
             of
             the
             Bases
             ,
             and
             Tangents
             of
             the
             Perpendiculars
             ,
             and
             contrarily
             ,
             are
             proportionall
             .
          
           It
           s
           Directory
           is
           Pubkutethepsaler
           ,
           which
           sheweth
           ,
           that
           
             Upalam
             ,
             Ubamen
             ,
             Vkelamb
             ,
             Etalum
             ,
             Ethaner
             ,
             epsoner
             ,
             Alamun
             ,
          
           and
           Erelam
           ,
           are
           the
           eight
           Enodandas
           the
           reupon
           depending
           .
        
         
           The
           third
           Axiom
           is
           ,
           that
           
             the
             Sines
             of
             the
             sides
             are
             proportionall
             to
             the
             Sines
             of
             their
             opposite
             Angles
             :
          
           the
           truth
           whereof
           holds
           in
           all
           Sphericall
           Triangles
           whatsoever
           ;
           which
           is
           proved
           partly
           out
           of
           the
           proportion
           betwixt
           the
           Sines
           of
           the
           Perpendiculars
           substerned
           under
           equall
           Angles
           ,
           and
           the
           Sines
           of
           the
           Hypotenusas
           :
           and
           partly
           ,
           by
           the
           Analogy
           ,
           that
           is
           betwixt
           the
           Sines
           of
           the
           Angles
           sustained
           by
           severall
           Perpendiculars
           ,
           demitted
           from
           one
           point
           ,
           and
           the
           Sines
           of
           the
           Perpendiculars
           themselves
           .
           The
           Directory
           of
           this
           Axiom
           is
           Vchedezexam
           ,
           whereby
           we
           know
           that
           
             Uchener
             ,
             Edamon
             ,
             Ezolum
             ,
             Exoman
             ,
          
           and
           Amaner
           ,
           are
           the
           five
           Enodandas
           thereof
           .
        
      
       
         
           The
           Orthogonosphericall
           Table
           consisteth
           of
           these
           six
           Figures
           :
           
             Valamenep
             ,
             Vemanore
             ,
             Enarulome
             ,
             Erolumane
             ,
             Achave
             ,
          
           and
           Esheva
           .
        
         
           THe
           first
           Figure
           ,
           Valamenep
           ,
           comprehendeth
           all
           those
           questions
           ,
           wherein
           the
           
             Subtendent
             ,
             and
             an
             Angle
             being
             given
             ,
             either
             another
             Angle
             ,
             or
             one
             of
             the
             Ambients
             is
             demanded
             .
          
        
         
           Of
           this
           Figure
           there
           be
           three
           Moods
           ,
           
             viz.
             Upalam
             ,
             Ubamen
          
           ,
           and
           Uphanep
           .
           The
           first
           ,
           to
           wit
           Upalam
           ,
           containeth
           all
           those
           Orthogonosphericall
           Problems
           ,
           wherein
           the
           
             Subtendent
             and
             one
             oblique
             Angle
             being
             given
             ,
             another
             oblique
             Angle
             is
             required
             ,
          
           and
           by
           its
           Resolver
           
             Torb
             —
             Tag
             —
             Nu
             ☞
             Mir
             ,
          
           sheweth
           ,
           that
           the
           summe
           of
           the
           Sine
           complement
           of
           the
           Subtendent
           side
           and
           Tangent
           of
           the
           Angle
           given
           ,
           (
           the
           Logarithms
           of
           these
           are
           alwayes
           to
           be
           understood
           )
           
           a
           digit
           being
           prescinded
           from
           the
           left
           ,
           is
           equall
           to
           the
           Tangent
           complement
           of
           the
           Angle
           required
           ;
           for
           the
           proposition
           goeth
           thus
           ,
           
             As
             the
             Radius
             ,
             to
             the
             Tangent
             of
             the
             Angle
             given
             :
             so
             the
             Sine
             complement
             of
             the
             Subtendent
             side
             ,
             to
             the
             Tangent
             complement
             of
             the
             Angle
             required
             :
          
           and
           because
           Tangents
           ,
           and
           Tangent
           complements
           are
           reciprocally
           proportionall
           ,
           instead
           of
           
             To
             —
             Tag
             —
             Nu
             ☞
             Mir
             ,
          
           or
           ,
           
             To
             —
             Lu
             —
             Mag
             ☞
             Tir
             ,
          
           which
           (
           for
           that
           the
           Radius
           is
           a
           meane
           proportionall
           betwixt
           the
           L.
           and
           N.
           the
           T.
           and
           M
           )
           is
           all
           one
           for
           inferring
           of
           the
           same
           fourth
           proportionall
           ,
           or
           foresaid
           quaesitum
           )
           we
           may
           say
           ,
           
             Mag
             —
             Nu
             —
             To
             ☞
             Mir
             ,
          
           that
           is
           ,
           
             As
             the
             Tangent
             complement
             of
             the
             given
             Angle
             to
             the
             Cosine
             of
             the
             Subtendent
             ,
             so
             the
             totall
             Sine
             to
             the
             Antitangent
             of
             the
             Angle
             demanded
          
           ;
           for
           the
           totall
           Sine
           being
           ,
           as
           I
           have
           told
           you
           ,
           a
           meane
           proportionall
           betwixt
           the
           Tangents
           and
           Cotangents
           ,
           the
           subtracting
           of
           the
           Cotangent
           ,
           or
           Tangent
           complement
           from
           the
           summe
           of
           the
           Radius
           ,
           and
           Antisine
           residuats
           a
           Logarithm
           equall
           to
           that
           of
           the
           remainder
           ,
           by
           abstracting
           the
           Radius
           from
           the
           sum
           of
           the
           Cosine
           of
           the
           subtendent
           ,
           and
           Tangent
           of
           the
           Angle
           given
           ,
           either
           of
           which
           will
           fall
           out
           to
           be
           the
           Antitangent
           of
           the
           required
           Angle
           .
        
         
           Notandum
           .
        
         
           [
           Here
           alwayes
           is
           to
           be
           observed
           ,
           that
           the
           subtracting
           of
           Logarithms
           may
           be
           avoyded
           ,
           by
           substituting
           the
           
             Arithmeticall
             complement
             thereof
          
           ,
           to
           be
           added
           to
           the
           Logarithms
           of
           the
           two
           middle
           proportionals
           (
           which
           Arithmeticall
           complement
           (
           according
           to
           Gellibrand
           )
           is
           nothing
           else
           ,
           but
           the
           difference
           between
           the
           Logarithm
           to
           be
           subtracted
           ,
           and
           another
           consisting
           of
           an
           unit
           ,
           or
           binarie
           with
           the
           addition
           of
           cyphers
           ,
           that
           is
           the
           single
           ,
           or
           double
           Radius
           )
           for
           so
           the
           sum
           of
           the
           three
           Logarithms
           ,
           cutting
           off
           an
           unit
           ,
           or
           binarie
           towards
           the
           left
           hand
           ,
           will
           still
           be
           the
           Logarithm
           of
           the
           fourth
           proportionall
           required
           .
        
         
           For
           the
           greater
           ease
           therefore
           in
           Trigonometricall
           computations
           ,
           such
           a
           Logarithmicall
           Canon
           is
           to
           be
           wished
           for
           ,
           wherein
           the
           Radius
           is
           left
           out
           of
           all
           the
           Secants
           ,
           and
           all
           the
           Tangents
           of
           Major
           Arches
           ,
           according
           to
           the
           method
           prescribed
           by
           Mr.
           Speidel
           ,
           who
           is
           willing
           to
           take
           the
           paines
           to
           make
           such
           a
           new
           Canon
           ,
           better
           then
           any
           that
           ever
           hitherto
           hath
           beene
           made
           use
           of
           ,
           so
           that
           the
           publike
           ,
           whom
           it
           most
           concerneth
           ,
           or
           some
           potent
           man
           ,
           well
           minded
           towards
           the
           Mathematicks
           ,
           would
           be
           so
           generous
           ,
           
           as
           to
           releeve
           him
           of
           the
           charge
           it
           must
           needs
           cost
           him
           ;
           which
           ,
           considering
           his
           great
           affection
           to
           ,
           and
           ability
           in
           those
           sciences
           ,
           will
           certainly
           be
           as
           small
           a
           summe
           ,
           as
           possibly
           he
           can
           bring
           it
           to
           .
           ]
        
         
           
             This
             Parenthesis
             ,
             though
             somewhat
             with
             the
             longest
             ,
             will
             not
             (
             I
             hope
             )
             be
             displeasing
             to
             the
             studious
             Reader
             .
          
        
         
           The
           second
           Mood
           of
           the
           first
           Figure
           is
           Ubamen
           ,
           which
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           the
           
             Subtendent
             ,
             and
             one
             oblique
             Angle
             being
             given
             ,
             the
             Ambient
             adjoyning
             the
             Angle
             given
             is
             required
             ,
          
           and
           by
           its
           Resolver
           ,
           
             Nag
             —
             Mu
             —
             Torp
             ☞
             Myr
             ,
          
           sheweth
           ,
           that
           ,
           
             if
             to
             the
             summe
             of
             the
             Logarithms
             of
             the
             two
             middle
             proportionals
             ,
             we
             adde
             the
             Arithmeticall
             complement
             of
             the
             first
             ,
             the
             cutting
             off
             the
             Index
             from
             the
             Aggregat
             of
             the
             three
             ,
             will
             residuat
             the
             Tangent
             complement
             of
             the
             side
             required
             :
          
           and
           therefore
           with
           the
           totall
           Sine
           in
           the
           first
           place
           ,
           it
           may
           be
           thus
           propounded
           ,
           
             Torp
             —
             Mu
             —
             Lag
             ☞
             Myr
          
           ;
           for
           the
           first
           Theorem
           being
           ,
           
             As
             the
             Sine
             complement
             of
             the
             Angle
             given
             ,
             to
             the
             Tangent
             complement
             of
             the
             subtendent
             side
             :
             so
             the
             totall
             Sine
             ,
             to
             the
             Tangent
             complement
             of
             the
             side
             required
             :
          
           just
           so
           the
           second
           Theorem
           ,
           which
           is
           that
           refined
           ,
           is
           ,
           
             As
             the
             totall
             Sine
             ,
             to
             the
             Tangent
             complement
             of
             the
             Subtendent
             :
             so
             the
             Secant
             of
             the
             given
             Angle
             ,
             to
             the
             Tangent
             complement
             of
             the
             demanded
             side
             .
          
           Here
           you
           must
           consider
           ,
           as
           I
           have
           told
           you
           already
           ,
           that
           of
           the
           whole
           Secant
           I
           take
           but
           its
           excesse
           above
           the
           Radius
           ,
           as
           I
           doe
           of
           all
           Tangents
           above
           45.
           
           Degrees
           ;
           because
           the
           cutting
           off
           the
           first
           digit
           on
           the
           left
           ,
           supplieth
           the
           subtraction
           ,
           requisite
           for
           the
           finding
           out
           of
           the
           fourth
           proportionall
           ;
           so
           that
           by
           addition
           onely
           the
           whole
           operation
           may
           be
           performed
           ,
           of
           all
           wayes
           the
           most
           succinct
           and
           ready
           .
           Otherwise
           ,
           because
           of
           the
           totall
           Sines
           meane
           proportionality
           betwixt
           the
           Sine
           complement
           ,
           and
           the
           Secant
           ;
           and
           betwixt
           the
           Tangent
           ,
           and
           Tangent
           complement
           ,
           it
           may
           be
           regulated
           thus
           ,
           
             To
             —
             Tu
             —
             Nag
             ☞
             Tyr
             ,
          
           that
           is
           ,
           
             As
             the
             Radius
             ,
             to
             the
             Tangent
             of
             the
             Subtendent
             ,
             so
             the
             Sine
             complement
             of
             the
             Angle
             given
             ,
             to
             the
             Tangent
             of
             the
             side
             required
             .
          
           The
           reason
           of
           the
           resolution
           both
           of
           this
           ,
           and
           of
           the
           former
           Datoquaere
           ,
           is
           grounded
           on
           the
           second
           Axiom
           ,
           and
           the
           proportion
           that
           ,
           in
           severall
           rectangled
           Sphericals
           which
           have
           the
           same
           acute
           Angle
           at
           the
           Base
           ,
           is
           found
           betwixt
           the
           Sines
           of
           their
           Perpendiculars
           ,
           and
           Tangents
           of
           their
           Bases
           ,
           as
           is
           shewne
           you
           by
           the
           two
           first
           Consonants
           of
           the
           Directory
           of
           Sbaprotca
           .
        
         
         
           The
           third
           and
           last
           Mood
           of
           the
           first
           Figure
           is
           Uphaner
           ,
           which
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           
             the
             Hypotenusa
             ,
             and
             one
             of
             the
             obliques
             being
             given
             ,
             the
             opposite
             Ambient
             is
             required
             ,
          
           and
           by
           its
           Resolver
           
             Tol
             —
             Sag
             —
             Su
             ☞
             Syr
             ,
          
           sheweth
           ,
           that
           ,
           
             if
             we
             adde
             the
             Logarithms
             of
             the
             Sine
             of
             the
             Angle
             ,
             and
             Sine
             of
             the
             Subtendent
             ,
             cutting
             off
             the
             left
             Supernumerarie
             digit
             from
             the
             summe
             ,
             it
             gives
             us
             the
             Logarithm
             of
             the
             Sine
             of
             the
             side
             demanded
          
           ;
           for
           it
           is
           ,
           
             As
             the
             totall
             Sine
             ,
             to
             the
             Sine
             of
             the
             Angle
             given
             :
             so
             the
             Sine
             of
             the
             subtendent
             side
             ,
             to
             the
             Sine
             of
             the
             side
             required
             :
          
           and
           because
           by
           the
           Axiom
           of
           Rulerst
           ,
           it
           was
           proved
           ,
           that
           
             when
             the
             Sine
             of
             any
             Arch
             is
             made
             Radius
             ,
             what
             was
             then
             the
             totall
             Sine
             ,
             becomes
             a
             Secant
          
           (
           and
           therefore
           Secant
           complement
           of
           that
           Arch
           )
           instead
           of
           
             Tol
             —
             Sag
             —
             Su
             ☞
             Syr
             ,
          
           we
           may
           say
           ,
           
             To
             —
             Ru
             —
             Rag
             ☞
             Ryr
             ,
          
           that
           is
           ,
           
             As
             the
             totall
             Sine
             ,
             is
             to
             the
             Secant
             complement
             of
             the
             subtendent
             :
             so
             the
             Secant
             complement
             of
             the
             Angle
             given
             ,
             to
             the
             Secant
             complement
             of
             the
             side
             demanded
             .
          
           The
           resolution
           of
           this
           Datoquaere
           by
           Sines
           ,
           is
           grounded
           on
           the
           first
           Axiom
           of
           Sphericals
           ,
           which
           elucidats
           the
           proportion
           betwixt
           the
           Sines
           of
           the
           Hypotenusas
           ,
           and
           Perpendiculars
           ,
           as
           it
           is
           declared
           to
           us
           by
           the
           first
           syllable
           of
           Suproscas
           Directory
           .
        
         
           The
           second
           Figure
           is
           Vemanore
           ,
           which
           containeth
           all
           those
           Orthogonosphericall
           questions
           ,
           wherein
           
             the
             subtendent
             ,
             and
             an
             Ambient
             being
             proposed
             ,
             either
             of
             the
             obliques
             ,
             or
             the
             other
             Ambient
             is
             required
             ,
          
           and
           hath
           three
           Moods
           ,
           
             viz.
             Ukelamb
             ,
             Ugemon
          
           ,
           and
           Uchener
           .
        
         
           The
           first
           Mood
           Ukelamb
           comprehendeth
           all
           those
           Orthogonosphericall
           Problems
           ,
           wherein
           the
           
             subtendent
             ,
             and
             one
             including
             side
             being
             given
             ,
             the
             interjacent
             Angle
             is
             demanded
             ,
          
           and
           by
           its
           Resolver
           
             Meg
             —
             Torp
             —
             Mu
             ☞
             Nir
          
           (
           or
           because
           of
           the
           totall
           Sines
           mean
           proportion
           betwixt
           the
           Tangent
           ,
           and
           Tangent
           complement
           )
           
             Torp-Teg
             —
             Mu
             ☞
             Nir
          
           (
           which
           is
           the
           same
           in
           effect
           )
           sheweth
           ,
           that
           if
           
             from
             the
             summe
             of
             the
             Logarithms
             of
             the
             middle
             termes
             ,
          
           (
           which
           in
           the
           first
           Analogy
           is
           the
           Radius
           ,
           and
           Tangent
           complement
           of
           the
           subtendent
           )
           
             we
             subtract
             the
             Tangent
             complement
             of
             the
             given
             Ambient
             :
          
           or
           ,
           in
           the
           second
           order
           of
           proportionals
           ,
           
             joyne
             the
             Tangent
             of
             the
             side
             given
             ,
             to
             the
             Tangent
             complement
             of
             the
             subtendent
             ,
             and
             from
             the
             sum
             cut
             off
             the
             Index
          
           (
           if
           need
           be
           )
           both
           will
           tend
           to
           the
           same
           end
           ,
           and
           
             produce
             for
             the
             fourth
             proportionall
             ,
             the
             Sine
             complement
             of
             the
             Angle
             required
          
           ;
           for
           to
           subtract
           a
           Tangent
           complement
           from
           the
           Radius
           ,
           and
           another
           
           number
           joyned
           together
           ,
           whether
           that
           Tangent
           complement
           be
           more
           or
           lesse
           then
           the
           Radius
           ,
           it
           is
           all
           one
           ,
           as
           if
           you
           should
           subtract
           the
           Radius
           from
           the
           said
           Tangent
           complement
           ,
           and
           that
           other
           number
           ;
           because
           the
           Tangent
           (
           or
           rather
           Logarithm
           of
           the
           Tangent
           ;
           for
           so
           it
           must
           be
           alwayes
           understood
           ,
           and
           not
           onely
           in
           Tangents
           ,
           but
           in
           Sines
           ,
           Secants
           ,
           Sides
           ,
           and
           Angles
           ,
           though
           for
           brevity
           sake
           the
           word
           Logarithm
           be
           oftentimes
           omitted
           )
           because
           I
           say
           ,
           the
           Logarithms
           of
           the
           Tangent
           ,
           and
           Tangent
           complement
           together
           ,
           being
           the
           double
           of
           the
           Radius
           )
           if
           first
           the
           Tangent
           complement
           surpasse
           the
           Radius
           ,
           and
           be
           to
           be
           subtracted
           from
           it
           ,
           and
           another
           number
           ,
           it
           is
           all
           one
           ,
           as
           if
           from
           the
           said
           number
           you
           would
           abstract
           the
           Radius
           ,
           and
           the
           Tangent
           complements
           excesse
           above
           it
           ,
           so
           that
           the
           Radius
           being
           in
           both
           ,
           there
           will
           remaine
           a
           Tangent
           with
           the
           other
           number-Likewise
           ,
           if
           a
           Tangent
           complement
           ,
           lesse
           then
           the
           Radius
           ,
           be
           to
           be
           subtracted
           from
           the
           summe
           of
           the
           Radius
           ,
           and
           another
           Logarithm
           ;
           it
           is
           yet
           all
           one
           ,
           as
           if
           you
           had
           subtracted
           the
           Radius
           from
           the
           same
           summe
           ;
           because
           ,
           though
           that
           Tangent
           complement
           be
           lesse
           then
           the
           Radius
           :
           yet
           ,
           that
           parcell
           of
           the
           Radius
           which
           was
           abstracted
           more
           then
           enough
           ,
           is
           recompensed
           in
           the
           Logarithm
           of
           the
           Tangent
           to
           be
           joyned
           with
           the
           other
           number
           ;
           for
           ,
           from
           which
           soever
           of
           the
           Tangents
           the
           Radius
           be
           subduced
           ,
           its
           Antitangent
           is
           remainder
           :
           both
           which
           cases
           may
           be
           thus
           illustrated
           in
           numbers
           ;
           and
           first
           ,
           where
           the
           Tangent
           complement
           is
           greater
           then
           the
           Radius
           ,
           as
           in
           these
           numbers
           6.
           4.
           3.
           1.
           and
           4.
           2.
           3.
           1.
           where
           ,
           let
           6.
           be
           the
           Tangent
           complement
           ,
           4.
           the
           Radius
           ,
           3.
           the
           number
           to
           be
           joyned
           with
           the
           Radius
           ,
           or
           either
           of
           the
           Tangents
           ,
           and
           1.
           the
           remainer
           ;
           for
           4.
           and
           3.
           making
           7.
           if
           you
           abstract
           6.
           there
           will
           remaine
           1.
           
           Likewise
           2.
           and
           3.
           making
           5.
           if
           you
           subtract
           4.
           there
           will
           remaine
           1.
           
           Next
           ,
           if
           the
           Tangent
           complement
           be
           lesse
           then
           the
           Radius
           ,
           as
           in
           2.
           4.
           3.
           5.
           and
           4.
           6.
           3.
           and
           5.
           where
           ,
           let
           2.
           be
           the
           Tangent
           complement
           ;
           for
           if
           from
           4.
           and
           3.
           joyned
           together
           ,
           you
           abstract
           2.
           there
           will
           remaine
           5.
           which
           will
           also
           be
           the
           remainder
           ,
           when
           you
           subtract
           4.
           from
           6.
           and
           3.
           added
           together
           .
           Now
           to
           make
           the
           same
           Resolver
           (
           the
           variety
           whereof
           I
           have
           beene
           so
           large
           in
           explaining
           )
           to
           runne
           altogether
           upon
           Tangents
           ,
           instead
           of
           
             Meg
             —
             To
             —
             Mu
             ☞
             Nir
             ,
          
           that
           is
           ,
           
             As
             the
             Tangent
             complement
             of
             the
             side
             given
             ,
             is
             to
             the
             totall
             Sine
             :
             so
             the
             Tangent
          
           
           
             complement
             of
             the
             subtendent
             side
             ,
             to
             the
             Sine
             complement
             of
             the
             Angle
             required
             ,
          
           we
           may
           say
           ,
           
             Tu
             —
             Teg
             —
             To
             ☞
             Nir
          
           ;
           that
           is
           ,
           
             As
             the
             Tangent
             for
             the
             subtendent
             ,
             is
             to
             the
             Tangent
             of
             the
             given
             side
             ;
             so
             the
             totall
             Sine
             ,
             to
             the
             Sine
             complement
             of
             the
             Angle
             required
             .
          
           All
           this
           is
           grounded
           on
           the
           second
           
             Axiom
             Sbaprotca
          
           ,
           and
           upon
           the
           reciprocall
           proportion
           of
           the
           Tangents
           and
           antitangents
           ,
           as
           is
           evident
           by
           the
           third
           characteristick
           of
           its
           Directory
           .
        
         
           The
           second
           Mood
           of
           Vemanore
           is
           Vgemon
           ,
           which
           comprehendeth
           all
           those
           orthogonosphericall
           problems
           ,
           wherein
           the
           
             subtendent
             ,
             with
             an
             Ambient
             being
             given
             ,
             an
             opposite
             oblique
             is
             required
             ,
          
           and
           by
           its
           Resolver
           ,
           
             Su-Seg-Tom
             ☞
             Sir
          
           ,
           or
           (
           by
           putting
           the
           Radius
           in
           the
           first
           place
           ,
           according
           to
           Uradesso
           ,
           the
           first
           branch
           of
           the
           first
           axiom
           of
           the
           Planorectangulars
           )
           
             To-Seg
             Ru
             ☞
             Sir
          
           ,
           sheweth
           ,
           that
           the
           
             summe
             of
             the
             side
             given
             ,
             and
             secant
             of
             the
             subtendent
          
           (
           the
           Supernumerarie
           digit
           being
           cut
           off
           )
           
             is
             the
             sine
             of
             the
             Angle
             required
          
           ;
           for
           the
           Theorem
           is
           ,
           
             As
             the
             sine
             of
             the
             subtendent
             ,
             to
             the
             sine
             of
             the
             side
             given
             :
             so
             the
             Radius
             ,
             to
             the
             sine
             of
             the
             Angle
             required
             :
          
           or
           ,
           
             As
             the
             totall
             sine
             ,
             to
             the
             sine
             of
             the
             side
             given
             :
             so
             the
             secant
             complement
             of
             the
             subtendent
             ,
             to
             the
             sine
             of
             the
             angle
             required
             :
          
           or
           ,
           changing
           the
           sines
           into
           secant
           complements
           ,
           and
           the
           secant
           complements
           into
           sines
           ,
           we
           may
           say
           ,
           
             To
             —
             Su
             —
             Reg
             ☞
             Rir
          
           ;
           because
           ,
           betwixt
           the
           sine
           and
           secant
           complement
           ,
           the
           Radius
           is
           a
           middle
           proportion
           .
           Other
           varieties
           of
           calculation
           in
           this
           ,
           as
           well
           as
           other
           problems
           ,
           may
           be
           used
           ;
           for
           ,
           besides
           that
           every
           proportion
           of
           the
           Radius
           to
           the
           sine
           ,
           Tangent
           ,
           or
           secant
           ,
           and
           contrarily
           ,
           may
           be
           varied
           three
           manner
           of
           wayes
           ,
           by
           the
           first
           Axiom
           of
           Plaine
           triangles
           ,
           the
           alteration
           of
           the
           middle
           termes
           may
           breed
           some
           diversity
           ,
           by
           a
           permutat
           ,
           or
           perturbed
           proportion
           ,
           which
           I
           thought
           good
           to
           admonish
           the
           Reader
           of
           here
           ,
           once
           for
           all
           ,
           because
           there
           is
           no
           problem
           ,
           whether
           in
           Plaine
           ,
           or
           Sphericall
           triangles
           ,
           wherein
           the
           Analogie
           admitteth
           not
           of
           so
           much
           change
           .
           The
           reasons
           of
           this
           Mood
           of
           Ugemon
           ,
           depend
           on
           the
           Axiom
           of
           Suprosca
           ,
           as
           the
           second
           characteristick
           of
           Vphugen
           seemeth
           to
           insinuate
           .
        
         
           The
           last
           Mood
           of
           the
           second
           figure
           is
           Vchener
           ,
           which
           comprehendeth
           all
           those
           problems
           ,
           wherein
           
             the
             subtendent
             ,
             &
             one
             Ambient
             being
             given
             ,
             the
             other
             Ambient
             is
             Required
             ,
          
           and
           by
           its
           Resolver
           ,
           
             Neg
             —
             To
             —
             Nu
             ☞
             Nyr
             ,
          
           or
           ,
           
             To
             —
             le-Nu
             ☞
             Nyr
          
           ,
           sheweth
           ,
           that
           
             the
             summe
             of
             the
             sine
             complement
             of
             the
             subtendent
             ,
             and
             the
             secant
             of
             the
          
           
           
             given
             side
          
           (
           which
           is
           the
           Arithmeticall
           complement
           of
           its
           Antisine
           )
           
             giveth
             us
             the
             sine
             complement
             of
             the
             side
             desired
             ,
          
           the
           Index
           being
           removed
           ;
           for
           the
           theorem
           is
           ,
           
             As
             the
             sine
             complement
             of
             the
             given
             side
             ,
             to
             the
             total
             sine
             ;
             so
             the
             sine
             complement
             of
             the
             subtendent
             ,
             to
             the
             sine
             complement
             of
             the
             side
             required
             :
          
           or
           more
           refinedly
           ,
           As
           
             the
             Radius
             ,
             to
             the
             sine
             complement
             of
             the
             subtendent
             :
             so
             the
             secant
             of
             the
             Leg
             given
             ,
             to
             the
             sine
             complement
             of
             the
             side
             required
             :
          
           and
           besides
           other
           varieties
           of
           Analogie
           ,
           according
           to
           the
           Axiom
           of
           Rulerst
           ,
           by
           making
           use
           of
           the
           reciprocall
           proportion
           of
           the
           sine-complements
           with
           the
           secants
           we
           may
           say
           ,
           
             To-Ne
             Lu
             -
             ☞
             Lyr
          
           ,
           that
           is
           ,
           As
           
             the
             totall
             sine
             ,
             is
             to
             the
             sine
             complement
             of
             the
             given
             side
             :
             so
             the
             secant
             of
             the
             subtendent
             ,
             to
             the
             secant
             of
             the
             side
             required
             .
          
           The
           reason
           of
           this
           Datoqueres
           Resolution
           is
           in
           Seproso
           the
           third
           Axiom
           of
           the
           Sphericals
           ,
           as
           is
           manifest
           by
           the
           first
           figurative
           of
           its
           Directorie
           Uchedezexam
           .
        
         
           The
           third
           figure
           is
           Enarrulome
           ,
           whose
           three
           Moods
           are
           
             Etalum
             ,
             Edamon
          
           ,
           and
           Ethaner
           .
        
         
           This
           figure
           comprehendeth
           all
           those
           orthogonosphericall
           questions
           ,
           wherein
           
             one
             of
             the
             Ambients
             with
             an
             Adjacent
             angle
             is
             given
             ,
             and
             the
             subtendent
             ,
             an
             opposite
             angle
             ,
             or
             the
             other
             containing
             side
             is
             required
             .
          
        
         
           Its
           first
           Mood
           Etalum
           ,
           involveth
           all
           those
           Orthogono
           sphericall
           problems
           ,
           wherein
           
             a
             containing
             side
             ,
             with
             an
             insident
             angle
             thereon
             is
             proposed
             ,
             and
             the
             hypotenusa
             demanded
             :
          
           and
           by
           its
           resolver
           
             Torp-me-nag
             ☞
             mur
          
           or
           (
           by
           inverting
           the
           demand
           upon
           the
           Scheme
           )
           
             Tolp.
             —
             me
             —
             nag
             ☞
             mur
          
           sheweth
           ,
           that
           the
           cutting
           of
           the
           first
           left
           digit
           ,
           from
           the
           summe
           of
           the
           Tangent
           complement
           of
           the
           Ambient
           proposed
           ,
           and
           the
           sine
           complement
           of
           the
           given
           angle
           ,
           affords
           us
           the
           Tangent
           complement
           of
           the
           subtendent
           required
           ;
           for
           the
           theorem
           goes
           thus
           ,
           As
           
             the
             totall
             sine
             ,
             to
             the
             tangent
             complement
             of
             the
             given
             side
             ;
             so
             the
             sine
             complement
             of
             the
             angle
             given
             ,
             to
             the
             tangent
             complement
             of
             the
             hypotenusa
             required
             .
          
           And
           because
           the
           totall
           sine
           ,
           hath
           the
           same
           proportion
           to
           the
           tangent
           complement
           ,
           which
           the
           sine
           ,
           hath
           to
           the
           sine
           complement
           ,
           we
           may
           as
           well
           say
           ,
           
             To-meg-Sa
             ☞
             nur
          
           ,
           that
           is
           ,
           As
           the
           
             Radius
             to
             the
             tangent
             complement
             of
             the
             Ambient
             side
             ;
             so
             the
             sine
             of
             the
             angle
             given
             ,
             to
             the
             sine
             complement
             of
             the
             subtendent
             required
             .
          
           The
           progresse
           of
           this
           Mood
           ,
           dependeth
           on
           the
           Axiom
           of
           Sbaprotca
           ,
           as
           you
           may
           perceive
           by
           the
           fourth
           consonant
           of
           its
           directorie
           Pubkutethepsaler
           .
        
         
         
           The
           second
           Mood
           of
           the
           third
           Figure
           ,
           is
           Edamon
           ,
           which
           comprehendeth
           all
           those
           Orthogonosphericall
           Problems
           ,
           
             wherein
             an
             Ambient
             and
             an
             Adjacent
             angle
             being
             given
             ,
             the
             opposite
             oblique
          
           (
           viz.
           
             the
             angle
             under
             which
             the
             Ambient
             is
             subtended
             )
             is
             required
             ,
          
           and
           by
           its
           Resolver
           
             To-Neg-Sa
             ☞
             Nir
          
           ,
           sheweth
           that
           the
           Addition
           of
           the
           Cosine
           of
           the
           Ambient
           ,
           and
           of
           the
           sine
           of
           the
           Angle
           proposed
           ,
           affordeth
           us
           (
           if
           we
           omit
           not
           the
           usuall
           presection
           )
           the
           Cosine
           of
           the
           Angle
           we
           seek
           for
           ;
           for
           it
           is
           ,
           
             As
             the
             Radius
             to
             the
             Cosine
             ,
             or
             sine
             complement
             of
             the
             given
             side
             :
             so
             the
             sine
             of
             the
             Angle
             proposed
             to
             the
             Antisins
             or
             sine
             complement
             of
             the
             Angle
             demanded
             :
          
           now
           the
           Radius
           being
           alwayes
           a
           meane
           proportionall
           betwixt
           the
           Sine
           complement
           ,
           and
           the
           Secant
           ,
           we
           may
           for
           
             To
             —
             Neg
             —
             Sa
             ☞
             Nir
             ,
             say
             ,
             To
             —
             Leg
             —
             Ra
             ☞
             Lir
             ,
          
           or
           
             To
             —
             Rag
             —
             Le
             ☞
             Lir
             :
          
           that
           is
           ,
           
             As
             the
             totall
             Sine
             to
             the
             Secant
             ,
             or
             cutter
             of
             the
             side
             given
             ,
             or
             to
             the
             Cosecant
             ,
             or
             Secant
             complement
             of
             the
             given
             Angle
             ;
             so
             is
             the
             Secant
             complement
             of
             the
             Angle
             ,
             or
             Secant
             of
             the
             side
             ,
             to
             the
             Secant
             ,
             or
             cutter
             of
             the
             Angle
             required
             .
          
           The
           reason
           of
           all
           this
           is
           grounded
           on
           Seproso
           ,
           because
           it
           runneth
           upon
           the
           proportion
           betwixt
           the
           Sines
           of
           the
           sides
           ,
           and
           the
           Sines
           of
           their
           opposite
           Angles
           ,
           as
           is
           perspicuous
           to
           any
           by
           the
           second
           syllable
           of
           the
           Directory
           of
           that
           Axiome
           .
        
         
           The
           last
           Mood
           of
           the
           third
           Figure
           is
           Ethaner
           ,
           which
           comprehendeth
           all
           those
           Orthogonosphericall
           Problems
           ,
           wherein
           an
           
             Ambient
             with
             an
             Oblique
          
           annexed
           thereto
           ,
           
             is
             given
             ,
             and
             the
             other
             Arch
             about
             the
             right
             Angle
             is
             required
             ,
          
           and
           by
           its
           Resolver
           ,
           
             Torb
             —
             Tag
             —
             Se
             ☞
             Tyr
             ,
          
           sheweth
           ,
           that
           if
           we
           joyne
           the
           Logarithms
           of
           the
           two
           middle
           proportionals
           ,
           which
           are
           the
           Tangent
           of
           the
           given
           Angle
           ,
           and
           the
           Sine
           of
           the
           side
           ,
           the
           usuall
           prefection
           being
           observed
           ,
           we
           shall
           thereby
           have
           the
           Tangent
           ,
           or
           toucher
           of
           the
           Ambient
           side
           desired
           ;
           for
           it
           is
           ,
           
             As
             the
             Radius
             to
             the
             Tangent
             of
             the
             Angle
             given
             ,
             so
             the
             Sine
             of
             the
             containing
             side
             proposed
             to
             the
             side
             required
             :
          
           And
           because
           the
           Tangent
           complement
           ,
           and
           Tangent
           are
           reciprocally
           proportionall
           ,
           the
           Sine
           likewise
           ,
           and
           Secant
           complement
           ,
           for
           
             To
             —
             Tag
             —
             Se
             ☞
             Tyr
             ,
          
           we
           may
           say
           ,
           keeping
           the
           same
           proportion
           ,
           
             To
             —
             Reg
             —
             Ma
             ☞
             Myrs
          
           that
           is
           ,
           
             As
             the
             Radius
             ,
             to
             the
             Secant
             complement
             of
             the
             given
             side
             :
             so
             the
             Tangent
             complement
             of
             the
             Angle
             proposed
             ,
             to
             the
             Tangent
             complement
             of
             the
             side
             required
             .
          
           The
           truth
           of
           all
           these
           operations
           dependeth
           on
           Sbaprotca
           ,
           the
           
           second
           Axiome
           of
           the
           Sphericals
           ,
           as
           is
           evidenced
           by
           θ.
           the
           fifth
           characteristick
           of
           its
           Directory
           Pubkutethepsaler
           .
        
         
           The
           fourth
           Figure
           is
           Erollumane
           ,
           which
           includeth
           all
           Orthogonosphericall
           questions
           ,
           wherein
           
             an
             Ambient
             ,
             and
             an
             opposite
             oblique
             being
             given
             ,
             the
             subtendent
             ,
             the
             other
             oblique
             ,
             or
             the
             other
             Ambient
             is
             demanded
             :
          
           It
           hath
           likewise
           ,
           conforme
           to
           the
           three
           former
           Figures
           ,
           three
           Moods
           belonging
           to
           it
           ;
           the
           first
           whereof
           is
           Ezolum
           .
        
         
           This
           Ezolum
           comprehendeth
           all
           those
           Orthogonosphericall
           ,
           Problems
           ,
           wherein
           
             one
             of
             the
             Legs
             ,
             with
             an
             opposite
             Angle
             being
             given
             ,
             the
             Subtendent
             is
             required
             ,
          
           and
           by
           its
           Resolver
           ,
           
             Sag
             —
             Sep
             —
             Rad
             ☞
             Sur
             ,
          
           or
           by
           putting
           the
           Radius
           in
           the
           first
           place
           ,
           
             To
             —
             Se-Rag
             ☞
             Sur
          
           ,
           sheweth
           ,
           that
           the
           abstracting
           of
           the
           Radius
           from
           the
           sum
           of
           the
           Sine
           of
           the
           side
           ,
           and
           Secant
           complement
           of
           the
           Angle
           given
           ,
           residuats
           the
           Sine
           of
           the
           hypotenusa
           required
           ;
           for
           it
           is
           ,
           
             As
             the
             Sine
             of
             the
             Angle
             given
             ,
             to
             the
             Sine
             of
             the
             opposite
             side
             :
             so
             the
             Radius
             to
             the
             Sine
             of
             the
             subtendent
             :
          
           or
           more
           refinedly
           ,
           
             As
             the
             totall
             Sine
             ,
             to
             the
             Sine
             of
             the
             side
             :
             so
             the
             Secant
             complement
             of
             the
             Angle
             given
             ,
             to
             the
             Sine
             of
             the
             subtendent
             side
             :
          
           And
           because
           of
           the
           Sines
           and
           Antisecants
           ,
           or
           Secant
           complements
           reciprocall
           proportionality
           ,
           
             To
             —
             Sag
             —
             Re
             ☞
             Ru
             ,
          
           that
           is
           ,
           
             As
             the
             Radius
             to
             the
             Sine
             of
             the
             Angle
             given
             :
             so
             the
             Secant
             complement
             of
             the
             proposed
             side
             ,
             to
             the
             Secant
             complement
             of
             the
             subtendent
             required
             .
          
           The
           reason
           of
           all
           this
           is
           grounded
           on
           the
           third
           Axiom
           Seproso
           ,
           as
           is
           made
           manifest
           by
           the
           third
           Syllable
           of
           its
           Directory
           .
        
         
           The
           second
           Mood
           of
           this
           Figure
           is
           Exoman
           ,
           which
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           
             a
             containing
             side
             ,
             and
             an
             opposite
             oblique
             being
             given
             ,
             the
             adjacent
             oblique
             is
             required
             :
          
           and
           by
           its
           Resolver
           ,
           
             Ne
             —
             To
             —
             Nag
             ☞
             Sir
             ,
          
           or
           more
           refinedly
           ,
           
             To
             —
             Le
             —
             Nag
             ☞
             Sir
             ,
          
           sheweth
           ,
           that
           the
           summe
           of
           the
           Sine
           of
           the
           Angle
           ,
           together
           with
           the
           Arithmeticall
           complement
           of
           the
           Antisine
           of
           the
           Leg
           ,
           (
           which
           in
           the
           Table
           I
           have
           so
           much
           recommended
           unto
           the
           Reader
           ,
           is
           set
           downe
           for
           a
           Secant
           )
           the
           usuall
           prefection
           being
           observed
           ,
           affordeth
           us
           the
           Sine
           of
           the
           Angle
           required
           ,
           and
           because
           of
           the
           reciprocall
           proportion
           betwixt
           the
           Sine
           complement
           ,
           and
           Secant
           ;
           and
           betwixt
           the
           Sine
           ,
           and
           Secant
           complement
           ,
           the
           Theorem
           may
           be
           composed
           thus
           :
           
             To
             —
             Neg
             —
             La
             ☞
             Rir
             ,
          
           that
           is
           ,
           
             As
             the
             Radius
             ,
             to
             the
             Sine
             complement
             of
             the
             given
             side
             :
             so
             the
             Secant
             of
             the
             Angle
             proposed
             ,
             to
             the
             Secant
             complement
             of
             the
          
           
           
             Angle
             demanded
          
           .
           The
           reason
           of
           this
           is
           likewise
           grounded
           on
           Seproso
           ,
           as
           you
           may
           perceive
           by
           the
           fourth
           characteristick
           of
           its
           Directory
           .
        
         
           The
           last
           Mood
           of
           this
           Figure
           is
           Epsoner
           ,
           which
           containeth
           all
           those
           Orthogonosphericall
           Problems
           ,
           wherein
           an
           Ambient
           and
           an
           opposite
           Oblique
           being
           given
           ,
           the
           other
           Ambient
           is
           demanded
           ,
           and
           by
           its
           Resolver
           ,
           
             Tag
             —
             Tolb
             —
             Te
             ☞
             Syr
             ,
          
           or
           more
           elabouredly
           ,
           
             Tolb
             —
             Mag
             —
             Te
             ☞
             Syr
             ,
          
           sheweth
           ,
           that
           the
           praescinding
           of
           the
           Radius
           from
           the
           summe
           of
           the
           Tangent
           of
           the
           side
           ,
           and
           Antitangent
           of
           the
           given
           Angle
           ,
           residuats
           the
           Sine
           of
           the
           side
           required
           ;
           for
           it
           is
           ,
           As
           
             the
             Tangent
             of
             the
             Angle
             proposed
             ,
             to
             the
             totall
             Sine
             :
             so
             the
             Tangent
             of
             the
             given
             side
             ,
             to
             the
             Sine
             of
             the
             side
             demanded
             :
          
           or
           ,
           
             As
             the
             Radius
             ,
             to
             the
             Tangent
             complement
             of
             the
             Angle
             given
             :
             so
             the
             Tangent
             of
             the
             given
             side
             ,
             to
             the
             Sine
             of
             the
             side
             required
             :
          
           and
           because
           of
           the
           reciprocall
           Analogy
           betwixt
           the
           Tangents
           ,
           and
           Co-tangents
           :
           and
           betwixt
           the
           Sines
           ,
           and
           Co-secants
           ,
           we
           may
           with
           the
           same
           confidence
           ,
           as
           formerly
           ,
           set
           it
           thus
           in
           the
           rule
           ,
           
             To
             —
             Meg
             —
             Ta
             ☞
             Ryr
             ,
          
           and
           it
           will
           find
           out
           the
           same
           quaesitum
           .
           The
           reason
           of
           the
           operations
           of
           this
           Mood
           because
           of
           the
           ingrediencie
           of
           Tangents
           dependeth
           on
           Sbaprotca
           ,
           as
           is
           perceivable
           by
           the
           sixth
           determinater
           of
           its
           Directory
           Pubkutethepsaler
           .
        
         
           The
           fifth
           Figure
           of
           the
           Orthogonosphericals
           is
           Achave
           ,
           which
           containeth
           all
           those
           Problems
           ,
           wherein
           the
           
             Angles
             being
             given
             ,
             the
             subtendent
             or
             an
             Ambient
             is
             desired
             ,
          
           and
           hath
           two
           Moods
           Alamun
           ,
           and
           Amaner
           .
        
         
           Alamun
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           the
           
             Angles
             being
             proposed
             ,
             the
             Hypotenusa
             is
             required
             ,
             and
             by
             its
             Resolver
             Tag
             —
             Torb
             —
             Ma
             ☞
             Nur
             ,
          
           or
           more
           compendiously
           ,
           
             Torb
             —
             Mag
             —
             Ma
             ☞
             Nur
             ,
          
           sheweth
           ,
           that
           the
           summe
           of
           the
           Co-tangents
           ,
           not
           exceeding
           the
           places
           of
           the
           Radius
           ,
           is
           the
           Sine
           complement
           of
           the
           subtendent
           required
           ;
           for
           it
           is
           ,
           As
           
             the
             Tangent
             of
             one
             of
             the
             Angles
             ,
             to
             the
             Radius
             :
             so
             the
             Tangent
             complement
             of
             the
             other
             Angle
             ,
             to
             the
             Sine
             complement
             of
             the
             Hypotenusa
             demanded
             :
          
           or
           ,
           As
           
             the
             totall
             Sine
             ,
             to
             the
             Tangent
             complement
             of
             one
             of
             the
             Angles
             :
             so
             the
             Tangent
             complement
             of
             the
             other
             Angle
             ,
             to
             the
             Sine
             complement
             of
             the
             subtendent
             we
             seek
             for
             .
          
           The
           running
           of
           this
           Mood
           upon
           Tangents
           ,
           notifieth
           its
           dependance
           on
           Sbaprotca
           ,
           as
           is
           evident
           by
           the
           seventh
           determinater
           of
           the
           Directory
           thereof
           .
        
         
         
           The
           second
           Mood
           of
           this
           Figure
           is
           Amaner
           ,
           which
           comprehendeth
           all
           those
           Orthogonosphericall
           Problems
           ,
           wherein
           
             the
             Angles
             being
             given
             ,
             an
             Ambient
             is
             demanded
             ,
          
           and
           by
           its
           Resolver
           ,
           
             Say
             —
             Nag
             —
             Tω
             ☞
             Nyr
             ,
          
           or
           more
           perspicuously
           ,
           
             Tω
             —
             Noy
             —
             Ray
             ☞
             Nyr
             ,
          
           sheweth
           ,
           that
           the
           summe
           of
           the
           Logarithms
           of
           the
           Antisine
           of
           the
           Angle
           opposite
           to
           the
           side
           required
           ,
           and
           the
           Arithmeticall
           complement
           of
           the
           Sine
           of
           the
           Angle
           ,
           adjoyning
           the
           said
           side
           ,
           which
           we
           call
           its
           Secant
           complement
           ,
           with
           the
           usuall
           presection
           ,
           is
           equall
           to
           the
           Sine
           complement
           of
           the
           same
           side
           demanded
           ;
           for
           it
           is
           ,
           As
           
             the
             Sine
             of
             the
             Angle
             adjoyning
             the
             side
             required
             ,
             to
             the
             Antisine
             of
             the
             other
             Angle
             :
             so
             the
             totall
             sine
             ,
             to
             the
             Antisine
             of
             the
             side
             demanded
             :
          
           or
           ,
           As
           
             the
             Radius
             ,
             to
             the
             Antisine
             of
             the
             Angle
             opposite
             to
             the
             demanded
             side
             :
             so
             the
             Antisecant
             of
             the
             Angle
             conterminat
             with
             that
             side
             ,
             to
             the
             Antisine
             of
             the
             side
             required
             :
          
           and
           because
           of
           the
           Analogy
           betwixt
           the
           Antisines
           ,
           and
           Secants
           :
           and
           likewise
           betwixt
           the
           Antisecants
           ,
           and
           Sines
           ,
           we
           may
           expresse
           it
           ,
           
             To
             —
             Say
             —
             La
             ☞
             Lyr
          
           ;
           that
           is
           ,
           As
           
             the
             Radius
             ,
             to
             the
             Sine
             of
             the
             Angle
             insident
             on
             the
             required
             side
             :
             so
             the
             Secant
             of
             the
             other
             given
             Angle
             ,
             to
             the
             Secant
             of
             the
             side
             that
             is
             demanded
             .
          
           Here
           the
           Angulary
           intermixture
           of
           proportions
           giveth
           us
           to
           understand
           ,
           that
           this
           Mood
           dependeth
           on
           Seproso
           ,
           as
           is
           manifested
           by
           the
           last
           characteristick
           of
           Uchedezexam
           the
           Directory
           of
           this
           Axiom
           .
        
         
           The
           sixth
           and
           last
           Figure
           is
           Escheva
           ,
           which
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           the
           two
           containing
           sides
           being
           given
           ,
           either
           the
           subtendent
           ,
           or
           an
           Angle
           is
           demanded
           :
           it
           hath
           two
           
             Moods
             ,
             Enerul
          
           and
           Erelam
           .
        
         
           The
           first
           Mood
           thereof
           Enerul
           ,
           containeth
           all
           such
           Problems
           as
           having
           the
           Ambients
           given
           ,
           require
           the
           subtendent
           ,
           and
           by
           its
           Resolver
           ,
           
             Ton
             —
             Neg
             —
             Ne
             ☞
             Nur
             ,
          
           sheweth
           ,
           that
           the
           
             summe
             of
             the
             Logarithms
             of
             the
             Cosines
             of
             the
             two
             Legs
             unradiated
             ,
             is
             the
             Logarithm
             of
             the
             Co-sine
             of
             the
             subtendent
          
           ;
           for
           it
           is
           ,
           As
           
             the
             totall
             Sine
             ,
             to
             the
             Co-sine
             of
             one
             of
             the
             Ambients
             :
             so
             the
             Co-sine
             of
             the
             other
             including
             Leg
             given
             ,
             to
             the
             Co-sine
             of
             the
             required
             subtendent
          
           ;
           and
           because
           of
           the
           Co-sinal
           ,
           and
           Secantine
           proportion
           ,
           we
           may
           safely
           say
           ,
           
             To
             —
             Leg
             —
             Le
             ☞
             Lur.
          
           That
           is
           ,
           As
           
             the
             Radius
             to
             the
             Secant
             of
             one
             shanke
             or
             Leg
             :
             so
             the
             secant
             of
             the
             other
             shanke
             or
             Leg
             ,
             to
             the
             secant
             of
             the
             Hypotenusa
             demanded
             .
          
           The
           coursing
           thus
           upon
           Sines
           ,
           and
           their
           proportionals
           evidenceth
           that
           this
           Mood
           dependeth
           on
           Suprosca
           ,
           
           the
           first
           of
           the
           Sphericall
           Axioms
           ,
           which
           is
           pointed
           at
           by
           the
           third
           and
           last
           characteristick
           of
           Vphugen
           the
           directorie
           thereof
           .
        
         
           The
           second
           Mood
           of
           the
           last
           figure
           ,
           and
           consequently
           the
           last
           Mood
           of
           al
           the
           Orthogonosphericals
           ,
           is
           Erelam
           ,
           which
           comprehendeth
           all
           those
           orthogonosphericall
           problems
           ,
           wherin
           the
           two
           containing
           sides
           being
           proposed
           ,
           an
           Angle
           is
           demanded
           ,
           and
           by
           its
           Resolver
           ,
           
             Sei
             —
             Teg
             —
             Torb
             ☞
             Tir
             ,
          
           or
           by
           primifying
           the
           Radius
           ,
           
             Torb
             —
             Tepi-Rexi
             ☞
             Tir
          
           ,
           giveth
           us
           to
           understand
           ,
           that
           the
           cutting
           off
           the
           Radius
           from
           the
           summe
           of
           the
           Tangent
           of
           the
           side
           opposite
           to
           the
           Angle
           demanded
           ,
           and
           the
           cosecant
           of
           the
           side
           conterminat
           with
           the
           said
           Angle
           ,
           residuats
           the
           touch-line
           of
           the
           Angle
           in
           question
           ;
           for
           it
           is
           ,
           
             As
             the
             sine
             of
             the
             side
             adjoyning
             the
             Angle
             required
             ,
             to
             the
             tangent
             of
             the
             other
             given
             side
             :
             so
             the
             Radius
             to
             the
             tangent
             of
             the
             Angle
             demanded
             :
          
           or
           ,
           
             As
             the
             totall
             sine
             to
             the
             Tangent
             of
             the
             Ambient
             opposite
             to
             the
             angle
             sought
             :
             so
             the
             Antisecant
             of
             the
             Leg
             adjacent
             to
             the
             said
             asked
             Angle
             ,
             to
             the
             Tangent
             or
             toucher
             thereof
             :
          
           and
           because
           Sines
           have
           the
           same
           proportion
           to
           cosecants
           ,
           which
           Tangents
           have
           to
           Cotangents
           ,
           we
           may
           say
           ,
           
             To
             —
             Sei
             —
             me
             ☞
             mir
             ,
          
           that
           is
           ,
           
             As
             the
             Radius
             to
             the
             sine
             of
             the
             side
             conterminat
             with
             the
             angle
             required
             :
             so
             the
             Cotangent
             of
             the
             other
             Leg
             ,
             to
             the
             Cotangent
             of
             the
             Angle
             searched
             after
             :
          
           or
           yet
           more
           profoundly
           by
           an
           Alternat
           proportion
           ,
           changing
           the
           relation
           of
           the
           fourth
           proportionall
           ,
           although
           the
           same
           formerly
           required
           Angle
           ,
           thus
           ,
           
             To
             —
             Rei
             —
             me
             ☞
             mor
             ,
          
           that
           is
           ,
           
             As
             the
             Radius
             to
             the
             Antisecant
             of
             the
             side
             adjacent
             to
             the
             Angle
             sought
             for
             ,
             so
             the
             Antitangent
             of
             the
             other
             side
             ,
             to
             the
             Antitangent
             of
             that
             sides
             opposit
             Angle
             ,
          
           which
           is
           the
           Angle
           demanded
           .
           The
           reason
           hereof
           is
           grounded
           on
           Sbaprotca
           ;
           for
           the
           Tangentine
           proportion
           of
           the
           terms
           of
           this
           Mood
           specifieth
           its
           dependance
           on
           the
           second
           Axiom
           ,
           which
           is
           showen
           unto
           us
           by
           the
           eight
           and
           last
           characteristick
           of
           its
           directorie
           Pubkutethepsaler
           .
        
         
           Here
           endeth
           the
           doctrine
           of
           the
           right-Angled
           sphericalls
           ,
           the
           whole
           diatyposis
           wherof
           is
           in
           the
           Equisolea
           or
           hippocrepidian
           diagram
           ,
           whose
           most
           intricate
           amfractuosities
           ,
           renvoys
           ,
           various
           mixture
           of
           analogies
           ,
           and
           perturbat
           situation
           of
           proportionall
           termes
           ,
           cannot
           choose
           but
           be
           pervious
           to
           the
           understanding
           of
           any
           judicious
           Reader
           that
           hath
           perused
           this
           Comment
           aright
           .
           And
           therefore
           let
           him
           give
           me
           leave
           (
           if
           he
           think
           fit
           )
           for
           his
           memorie
           sake
           ,
           to
           remit
           him
           to
           it
           ,
           before
           he
           proceed
           any
           further
           .
        
      
       
         
         
           The
           Loxogonosphericall
           Triangles
           ,
           whether
           Amblygonosphericall
           or
           Oxygonosphericall
           ,
           are
           either
           Monurgetick
           or
           Disergetick
           .
        
         
           THe
           Monurgetick
           have
           two
           figures
           ,
           Datamista
           and
           Datapura
           .
        
         
           Datamista
           is
           of
           all
           those
           Loxogonospherical
           Monurgetick
           problems
           ,
           wherein
           the
           Angles
           and
           sides
           being
           intermixedly
           given
           ,
           (
           and
           therefore
           one
           of
           them
           being
           alwaies
           of
           another
           kind
           from
           the
           other
           two
           )
           either
           an
           Angle
           ,
           or
           a
           side
           is
           demanded
           :
           it
           hath
           two
           Moods
           ,
           Lamaneprep
           ,
           and
           Menerolo
           .
        
         
           The
           first
           Mood
           Lamaneprep
           ,
           comprehendeth
           all
           those
           Loxogonosphericall
           problems
           ,
           wherein
           two
           angles
           being
           given
           ,
           and
           an
           
             opposit
             side
          
           ,
           another
           
             opposit
             side
          
           is
           demanded
           ,
           and
           by
           its
           Resolver
           ,
           
             Sapeg
             —
             Se
             —
             Sapy
             —
             ☞
             Syr
             ,
          
           sheweth
           ,
           that
           if
           to
           the
           Logarithms
           of
           the
           sine
           of
           the
           side
           given
           ,
           and
           sine
           of
           the
           Angle
           opposit
           ,
           to
           the
           side
           required
           ,
           we
           joyne
           the
           Arithmeticall
           complement
           of
           the
           sine
           of
           the
           Angle
           opposit
           ,
           to
           the
           proposed
           side
           (
           which
           is
           the
           refined
           Antisecant
           )
           we
           will
           thereby
           attain
           to
           the
           knowledge
           of
           the
           sine
           of
           the
           side
           demanded
           .
           The
           reason
           of
           this
           is
           grounded
           on
           the
           third
           Axiom
           ,
           Seprosa
           ,
           as
           you
           may
           perceive
           by
           the
           first
           syllable
           of
           the
           Obliquangularie
           directory
           ,
           Lame
           .
        
         
           The
           second
           Mood
           of
           this
           figure
           is
           Menerolo
           ,
           wich
           comprehendeth
           all
           those
           Amblygonosphericall
           problems
           ,
           wherein
           two
           sides
           being
           given
           with
           an
           opposit
           angle
           ,
           another
           opposit
           angle
           is
           demanded
           ,
           and
           by
           its
           Resolver
           
             Sepag
             —
             Sa
             —
             Sepi
             ☞
             Sir
             ,
          
           sheweth
           ,
           that
           if
           to
           the
           summe
           of
           the
           Logarithms
           of
           the
           sine
           of
           the
           given
           angle
           ,
           and
           sine
           of
           the
           side
           opposit
           to
           the
           angle
           required
           ,
           we
           joyne
           the
           Arithmeticall
           complement
           of
           the
           sine
           of
           the
           side
           opposit
           to
           the
           given
           angle
           (
           which
           is
           the
           refined
           Cosecant
           of
           the
           said
           angle
           )
           it
           will
           afford
           us
           the
           sine
           of
           the
           angle
           required
           .
           The
           reason
           of
           this
           operation
           is
           grounded
           on
           the
           third
           Axiom
           of
           Sphericalls
           ,
           a
           progresse
           in
           sines
           shewing
           clearely
           ,
           how
           that
           both
           this
           ,
           and
           the
           former
           ,
           doe
           totally
           depend
           on
           the
           Axiom
           of
           Seprosa
           ,
           as
           is
           evident
           by
           the
           second
           syllable
           of
           its
           directorie
           ,
           Lame
           .
        
         
           The
           second
           figure
           of
           the
           Monurgetick
           Loxogonosphericalls
           
           treateth
           of
           all
           those
           questions
           ,
           wherein
           the
           Datas
           being
           either
           sides
           alone
           ,
           or
           Angles
           alone
           ,
           an
           Angle
           or
           a
           side
           is
           demanded
           .
           This
           Figure
           of
           Datapura
           is
           divided
           into
           two
           Moods
           ,
           viz.
           Nerelema
           ,
           and
           Ralamane
           ,
           which
           are
           of
           such
           affinity
           ,
           that
           upon
           one
           and
           the
           same
           Theorem
           dependeth
           the
           Analogy
           that
           resolveth
           both
           .
        
         
           The
           first
           Mood
           thereof
           ,
           Nerelema
           ,
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           the
           three
           sides
           being
           given
           ,
           an
           Angle
           is
           demanded
           ,
           and
           is
           the
           third
           of
           the
           Monurgeticks
           ,
           as
           by
           its
           Characteristick
           the
           third
           Liquid
           is
           perceivable
           .
        
         
           The
           curteous
           Reader
           may
           be
           pleased
           to
           take
           notice
           ,
           that
           in
           both
           the
           Moods
           of
           the
           Datapurall
           Figure
           ,
           I
           am
           in
           some
           measure
           necessitated
           for
           the
           better
           order
           sake
           ,
           to
           couch
           two
           precepts
           ,
           or
           documents
           ,
           for
           the
           Faciendas
           thereof
           ,
           and
           to
           premise
           that
           one
           concerning
           the
           three
           Legs
           given
           ,
           before
           I
           make
           any
           mention
           of
           the
           maine
           Resolver
           ,
           whereupon
           both
           the
           foresaid
           Moods
           are
           founded
           ,
           to
           which
           Resolver
           ,
           because
           of
           both
           their
           dependences
           on
           it
           ,
           I
           have
           allowed
           here
           in
           the
           Glosse
           ,
           the
           same
           middle
           place
           ,
           which
           it
           possesseth
           in
           the
           Table
           of
           my
           Trissotetras
           .
        
         
           The
           precept
           of
           Nerelema
           is
           Halbasalzes
           *
           Ad*
           Ab*
           
             Sadsabreregalsbis
             Ir
          
           :
           that
           is
           to
           say
           ,
           for
           the
           finding
           out
           of
           an
           Angle
           when
           the
           three
           Legs
           are
           given
           ,
           as
           soone
           as
           we
           have
           constituted
           the
           
             sustentative
             Leg
          
           of
           that
           Angle
           a
           Base
           ,
           the
           halfe
           thereof
           must
           be
           taken
           ,
           and
           to
           that
           halfe
           we
           must
           
             adde
             halfe
          
           the
           difference
           of
           the
           other
           two
           Legs
           ,
           and
           likewise
           from
           that
           
             halfe
             subtract
          
           the
           
             half
             difference
          
           of
           the
           foresaid
           two
           Legs
           ,
           then
           the
           summe
           and
           the
           residue
           being
           two
           Arches
           ,
           we
           must
           ,
           to
           the
           Logarithms
           of
           the
           Sine
           of
           the
           summe
           ,
           and
           Sine
           of
           the
           Remainer
           ,
           joyne
           the
           Logarithms
           of
           the
           Arithmeticall
           complements
           of
           the
           Sines
           of
           the
           sides
           ,
           which
           are
           the
           refined
           Antisecants
           of
           the
           said
           Legs
           ,
           and
           halfe
           that
           summe
           will
           afford
           ,
           us
           the
           Logarithm
           of
           the
           Sine
           of
           an
           Arch
           ,
           which
           doubled
           ,
           is
           the
           verticall
           Angle
           ,
           we
           demand
           ;
           for
           out
           of
           its
           Resolver
           ,
           
             Parses
             —
             Powto
             —
             Parsadsab
             ☞
             Powsalvertir
             ,
          
           is
           the
           Analogy
           of
           the
           former
           work
           made
           cleare
           ,
           the
           Theorem
           being
           ,
           
             As
             the
             Oblong
             or
             Parallelogram
             contained
             under
             the
             Sines
             of
             the
             Legs
             ,
             to
             the
             square
             ,
             power
             or
             quadrat
             of
             the
             totall
             Sine
             :
             so
             the
             Rectangle
             ,
             or
             Oblong
             made
             of
             the
             right
             Sines
             of
             the
          
           sum
           ,
           and
           difference
           
             of
             the
             halfe
             Base
             ,
             and
             difference
             of
             the
             Legs
             ,
             to
             the
             square
             of
             the
             right
             Sine
             of
             halfe
             the
             verticall
             Angle
             .
          
        
         
           The
           reason
           hereof
           will
           be
           manifest
           enough
           to
           the
           industrious
           
           Reader
           ,
           if
           when
           by
           a
           peculiar
           Diagram
           ,
           of
           whose
           equiangular
           Triangles
           the
           foresaid
           Sines
           and
           differences
           are
           made
           the
           constitutive
           sides
           ,
           he
           hath
           evinced
           their
           Analogy
           to
           one
           another
           ,
           he
           be
           then
           pleased
           to
           perpend
           ,
           how
           ,
           in
           two
           rowes
           of
           proportionall
           numbers
           ,
           the
           products
           arising
           of
           the
           homologall
           roots
           ,
           are
           in
           the
           same
           proportion
           amongst
           themselves
           ,
           that
           the
           said
           roots
           towards
           one
           another
           are
           ;
           wherewithall
           if
           he
           doe
           consider
           ,
           how
           the
           halfs
           must
           needs
           keep
           the
           same
           proportion
           that
           their
           wholes
           ;
           and
           then
           ,
           in
           the
           work
           it selfe
           of
           collationing
           severall
           orders
           of
           proportionall
           termes
           ,
           both
           single
           and
           compound
           ,
           be
           carefull
           to
           dash
           out
           a
           divider
           against
           a
           multiplyer
           ,
           and
           afterwards
           proceed
           in
           all
           the
           rest
           ,
           according
           to
           the
           ordinary
           rules
           of
           Aequation
           ,
           and
           Analogy
           ,
           he
           cannot
           choose
           but
           extricat
           himselfe
           with
           ease
           forth
           of
           all
           the
           windings
           of
           this
           elaboured
           proposition
           .
        
         
           Upon
           this
           Theorem
           (
           as
           I
           have
           told
           you
           )
           dependeth
           likewise
           the
           Document
           for
           the
           faciendum
           of
           Ralamane
           ,
           which
           is
           the
           second
           Mood
           of
           Datapura
           ,
           and
           the
           last
           of
           the
           Monurgetick
           Loxogonosphericals
           ,
           as
           is
           pointed
           at
           by
           Nera
           the
           Directory
           therof
           .
           This
           
             Mood
             Ralamane
          
           comprehendeth
           all
           those
           Loxogonosphericall
           Problems
           ,
           wherein
           the
           three
           Angles
           being
           given
           ,
           a
           side
           is
           demanded
           .
           And
           by
           its
           Resolver
           ,
           
             Parses
             —
             Powto
             —
             Parsadsab
             ☞
             Powsalvertir
             ,
          
           according
           to
           the
           peculiar
           precept
           of
           this
           
             Mood
             Kourbfasines
             (
             Ereled
             )
             Koufbraxypopyx
             ,
          
           sheweth
           ,
           that
           if
           we
           take
           the
           complement
           to
           a
           semicircle
           of
           the
           Angle
           opposite
           to
           the
           side
           required
           ,
           which
           for
           distinction
           sake
           we
           doe
           here
           call
           the
           Base
           ;
           and
           frame
           ,
           of
           the
           foresaid
           complement
           to
           a
           semicircle
           ,
           a
           second
           Base
           for
           the
           fabrick
           of
           a
           new
           Triangle
           ,
           whose
           other
           two
           sides
           have
           the
           graduall
           measure
           of
           the
           former
           Triangles
           other
           two
           Angles
           :
           (
           and
           so
           the
           three
           Angles
           being
           converted
           into
           sides
           )
           the
           complement
           to
           a
           Semicircle
           of
           the
           new
           Verticall
           ,
           or
           Angle
           opposite
           to
           the
           new
           Base
           ,
           will
           be
           the
           measure
           of
           the
           true
           Base
           or
           Leg
           required
           ,
           and
           the
           Angle
           insident
           on
           the
           right
           end
           of
           the
           new
           Base
           in
           the
           second
           Triangle
           ,
           falleth
           to
           be
           the
           side
           conterminall
           with
           the
           left
           end
           of
           the
           true
           Base
           in
           the
           first
           Triangle
           ,
           and
           the
           Angle
           adjoyning
           the
           left
           end
           of
           the
           false
           Base
           in
           the
           second
           Triangle
           ,
           becomes
           the
           side
           adjacent
           to
           the
           right
           end
           of
           the
           old
           Base
           in
           the
           first
           Triangle
           .
           So
           that
           thus
           by
           the
           Angles
           all
           andeach
           of
           the
           sides
           are
           found
           out
           ,
           all
           which
           works
           are
           to
           be
           performed
           by
           the
           preceding
           
           Mood
           ,
           upon
           the
           Theorem
           ,
           whereof
           the
           reason
           of
           all
           these
           operations
           doth
           depend
           .
        
      
       
         
           The
           Disergetick
           Loxogonosphericals
           are
           grounded
           on
           foure
           Axioms
           ,
           viz.
           
        
         
           1.
           
           NAbadprosver
           .
           2.
           
           
             Naverprortes
             ,
             Siubprortab
          
           ,
           and
           Niubprodnesver
           :
           the
           foure
           Directories
           whereof
           ,
           each
           in
           order
           to
           its
           owne
           Axiome
           ,
           are
           
             Alama
             ,
             Allera
             ,
             Ammena
          
           ,
           and
           Ennerra
           .
        
         
           The
           first
           Axiome
           is
           ,
           Nabadprosver
           ,
           that
           is
           ,
           In
           Obliquangular
           Sphericals
           ,
           if
           a
           Perpendicular
           be
           demitted
           from
           the
           verticall
           Angle
           to
           the
           opposite
           side
           ,
           continued
           if
           need
           be
           ,
           
             The
             Sines
             complements
             of
             the
             Angles
             at
             the
             Base
             ,
             will
             be
             directly
             proportionall
             to
             the
             Sines
             of
             the
             verticall
             Angles
             ,
          
           and
           contrary
           :
           the
           reason
           hereof
           is
           inferred
           out
           of
           the
           proportion
           ,
           which
           the
           Sines
           of
           Angles
           ,
           substerned
           by
           Perpendiculars
           ,
           have
           to
           the
           Sines
           of
           the
           said
           perpendiculars
           ,
           so
           that
           they
           belong
           to
           the
           Arches
           of
           great
           Circles
           ,
           concurring
           in
           the
           same
           point
           ,
           and
           that
           from
           some
           point
           of
           the
           one
           ,
           they
           be
           let
           fall
           on
           the
           other
           Arches
           ▪
           which
           proportion
           of
           the
           Sines
           of
           the
           said
           Perpendiculars
           ,
           to
           the
           Sines
           of
           the
           Angles
           subtended
           by
           them
           ,
           stoweth
           immediatly
           from
           the
           
             proportion
             ,
             which
          
           (
           in
           severall
           Orthogonosphericals
           ,
           having
           the
           same
           acute
           Angle
           at
           the
           Base
           )
           
             is
             betwixt
             the
             Sines
             of
             the
             Hypotenusas
             ,
             and
             the
             Sines
             of
             the
             perpendiculars
          
           ;
           the
           demonstration
           whereof
           is
           plainly
           set
           downe
           in
           my
           Glosse
           on
           Suprosca
           ,
           the
           first
           generall
           Axiome
           of
           the
           Sphericals
           ,
           of
           which
           this
           Axiome
           of
           Nabadprosver
           is
           a
           consectary
           .
        
         
           The
           Directory
           of
           this
           Axiome
           is
           Alama
           ,
           which
           sheweth
           ,
           that
           the
           Moods
           of
           Alamebna
           and
           Amanepra
           are
           grounded
           on
           it
           .
        
         
           The
           second
           Disergetick
           Axiome
           is
           Naverprortes
           ,
           that
           is
           to
           say
           ,
           
             the
             Sines
             complements
             of
             the
             verticall
             Angles
             ,
             in
             obliquangular
             Triangles
          
           (
           a
           Perpendicular
           being
           let
           fall
           from
           the
           double
           verticall
           on
           the
           opposite
           side
           )
           
             are
             reciprocally
             proportionall
             to
             the
             Tangents
             of
             the
             sides
             :
          
           the
           reason
           hereof
           proceedeth
           from
           Sbaprotca
           ,
           the
           second
           generall
           Axiome
           of
           the
           Sphericals
           ;
           according
           to
           which
           ,
           if
           we
           doe
           but
           regulate
           ,
           after
           the
           customary
           Analogicall
           manner
           ,
           two
           quaternaries
           
           of
           proportionals
           of
           the
           former
           Sines
           complements
           ,
           and
           Tangents
           proposed
           ,
           we
           will
           find
           by
           the
           extremes
           alone
           (
           excluding
           all
           the
           intermediate
           termes
           )
           that
           
             the
             Sines
             complements
             of
             the
             verticall
             Angles
             (
             both
             forwardly
             ,
             and
             inversedly
             )
             are
             reciprocally
             proportioned
             to
             the
             Tangents
             of
             the
             sides
             ,
          
           and
           contrariwise
           from
           the
           Tangents
           ,
           to
           the
           Sines
           .
           The
           Directory
           of
           this
           Axiome
           is
           Allera
           ,
           which
           evidenceth
           ,
           that
           the
           Moods
           of
           Allamebne
           ,
           and
           Erelomab
           depend
           upon
           it
           .
        
         
           The
           third
           Disergetick
           Axiome
           is
           Sinbprortab
           ,
           that
           is
           to
           say
           ,
           that
           
             in
             Obliquangular
             Sphericals
          
           (
           if
           a
           perpendicular
           be
           drawne
           from
           the
           verticall
           Angle
           unto
           the
           opposite
           side
           ,
           continued
           if
           need
           be
           )
           
             the
             Sines
             of
             the
             segments
             of
             the
             Base
             ,
             are
             reciprocally
             proportionall
             to
             the
             Tangents
             of
             the
             Angles
             conterminate
             at
             the
             Base
             ,
             and
             contrary
             :
          
           the
           proofe
           of
           this
           ,
           as
           well
           as
           that
           of
           the
           former
           confectary
           dependeth
           on
           Sbaprotca
           ,
           the
           second
           generall
           Axiome
           of
           the
           Sphericals
           ,
           according
           to
           which
           ,
           if
           we
           so
           Diagrammatise
           an
           Ambly
           gonosphericall
           Triangle
           ,
           by
           Quadranting
           the
           Perpendicular
           ,
           and
           all
           the
           sides
           ,
           and
           describing
           from
           the
           Basangulary
           points
           two
           Quadrantall
           Arches
           ,
           till
           we
           hit
           upon
           two
           rowes
           of
           proportionall
           Sines
           of
           Bases
           to
           Tangents
           of
           Perpendiculars
           ,
           then
           shall
           we
           be
           sure
           (
           if
           we
           exclude
           the
           intermediate
           termes
           )
           to
           fall
           upon
           a
           reciprocall
           Analogy
           of
           Sines
           ,
           and
           Tangents
           ,
           which
           alternatly
           changed
           ,
           will
           afford
           the
           reciprocall
           proportion
           of
           the
           Sines
           of
           the
           Segments
           of
           the
           Base
           ,
           to
           the
           Tangents
           of
           the
           Angles
           conterminat
           thereat
           ,
           the
           thing
           required
           .
        
         
           The
           Directory
           of
           this
           Axiome
           is
           Ammena
           ,
           which
           certifieth
           that
           Ammanepreb
           and
           Enerablo
           are
           founded
           thereon
           .
        
         
           The
           fourth
           and
           last
           Disergetick
           Axiome
           is
           Niubprodnesver
           ,
           that
           is
           to
           say
           ,
           that
           in
           all
           
             Loxogonosphericals
             (
             where
             the
             Cathetus
             is
             regularly
             demitted
             )
             the
             Sines
             complements
             of
             the
             Segments
             of
             the
             Base
             ,
             are
             directly
             proportionall
             to
             the
             Sines
             complements
             of
             the
             sides
             of
             the
             verticall
             Angles
             ,
             and
             contrary
             .
          
           The
           reason
           hereof
           is
           made
           manifest
           ,
           by
           the
           proportion
           that
           is
           betwixt
           the
           Sines
           of
           Angles
           ,
           subtended
           by
           Perpendiculars
           and
           the
           Sines
           of
           these
           Perpendiculars
           ;
           out
           of
           which
           we
           collation
           severall
           proportions
           ,
           till
           ,
           both
           forwardly
           and
           inversedly
           we
           pitch
           at
           last
           upon
           the
           direct
           proportion
           required
           .
        
         
           The
           Directory
           of
           this
           Axiome
           is
           Ennerra
           ,
           which
           declareth
           that
           Ennerable
           ,
           and
           Errelome
           are
           its
           dependents
           .
        
      
       
         
         
           Of
           the
           Disergetick
           Loxogonosphericals
           there
           be
           in
           all
           foure
           Figures
           ;
           two
           Angulary
           ,
           and
           two
           Laterall
           .
        
         
           THe
           two
           Angulary
           are
           Ahalebmane
           and
           Ahamepnare
           :
           The
           two
           laterall
           are
           Ehenabrole
           and
           Eheromabne
           .
        
         
           The
           first
           Angulary
           Disergetick
           Loxogonosphericall
           Figure
           ,
           Ahalebmane
           ,
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           two
           Angles
           being
           given
           with
           a
           side
           betweene
           ,
           either
           the
           third
           Angle
           ,
           or
           an
           opposite
           side
           is
           demanded
           ;
           and
           accordingly
           hath
           two
           Moods
           ,
           the
           first
           whereof
           is
           Alamebna
           ,
           and
           the
           second
           Allamebne
           .
        
         
           Alamebna
           concerneth
           all
           those
           Loxogonosphericall
           Disergetick
           Problems
           ,
           wherein
           two
           Angles
           being
           proposed
           ,
           with
           an
           interjacent
           side
           ,
           the
           third
           Angle
           is
           required
           ;
           which
           Angle
           ,
           according
           to
           the
           severall
           Cases
           of
           this
           Mood
           ,
           is
           alwayes
           one
           of
           the
           Angles
           at
           the
           Base
           ,
           that
           is
           to
           say
           (
           in
           the
           termes
           of
           my
           
             Trissotetr
             as
          
           )
           a
           prime
           ,
           or
           next
           opposite
           ,
           or
           at
           least
           one
           of
           the
           co-opposites
           ,
           to
           the
           Perpendicular
           to
           be
           demitted
           .
           And
           therefore
           ,
           conforme
           to
           the
           nature
           of
           the
           Case
           of
           the
           Datoquaere
           in
           hand
           ,
           and
           that
           it
           may
           the
           more
           conveniently
           fall
           within
           the
           compasse
           of
           the
           Axiome
           of
           Nabadprosver
           ,
           an
           Angle
           by
           the
           first
           operation
           of
           this
           Disergetick
           is
           to
           be
           found
           out
           ,
           which
           must
           either
           be
           a
           
             double
             verticall
             ,
             a
             verticall
          
           in
           the
           little
           rectangle
           ,
           or
           a
           verticall
           ,
           or
           co-verticall
           (
           as
           sometimes
           I
           call
           it
           )
           in
           one
           of
           the
           correctangles
           .
        
         
           Thus
           much
           I
           have
           thought
           fit
           to
           premise
           of
           the
           praenoscendum
           of
           this
           Mood
           ,
           before
           I
           come
           to
           its
           Cathetothesis
           ;
           because
           ,
           in
           my
           Trissotetrall
           Table
           ,
           to
           avoid
           the
           confusion
           of
           homogeneall
           termes
           (
           though
           the
           order
           of
           doctrine
           would
           seeme
           to
           require
           another
           Method
           )
           the
           first
           and
           prime
           Orthogonosphericall
           work
           is
           totally
           unfolded
           ,
           before
           I
           speak
           any
           thing
           of
           the
           variety
           of
           the
           Perpendiculars
           demission
           ,
           to
           which
           ,
           owing
           its
           rectangularity
           ,
           it
           thereby
           obtaineth
           an
           infallible
           progresse
           to
           the
           quaesitum
           :
           but
           ,
           seeing
           in
           the
           Glosse
           I
           am
           not
           to
           astrict
           my self
           to
           so
           little
           bounds
           ,
           as
           in
           my
           Table
           ,
           I
           will
           observe
           the
           order
           that
           is
           most
           expedient
           ;
           and
           ,
           before
           the
           resolution
           of
           any
           operation
           in
           this
           Mood
           ,
           deduce
           the
           diversity
           of
           the
           Perpendiculars
           prosiliencie
           in
           the
           severall
           Cases
           thereof
           .
        
         
         
           Let
           the
           Reader
           then
           be
           pleased
           to
           consider
           ,
           that
           the
           generall
           Maxim
           for
           the
           Cathetothesis
           of
           this
           Mood
           is
           Cafregpiq
           ,
           the
           meaning
           whereof
           is
           ,
           that
           ,
           whether
           the
           side
           whereon
           the
           Perpendicular
           is
           demitted
           be
           increased
           or
           not
           ,
           that
           is
           to
           say
           ,
           whether
           the
           Perpendicular
           fall
           outwardly
           ,
           or
           inwardly
           ,
           it
           must
           fall
           from
           the
           extremity
           of
           the
           
             given
             side
          
           ,
           and
           opposite
           to
           the
           
             Angle
             required
          
           :
           however
           it
           is
           to
           be
           remarked
           ,
           that
           in
           this
           Mood
           ,
           whatever
           be
           the
           affection
           of
           the
           Angles
           (
           unlesse
           they
           be
           all
           three
           alike
           )
           the
           Perpendicular
           may
           fall
           out
           wardly
           .
        
         
           The
           generall
           maxim
           for
           the
           Cathetothesis
           of
           this
           Mood
           ,
           as
           well
           as
           for
           that
           of
           all
           the
           rest
           ,
           is
           divided
           into
           foure
           Tenets
           ,
           according
           to
           the
           number
           of
           the
           Cases
           of
           every
           Mood
           .
        
         
           Here
           must
           I
           admonish
           the
           Reader
           ,
           that
           he
           startle
           not
           at
           the
           mentioning
           of
           foure
           especiall
           
             Cathetothetick
             Tenets
          
           ,
           and
           foure
           severall
           Cases
           belonging
           to
           each
           
             Disergetick
             Mood
          
           ,
           seeing
           ,
           to
           the
           most
           observant
           eye
           ,
           there
           be
           but
           three
           of
           either
           perceptible
           in
           my
           
             Trissotetr
             as
          
           ;
           for
           ,
           the
           fourth
           both
           Tenet
           and
           Case
           being
           the
           same
           by
           way
           of
           expression
           in
           all
           the
           Moods
           ,
           and
           being
           fully
           resolved
           by
           the
           third
           Case
           of
           every
           Mood
           ,
           it
           shall
           suffice
           to
           speak
           thereof
           here
           once
           for
           all
           :
           The
           Tenet
           of
           this
           common
           Case
           is
           Simomatin
           ,
           that
           is
           to
           say
           ,
           when
           all
           the
           three
           Angles
           in
           any
           of
           those
           Disergeticks
           are
           of
           the
           same
           affection
           ,
           either
           all
           acute
           ,
           or
           all
           oblique
           ,
           the
           Perpendicular
           falleth
           inwardly
           ,
           whether
           the
           double
           verticall
           be
           an
           Angle
           given
           ,
           an
           Angle
           demanded
           ,
           or
           neither
           .
           Yet
           here
           it
           is
           to
           be
           considered
           ,
           that
           seeing
           Triangles
           may
           be
           either
           calculated
           by
           their
           reall
           and
           naturall
           ,
           or
           by
           their
           circular
           parts
           ,
           or
           by
           both
           together
           ,
           and
           that
           for
           the
           more
           facility
           we
           oftentimes
           ,
           instead
           of
           the
           proposed
           Triangle
           ,
           resolve
           its
           opposite
           ;
           it
           is
           not
           the
           reall
           and
           given
           Triangle
           ,
           that
           in
           this
           case
           we
           so
           much
           take
           notice
           of
           ,
           as
           of
           its
           resolvable
           ,
           and
           equivalent
           ,
           the
           opposite
           Triangle
           :
           as
           for
           example
           ,
           If
           a
           Sphericall
           Triangle
           ,
           with
           two
           obtuse
           Angles
           ,
           and
           one
           acute
           ,
           be
           given
           you
           to
           resolve
           ,
           it
           will
           fall
           within
           the
           compasse
           of
           Simomatin
           ;
           because
           its
           opposite
           Sphericall
           is
           simply
           acute
           angled
           :
           and
           also
           if
           you
           be
           desired
           to
           calculate
           a
           Sphericall
           Triangle
           with
           two
           acute
           Angles
           ,
           and
           one
           obtuse
           ,
           it
           will
           likewise
           fall
           within
           the
           reach
           of
           the
           same
           Case
           ;
           because
           its
           opposite
           Sphericall
           is
           simply
           obtusangled
           .
           The
           reason
           of
           both
           the
           premisses
           is
           from
           the
           equality
           of
           the
           
             opposite
             Angles
          
           of
           concurring
           Quadrants
           ,
           
           which
           that
           they
           are
           equall
           ,
           no
           man
           needs
           to
           doubt
           ,
           that
           will
           take
           the
           paines
           to
           let
           fall
           a
           Perpendicular
           from
           the
           middle
           of
           the
           one
           Quadrant
           upon
           the
           other
           ;
           for
           so
           there
           will
           be
           two
           Triangles
           made
           equilaterall
           :
           and
           seeing
           it
           is
           an
           universally
           received
           truth
           ,
           that
           
             equall
             sides
             sustaine
             equall
             Angles
          
           ,
           the
           identitie
           of
           the
           Perpendicular
           in
           both
           the
           foresaid
           Triangles
           ,
           must
           needs
           manifest
           the
           equality
           of
           the
           two
           opposite
           Angles
           .
        
         
           I
           have
           beene
           the
           ampler
           in
           the
           elucidating
           of
           this
           Case
           ,
           that
           ,
           it
           over-running
           all
           the
           Moods
           of
           the
           Disergetick
           Loxogonosphericals
           ,
           the
           Reader
           ,
           in
           what
           Mood
           or
           Datoquare
           soever
           he
           please
           to
           resolve
           this
           foresaid
           Case
           ,
           may
           for
           that
           purpose
           to
           this
           place
           have
           recourse
           ;
           to
           the
           which
           ,
           without
           any
           further
           intended
           reiteration
           of
           this
           Tenet
           ,
           I
           doe
           heartily
           remit
           him
           .
        
         
           The
           first
           especiall
           Tenet
           of
           the
           generall
           Maxim
           of
           the
           Cathetothesis
           of
           this
           Mood
           is
           Dasimforaug
           ,
           that
           is
           ,
           
             When
             the
             given
             Angles
             are
             of
             the
             same
             nature
             ,
             but
             different
             from
             the
             required
             ,
             the
             Perpendicular
             falleth
             outwardly
             ,
             and
             the
             first
             verticall
             is
             a
             given
             Angle
             :
          
           the
           second
           Tenet
           belonging
           to
           the
           second
           Case
           of
           this
           Mood
           is
           Dadisforeug
           ,
           that
           is
           ,
           
             When
             the
             proposed
             Angles
             are
             of
             different
             affections
             ,
             the
             Perpendicular
             is
             externally
             demitted
             ,
             and
             one
             of
             the
             given
             Angles
             is
             a
             second
             verticall
             :
          
           Yet
           this
           discrepance
           is
           to
           be
           observed
           betweene
           the
           externall
           prosiliencie
           of
           the
           Perpendicular
           Arch
           in
           this
           Case
           ,
           and
           that
           other
           of
           the
           former
           ;
           that
           in
           the
           former
           ,
           it
           is
           no
           matter
           from
           which
           of
           the
           ends
           of
           the
           proposed
           side
           ,
           the
           Perpendicular
           be
           let
           fall
           upon
           one
           of
           the
           comprehending
           Legs
           of
           the
           Angle
           required
           ,
           which
           Leg
           must
           be
           increased
           ;
           for
           it
           is
           a
           generall
           Notandum
           ,
           that
           
             the
             sustentative
             Leg
             of
             a
             perpendiculars
             exterior
             demission
             must
             alwayes
             be
             continued
             :
          
           but
           in
           this
           Case
           ,
           the
           outward
           falling
           of
           the
           Perpendicular
           is
           onely
           from
           one
           extremity
           of
           the
           given
           side
           ;
           for
           ,
           if
           it
           be
           demitted
           likewise
           from
           the
           other
           end
           ,
           it
           falleth
           then
           inwardly
           ,
           and
           so
           produceth
           the
           third
           Tenet
           of
           this
           Mood
           ,
           which
           is
           Dadisgatin
           ,
           that
           is
           ,
           
             If
             the
             given
             Angles
             be
             of
             a
             different
             quality
             ,
             and
             that
             the
             Perpendicular
             be
             internally
             demitted
             ,
             the
             double
             verticall
             is
             one
             of
             the
             proposed
             Angles
             .
          
        
         
           The
           nature
           of
           the
           Perpendiculars
           demission
           in
           all
           the
           Cases
           of
           this
           Mood
           being
           thus
           to
           the
           full
           explained
           ,
           we
           may
           without
           impediment
           proceed
           to
           the
           performance
           of
           all
           the
           Orthogonosphericall
           operations
           ,
           each
           in
           its
           owne
           order
           thereto
           belonging
           .
        
         
         
           To
           begin
           therefore
           at
           the
           first
           ,
           whose
           quasitum
           (
           as
           I
           have
           told
           you
           already
           )
           is
           a
           
             verticall
             Angle
          
           ,
           we
           must
           know
           ,
           seeing
           the
           work
           is
           Orthogonospherically
           to
           be
           performed
           ,
           that
           the
           forementioned
           praenoscendum
           cannot
           be
           obtained
           without
           the
           help
           of
           one
           of
           the
           sixteen
           Datoquaeres
           ;
           and
           therefore
           in
           my
           Trissotetras
           (
           considering
           the
           nature
           of
           what
           is
           given
           ,
           and
           asked
           in
           the
           Cases
           of
           this
           Mood
           )
           I
           have
           appointed
           Upalam
           to
           be
           the
           subservient
           of
           its
           praenoscendum
           ;
           for
           ,
           by
           the
           Resolver
           thereof
           ,
           
             To
             —
             Tag
             —
             Nu
             ☞
             Mir
             ,
          
           (
           the
           subtendent
           and
           an
           Angle
           being
           given
           ;
           for
           one
           of
           the
           given
           sides
           of
           every
           Loxogonosphericall
           ,
           if
           the
           Perpendicular
           be
           rightly
           demitted
           ,
           becomes
           a
           subtendent
           ,
           and
           sometimes
           two
           given
           sides
           are
           subtendents
           both
           )
           we
           frame
           these
           three
           peculiar
           Problems
           ,
           for
           the
           three
           praenoscendas
           ;
           to
           wit
           ,
           Utopat
           ,
           for
           the
           
             double
             verticall
          
           ,
           by
           the
           meanes
           of
           the
           great
           Subtendent
           side
           ,
           and
           the
           
             prime
             opposite
             Angle
          
           :
           secondly
           ,
           Udobaud
           ,
           for
           obtaining
           of
           the
           
             first
             verticall
          
           in
           the
           little
           rectangle
           ,
           by
           vertue
           of
           the
           
             lesser
             subtendent
          
           in
           the
           same
           rectangle
           ,
           and
           the
           
             next
             opposite
             Angle
          
           :
           lastly
           ,
           Uthophauth
           ,
           for
           the
           
             first
             Co-verticall
          
           ,
           by
           meanes
           of
           the
           
             first
             Co-subtendent
          
           ,
           and
           
             first
             coopposite
             Angle
          
           :
           all
           which
           is
           at
           large
           set
           downe
           in
           the
           first
           partition
           of
           Alamebna
           in
           my
           Table
           .
        
         
           The
           first
           and
           chiefe
           operation
           being
           thus
           perfected
           ,
           the
           verticall
           Angles
           so
           found
           out
           must
           concurre
           with
           each
           its
           correspondent
           opposite
           for
           the
           obtaining
           of
           the
           Perpendicular
           ,
           necessary
           for
           the
           accomplishment
           of
           the
           second
           operation
           in
           every
           one
           of
           the
           Cases
           of
           the
           foresaid
           Mood
           ;
           to
           which
           effect
           Amaner
           is
           made
           the
           Subservient
           ,
           by
           whose
           Resolver
           
             Say
             —
             Nag
             —
             Tw
             ☞
             Nyr
             ,
          
           these
           three
           Datoquaeres
           ,
           
             Opatca
             ,
             Obautca
          
           ,
           and
           Ophauthca
           come
           to
           light
           ,
           and
           is
           manifestly
           shown
           how
           ,
           by
           any
           paire
           of
           three
           severall
           couples
           of
           different
           Angles
           ,
           the
           Perpendicular
           is
           acquirable
           .
        
         
           Now
           ,
           though
           of
           this
           work
           (
           as
           it
           is
           a
           single
           one
           )
           no
           more
           then
           of
           the
           other
           succeeding
           it
           in
           the
           same
           Mood
           ,
           nor
           of
           the
           last
           two
           in
           any
           of
           the
           Disergeticks
           in
           their
           full
           Analogy
           ,
           I
           doe
           not
           make
           any
           mention
           at
           all
           in
           my
           Table
           ;
           but
           ,
           after
           the
           couching
           of
           the
           first
           operation
           for
           the
           Praenoscendas
           ,
           supply
           the
           roomes
           of
           the
           other
           two
           ,
           with
           an
           equivalent
           row
           of
           proportionals
           out
           of
           them
           specified
           ,
           for
           attaining
           to
           the
           knowledge
           of
           the
           maine
           quaesitum
           :
           yet
           in
           this
           Comment
           upon
           that
           Table
           ,
           for
           the
           more
           perspicuities
           sake
           ,
           and
           that
           the
           Reader
           may
           as
           well
           know
           ,
           what
           way
           the
           rule
           
           is
           made
           ,
           as
           how
           thereby
           a
           demand
           is
           to
           be
           found
           out
           ,
           I
           have
           thought
           fit
           to
           expatiat
           my selfe
           for
           his
           satisfaction
           on
           each
           operation
           apart
           ,
           and
           Analytically
           to
           display
           in
           the
           glosse
           ,
           what
           is
           compounded
           in
           the
           Trissotetras
           .
        
         
           And
           therefore
           ,
           according
           to
           that
           prescribed
           method
           ,
           to
           proceed
           in
           this
           Mood
           ,
           the
           perpendicular
           by
           the
           second
           operation
           being
           already
           obtained
           ,
           it
           is
           requisite
           for
           the
           promoving
           of
           a
           third
           work
           ,
           that
           the
           said
           Perpendicular
           be
           made
           to
           joyne
           with
           
             the
             second
             verticaline
             ,
             the
             double
             verticall
             ,
             and
             second
             co-verticall
          
           conforme
           to
           the
           quality
           of
           the
           three
           Cases
           ,
           thereby
           to
           obtaine
           the
           Angles
           
             at
             the
             Base
          
           ,
           for
           the
           which
           all
           these
           operations
           have
           beene
           set
           on
           foot
           ;
           to
           wit
           ,
           
             the
             next
             cathetopposite
          
           ,
           (
           whose
           complement
           to
           a
           Semicircle
           is
           alwayes
           the
           Angle
           required
           )
           
             the
             prime
             cathetopposite
          
           ,
           and
           
             the
             second
             Cocathetopposite
          
           :
           for
           the
           prosecuting
           of
           this
           last
           work
           ,
           Edoman
           is
           the
           subservient
           ,
           by
           whose
           Resolver
           ,
           
             To
             —
             Neg
             —
             Sa
             ☞
             Nir
             ,
          
           we
           are
           instructed
           how
           to
           regulate
           the
           Problemets
           of
           
             Catheudob
             ,
             Cathatop
          
           ,
           and
           Catheuthops
           .
        
         
           Now
           these
           two
           last
           operations
           being
           thus
           made
           patent
           in
           their
           severall
           structures
           ,
           it
           is
           not
           amisse
           that
           we
           ponder
           how
           appositely
           they
           may
           be
           conflated
           into
           one
           ,
           to
           the
           end
           ,
           that
           the
           verity
           of
           all
           the
           finall
           Resolvers
           of
           the
           Disergeticks
           in
           my
           Trissotetras
           (
           which
           are
           all
           and
           each
           of
           them
           composed
           of
           the
           ingredient
           termes
           of
           two
           different
           works
           )
           may
           be
           the
           more
           evidently
           knowne
           ,
           and
           obvious
           to
           the
           reach
           of
           any
           ordinary
           capacity
           ,
           for
           the
           performance
           hereof
           ,
           the
           Resolvers
           of
           these
           two
           operations
           are
           to
           be
           laid
           before
           us
           ,
           
             Say
             —
             Nag
             —
             Ta
             ☞
             Nyr
             ,
          
           and
           
             To
             —
             Neg
             —
             Sa
             ☞
             Nir
             :
          
           and
           ,
           seeing
           out
           of
           both
           these
           orders
           of
           proportionals
           ,
           there
           must
           result
           but
           one
           ,
           it
           is
           to
           be
           considered
           ,
           which
           be
           the
           foure
           ejectitious
           termes
           ,
           and
           which
           those
           foure
           we
           should
           reserve
           for
           the
           Analogy
           required
           ;
           all
           which
           ,
           that
           it
           may
           be
           the
           better
           understood
           by
           the
           industrious
           Reader
           ,
           I
           will
           interpret
           the
           Resolvers
           so
           farre
           forth
           as
           is
           requisite
           :
           and
           therefore
           
             Say
             —
             Nag
             —
             To
             ☞
             Nyr
             ,
          
           being
           ,
           
             As
             the
             Sine
             of
             one
             of
             the
             Angles
             at
             the
             Base
             ,
             or
             Cathetopposite
             ,
             is
             to
             the
             Sine
             complement
             of
             a
             verticall
             :
             so
             the
             Radius
             to
             the
             Sine
             complement
             of
             the
             Perpendicular
             :
          
           And
           the
           other
           ,
           
             To
             —
             Neg
             —
             Sa
             ☞
             Nir
             ,
          
           being
           ,
           
             As
             the
             Radius
             ,
             to
             the
             Sine
             complement
             of
             the
             Perpendicular
             :
             so
             the
             Sine
             of
             a
             verticall
             ,
             to
             the
             Sine
             complement
             of
             a
             Cathetopposite
             or
             Angle
             at
             the
             Base
          
           ;
           it
           is
           perceivable
           enough
           
           how
           both
           the
           Radius
           ,
           and
           the
           Perpendicular
           are
           in
           both
           the
           rows
           :
           nor
           can
           it
           well
           escape
           the
           knowledge
           of
           one
           never
           so
           little
           versed
           in
           the
           elements
           of
           Arithmetick
           ,
           that
           the
           Perpendiculars
           being
           the
           fourth
           terme
           in
           the
           first
           order
           of
           proportionals
           ,
           is
           nothing
           else
           but
           that
           it
           is
           the
           quotient
           of
           the
           product
           of
           the
           middle
           termes
           ,
           divided
           by
           the
           first
           ,
           or
           Logarithmically
           the
           remainder
           of
           the
           first
           termes
           abstraction
           from
           the
           summe
           of
           the
           middle
           two
           ;
           so
           that
           the
           whole
           power
           thereof
           is
           inclosed
           in
           these
           three
           termes
           ,
           whereby
           it
           is
           most
           evident
           ,
           that
           with
           what
           terme
           soever
           the
           foresaid
           Perpendicular
           be
           employed
           to
           concurre
           in
           operation
           ,
           the
           same
           effect
           will
           be
           produced
           by
           the
           concurrence
           of
           its
           ingredients
           with
           the
           said
           terme
           ,
           and
           therefore
           in
           the
           second
           row
           of
           proportionals
           ,
           where
           it
           is
           made
           use
           of
           for
           a
           fellow
           multiplyer
           with
           the
           third
           terme
           to
           produce
           a
           factus
           ,
           which
           divided
           may
           quote
           the
           maine
           quaesitum
           ,
           or
           Logarithmically
           to
           joyne
           with
           the
           third
           terme
           ,
           for
           the
           summing
           of
           an
           Aggregat
           from
           which
           the
           first
           terme
           being
           abstracted
           ,
           may
           residuat
           the
           terme
           demanded
           ,
           it
           is
           all
           one
           ,
           whether
           the
           work
           be
           performed
           by
           it selfe
           ,
           or
           by
           its
           equivalent
           ,
           viz.
           the
           three
           first
           termes
           of
           the
           first
           order
           of
           proportionals
           ,
           in
           whose
           potentia
           it
           is
           :
           whereupon
           the
           fourth
           terme
           in
           the
           second
           row
           being
           that
           ,
           for
           the
           obtaining
           whereof
           ,
           both
           the
           Analogies
           are
           made
           ,
           we
           need
           not
           waste
           any
           labour
           about
           the
           finding
           out
           of
           the
           Perpendicular
           (
           though
           a
           subservient
           to
           the
           chiefe
           quaesitum
           )
           but
           leaving
           roome
           for
           it
           in
           both
           the
           rows
           ,
           that
           the
           equipollencie
           of
           its
           conflaters
           may
           the
           better
           appeare
           ,
           go
           on
           in
           work
           without
           it
           ,
           and
           ,
           by
           the
           meanes
           of
           its
           constructive
           parts
           ,
           with
           as
           much
           certainty
           effectuat
           the
           same
           designe
           .
        
         
           Thus
           may
           you
           see
           then
           how
           the
           eight
           termes
           of
           the
           forementioned
           Resolvers
           ,
           are
           reduced
           unto
           six
           ;
           but
           there
           remaining
           yet
           two
           more
           to
           be
           ejected
           ,
           that
           both
           the
           orders
           may
           be
           brought
           unto
           a
           compound
           row
           of
           foure
           proportionals
           :
           let
           us
           consider
           the
           Radius
           ,
           which
           ,
           being
           in
           both
           the
           rows
           as
           I
           have
           once
           told
           you
           already
           ,
           may
           peradventure
           ,
           without
           any
           prejudice
           to
           the
           work
           ,
           be
           spared
           out
           of
           both
           .
           Thus
           much
           thereof
           to
           any
           is
           perceiveable
           ,
           that
           in
           the
           first
           Resolver
           ,
           it
           is
           the
           third
           proportionall
           ;
           and
           in
           the
           second
           ,
           the
           first
           ,
           and
           consequently
           a
           multiplyer
           in
           the
           one
           ,
           and
           in
           the
           other
           a
           divider
           :
           or
           Logarithmically
           in
           the
           second
           a
           subtractor
           ,
           and
           in
           the
           first
           an
           adder
           :
           now
           it
           being
           well
           known
           that
           division
           overthrows
           
           the
           structure
           of
           multiplication
           ,
           and
           that
           what
           is
           made
           up
           by
           addition
           ,
           is
           by
           subtraction
           cast
           down
           ;
           we
           need
           not
           undergoe
           the
           laboriousnesse
           of
           such
           a
           Penelopaean
           task
           ,
           and
           by
           the
           division
           and
           abstraction
           of
           what
           we
           did
           adde
           and
           multiply
           ,
           weave
           and
           unweave
           ,
           build
           up
           ,
           and
           throw
           downe
           the
           self
           same
           thing
           :
           but
           choose
           rather
           (
           seeing
           the
           Radius
           undoeth
           in
           the
           one
           ,
           what
           it
           doth
           in
           the
           other
           (
           which
           ineffect
           is
           to
           doe
           nothing
           at
           all
           )
           to
           dash
           the
           one
           against
           the
           other
           ,
           and
           race
           it
           out
           of
           both
           )
           then
           idely
           to
           expend
           time
           ,
           and
           have
           the
           proportion
           pestred
           with
           unnecessarie
           termes
           .
           Thus
           from
           those
           two
           resolvers
           ,
           foure
           termes
           being
           with
           reason
           ejected
           ,
           we
           must
           ,
           for
           the
           finding
           out
           of
           the
           last
           in
           the
           second
           Resolver
           ,
           effectuat
           as
           much
           by
           three
           ,
           as
           formerly
           was
           on
           seven
           incumbent
           ,
           which
           three
           being
           the
           first
           ,
           and
           second
           termes
           in
           the
           first
           row
           of
           proportionalls
           ;
           and
           the
           third
           in
           the
           second
           ,
           the
           two
           Resolvers
           
             Say
             —
             Nag
             —
             To
             ☞
             Nyr
             ,
          
           &
           
             To
             —
             Neg
             —
             Sa
             ☞
             nir
          
           are
           comprehended
           by
           this
           one
           
             Say
             —
             Nag
             —
             Sa
             —
             ☞
             Nir
             ,
          
           that
           is
           ,
           
             As
             the
             sine
             of
             one
             verticall
             to
             the
             Antisine
             of
             an
             opposite
             ;
             so
             the
             sine
             of
             another
             verticall
             to
             the
             Antisine
             of
             another
             opposite
             :
          
           and
           though
           the
           second
           Resolver
           doth
           import
           ,
           that
           this
           other
           opposite
           is
           to
           be
           found
           out
           by
           the
           Antisine
           of
           the
           perpendicular
           ,
           and
           sine
           of
           a
           secondarie
           verticall
           ,
           yet
           doth
           it
           in
           nothing
           evince
           the
           coincidence
           of
           the
           two
           operations
           in
           one
           ;
           because
           the
           first
           two
           termes
           of
           the
           Resultative
           Analogie
           ,
           doe
           adaequatly
           stand
           for
           the
           perpendicular
           ,
           which
           I
           have
           proved
           already
           ,
           and
           therefore
           these
           two
           in
           their
           proper
           places
           co-working
           with
           the
           third
           terme
           ,
           according
           to
           the
           rule
           of
           proportion
           ,
           have
           the
           selfe
           same
           influence
           ,
           that
           the
           Perpendicular
           so
           seconded
           ,
           hath
           upon
           the
           operatum
           .
        
         
           Now
           ,
           to
           contract
           the
           generality
           of
           this
           finall
           Resolver
           ,
           
             Say
             —
             Nag
             —
             Sa
             ☞
             Nir
             ,
          
           to
           all
           the
           particular
           Cases
           of
           this
           Mood
           ,
           we
           must
           say
           ,
           
             When
             the
             given
             Angles
             are
             of
             the
             same
             affection
             ,
             and
             the
             required
             diverse
             ,
          
           as
           in
           Dasimforaug
           ,
           the
           first
           case
           ,
           
             Sat
             —
             Nop
             —
             Seud
             ☞
             Nob*
             Kir
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             of
             the
             double
             verticall
             to
             the
             Antisine
             of
             the
             prime
             Cathetopposite
             :
             so
             the
             Sine
             of
             the
             second
             verticaline
          
           (
           or
           verticall
           in
           the
           lesser
           rectangle
           )
           
             to
             the
             Antisine
             of
             the
             next
             Cathetopposite
             ,
          
           whose
           complement
           to
           a
           semicircle
           is
           the
           Angle
           required
           .
        
         
           But
           when
           ,
           the
           affection
           of
           the
           given
           Angles
           being
           different
           ,
           the
           perpendicular
           is
           made
           to
           fall
           without
           ,
           as
           in
           Dadisforeug
           ,
           the
           second
           Case
           of
           this
           Mood
           ,
           the
           Resolver
           thereof
           is
           particularised
           
           thus
           ,
           
             Saud
             —
             Nob
             —
             Sat
             ☞
             Nop
             *
             Ir
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             of
             the
             first
             verticaline
          
           (
           or
           verticall
           in
           the
           rectanglet
           ,
           )
           
             to
             the
             Sine
             complement
             of
             the
             nearest
             Cathetopposite
             :
             so
             the
             Sine
             of
             the
             double
             verticall
             ,
             to
             the
             Sine
             complement
             of
             the
             prime
             Cathetopposite
             ,
          
           which
           is
           the
           Angle
           required
           .
        
         
           And
           lastly
           ,
           if
           with
           the
           different
           qualities
           of
           the
           given
           and
           demanded
           Angles
           ,
           the
           Perpendicular
           be
           let
           fall
           within
           ,
           as
           in
           Dadisgatin
           ,
           the
           third
           Case
           of
           this
           Mood
           ,
           then
           is
           the
           finall
           Resolver
           to
           be
           determined
           thus
           ,
           
             Sauth
             —
             Noph
             —
             Seuth
             ☞
             Nops
             *
             Ir
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             of
             the
             first
             coverticall
             ,
             to
             the
             Co-sine
             of
             the
             first
             Co-opposite
             :
             so
             is
             the
             Sine
             of
             the
             second
             coverticall
             ,
             to
             the
             Co-sine
             of
             the
             second
             Co-opposite
          
           which
           is
           the
           Angle
           required
           .
        
         
           The
           originall
           reason
           of
           all
           these
           operations
           is
           grounded
           on
           the
           Axiome
           of
           Nabadprosver
           ,
           as
           the
           first
           syllable
           of
           its
           Directory
           Alama
           giveth
           us
           to
           understand
           ,
           which
           we
           may
           easily
           perceive
           by
           the
           Analogy
           ,
           that
           is
           onely
           amongst
           the
           Angles
           without
           any
           intermixture
           of
           sides
           in
           the
           termes
           of
           the
           proportion
           .
        
         
           The
           second
           Mood
           of
           the
           first
           Angulary
           Figure
           (
           that
           is
           to
           say
           ,
           the
           first
           two
           termes
           of
           whose
           datas
           are
           Angles
           )
           is
           Allamebne
           ,
           which
           comprehendeth
           all
           those
           Disergetick
           questions
           ,
           wherein
           two
           Angles
           being
           given
           and
           a
           side
           betweene
           ,
           one
           of
           the
           other
           sides
           is
           demanded
           ,
           which
           side
           (
           the
           perpendicular
           being
           let
           fall
           )
           is
           alwayes
           one
           of
           the
           
             second
             Subtendents
          
           ,
           viz.
           in
           the
           first
           Case
           a
           
             second
             Subtendent
          
           of
           the
           lesser
           Triangle
           ,
           in
           the
           second
           a
           
             second
             Subtendent
          
           in
           the
           great
           rectangle
           ,
           and
           in
           the
           last
           a
           
             second
             Co-subtendent
          
           .
        
         
           To
           the
           knowledge
           of
           all
           these
           ,
           that
           we
           may
           the
           more
           easily
           attaine
           ,
           we
           must
           consider
           the
           generall
           maxim
           of
           the
           Cathetothesis
           of
           this
           Mood
           ,
           which
           is
           Cafyxegeq
           that
           is
           to
           say
           ,
           that
           
             in
             all
             the
             Cases
             of
             Allamebne
             the
             Perpendicular
             falleth
             from
             the
             side
             required
             ,
             and
             from
             that
             point
             thereof
             ,
             where
             it
             conterminats
             with
             the
             given
             side
             upon
             the
             third
             side
             ,
             continued
             if
             need
             be
          
           ;
           and
           according
           to
           the
           variety
           of
           the
           
             second
             subtendent
          
           ,
           which
           is
           the
           side
           demanded
           ,
           there
           be
           these
           three
           especiall
           Tenets
           of
           this
           generall
           Maxim
           ,
           to
           wit
           ,
           
             Dasimforauxy
             ,
             Dadiscracforeng
          
           ,
           and
           Dadiscramgatin
           .
        
         
           Dasimforauxy
           ,
           the
           first
           especiall
           Tenet
           of
           the
           generall
           Maxim
           of
           the
           Cathetothesis
           of
           this
           Mood
           sheweth
           ,
           that
           ,
           
             when
             the
             proposed
             Angles
             are
             of
             the
             same
             quality
             and
             homogeneall
             ,
             the
             Perpendicular
             falleth
             externally
             ,
             and
             the
             first
             verticall
             is
             one
             of
             the
             given
          
           
           Angles
           ,
           and
           annexed
           to
           the
           required
           side
           .
        
         
           The
           second
           Tenet
           ,
           Dadiscracforeug
           ,
           which
           pertaineth
           to
           the
           second
           Case
           of
           this
           Mood
           ,
           sheweth
           ,
           that
           
             when
             the
             given
             Angles
             are
             of
             a
             discrepant
             nature
             ,
             and
             heterogeneall
             ,
             and
             that
             the
             concurse
             of
             the
             proposed
             and
             required
             sides
             is
             at
             an
             acute
             Angle
             ,
             that
             then
             the
             Perpendicular
             must
             be
             demitted
             outwardly
             ,
             and
             one
             of
             the
             proposed
             Angles
             becomes
             a
             second
             verticall
             .
          
        
         
           The
           third
           Tenet
           is
           Dadiscramgatin
           ,
           whereby
           we
           learne
           ,
           that
           if
           
             with
             the
             various
             affection
             of
             the
             Angles
             given
             ,
             the
             concurse
          
           (
           mentioned
           in
           the
           preceding
           Tenet
           )
           
             be
             at
             an
             obtuse
             Angle
             ,
             the
             Perpendicular
             falleth
             inwardly
             ,
             and
             that
             one
             of
             the
             foresaid
             Angles
             is
             a
             double
             verticall
             .
          
           This
           is
           the
           onely
           Case
           of
           Allamebne
           ,
           wherein
           the
           Perpendicular
           is
           demitted
           inwardly
           ,
           save
           when
           the
           three
           Angles
           are
           qualified
           all
           alike
           ,
           of
           which
           Case
           ,
           because
           it
           falleth
           in
           all
           the
           Moods
           of
           the
           Loxogonosphericall
           Disergeticks
           ,
           and
           that
           in
           Alamebna
           I
           have
           spoke
           at
           large
           thereof
           ,
           I
           shall
           not
           need
           (
           I
           hope
           )
           to
           make
           any
           more
           mention
           hereafter
           .
        
         
           Having
           thus
           unfolded
           the
           mysteries
           of
           the
           Perpendiculars
           demission
           in
           all
           the
           Cases
           of
           this
           Mood
           (
           as
           I
           must
           doe
           in
           all
           those
           of
           every
           one
           of
           the
           other
           Loxogonosphericall
           Disergeticks
           ;
           because
           such
           Obliquangulars
           ,
           till
           they
           be
           reduced
           to
           a
           rectangularity
           (
           which
           without
           the
           Perpendicular
           is
           not
           performable
           )
           can
           never
           Logarithmically
           be
           resolved
           )
           I
           may
           safely
           go
           on
           ,
           without
           any
           let
           to
           the
           Reader
           ,
           to
           the
           three
           severall
           Orthogonosphericall
           operations
           thereof
           ,
           as
           they
           stand
           in
           order
           .
        
         
           The
           quaesitas
           of
           the
           first
           operation
           ,
           which
           are
           alwayes
           the
           praenoscendas
           of
           the
           Mood
           ,
           are
           in
           this
           Mood
           the
           same
           that
           they
           were
           in
           the
           last
           ,
           to
           wit
           ,
           the
           double
           
             verticall
             ,
             the
             first
             verticaline
             ,
             and
             the
             first
             coverticall
             :
          
           and
           are
           likewise
           to
           be
           found
           out
           by
           the
           same
           Datas
           both
           of
           side
           ,
           and
           Angle
           here
           ,
           that
           they
           were
           in
           the
           former
           Mood
           ;
           that
           is
           ,
           for
           the
           side
           ,
           by
           the
           
             first
             and
             great
             Subtendent
             :
             the
             first
             but
             little
             Subtendent
             :
             and
             the
             first
             Co-subtendent
             :
          
           and
           for
           the
           Angle
           ,
           by
           the
           
             prime
             Cathetopposite
             ,
             the
             nearest
             Cathetopposite
             ,
             and
             the
             first
             Co-cathetopposite
             :
          
           so
           that
           the
           Datoquaere
           sounding
           thus
           ,
           
             the
             Subtendent
             ,
             and
             an
             Oblique
             Angle
             being
             given
             ,
             to
             find
             the
             other
             Oblique
             ,
          
           the
           Subservient
           of
           this
           Computation
           must
           needs
           be
           Upalam
           ,
           and
           its
           Resolver
           ,
           
             To
             —
             Tag
             —
             Nu
             ☞
             Mir
             ,
          
           which
           sheweth
           ,
           that
           the
           subducing
           of
           the
           Logarithm
           of
           the
           Radius
           from
           the
           
           summe
           of
           the
           Logarithms
           of
           the
           Sine
           complement
           of
           one
           of
           the
           first
           Subtendents
           ,
           and
           Tangent
           of
           one
           of
           the
           Angles
           at
           the
           Base
           ,
           residuats
           the
           Logarithm
           of
           the
           Tangent
           complement
           of
           one
           of
           the
           verticals
           required
           ,
           and
           consequently
           involveth
           within
           so
           much
           generality
           the
           particular
           resolutions
           of
           the
           Sub-problems
           of
           Upalam
           ,
           viz.
           
             Utopat
             ,
             Vdoband
          
           ,
           and
           Vthophauth
           ,
           diversified
           thus
           according
           to
           the
           variety
           of
           their
           praenoscendas
           ,
           whereon
           ,
           to
           speak
           ingenuously
           ,
           I
           intend
           to
           insist
           no
           longer
           ;
           for
           ,
           besides
           that
           the
           peculiar
           enodation
           of
           all
           the
           three
           apart
           is
           clearly
           set
           downe
           in
           my
           glosse
           on
           the
           last
           Mood
           ,
           they
           are
           in
           both
           the
           first
           partitions
           of
           the
           Moods
           of
           Ahalebmane
           to
           the
           full
           expressed
           in
           the
           Table
           of
           my
           Trissotetras
           .
        
         
           The
           verticall
           Angles
           ,
           according
           to
           the
           diversity
           of
           the
           three
           Cases
           being
           by
           the
           foresaid
           Datas
           thus
           obtained
           ,
           must
           concurre
           with
           each
           its
           correspondent
           first
           Subtendent
           (
           notified
           by
           the
           Characteristicks
           of
           τ.
           δ.
           θ.
           )
           for
           finding
           out
           of
           the
           perpendicular
           ,
           requisite
           for
           the
           performance
           of
           the
           second
           work
           in
           every
           one
           of
           the
           Cases
           of
           this
           Mood
           .
           And
           to
           this
           effect
           Ubamen
           is
           made
           the
           Subservient
           ,
           by
           whose
           Resolver
           
             Nag
             —
             Mu
             —
             ☞
             Torp
             ☞
             Myr
             ,
          
           these
           three
           Problems
           ,
           
             Vtatatca
             ,
             Vdaudca
          
           ,
           and
           Uthauthca
           ,
           are
           made
           manifest
           ,
           and
           the
           same
           quaesitum
           attained
           unto
           by
           the
           Datas
           of
           three
           severall
           Subtendents
           ,
           and
           verticals
           .
           *
        
         
           The
           Perpendicular
           being
           thus
           found
           out
           ,
           must
           ,
           for
           the
           surtherance
           of
           the
           third
           operation
           ,
           joyne
           with
           the
           
             second
             verticaline
          
           ,
           the
           
             double
             verticall
          
           ,
           and
           
             second
             co-verticall
          
           ,
           according
           to
           the
           nature
           of
           the
           Case
           in
           question
           ,
           (
           the
           Datas
           being
           the
           same
           with
           those
           of
           the
           third
           work
           of
           the
           last
           Mood
           )
           thereby
           to
           attaine
           unto
           the
           knowledge
           of
           the
           second
           
             little
             Subtendent
          
           ,
           the
           second
           
             great
             Subtendent
          
           ,
           and
           the
           second
           Co-subtendent
           ,
           the
           which
           are
           all
           the
           maine
           quaesitas
           of
           this
           Mood
           :
           To
           the
           performance
           of
           this
           last
           operation
           ,
           Etalum
           is
           the
           subservient
           ,
           whose
           Resolver
           
             Torp
             —
             Me
             —
             Nag
             ☞
             Mur
             ,
          
           teacheth
           us
           how
           to
           deale
           with
           the
           
             under
             datoquaeres
          
           of
           
             Catheudwd
             ,
             Cathatwt
          
           ,
           and
           Catheuthwth
           .
        
         
           Now
           ,
           the
           coalescencie
           of
           these
           last
           two
           operations
           in
           one
           ,
           proceeding
           from
           the
           casting
           out
           of
           the
           Radius
           in
           both
           the
           orders
           of
           proportionals
           ,
           and
           leaing
           roome
           for
           the
           perpendicular
           ,
           without
           taking
           the
           paines
           to
           know
           its
           value
           ,
           as
           hath
           beene
           shewne
           already
           in
           the
           first
           Mood
           of
           the
           same
           Figure
           ;
           it
           cannot
           be
           much
           
           amisse
           in
           this
           place
           to
           give
           a
           further
           illustration
           thereof
           ,
           and
           make
           the
           Reader
           ,
           by
           an
           Arithmeticall
           demonstration
           ,
           feele
           (
           as
           it
           were
           )
           how
           palpable
           the
           truth
           is
           of
           compacting
           eight
           proportionals
           into
           foure
           ;
           let
           there
           be
           then
           these
           two
           orders
           of
           numbers
           ,
           4
           —
           6
           —
           8
           ☞
           12.
           and
           8
           —
           12.
           —
           14
           ☞
           21.
           
           Where
           ,
           we
           may
           suppose
           eight
           to
           be
           the
           Radius
           ,
           and
           twelve
           the
           Perpendicular
           (
           for
           such
           like
           suppositions
           can
           inferre
           no
           great
           absurdity
           )
           and
           then
           let
           us
           consider
           how
           those
           termes
           doe
           beare
           to
           one
           another
           ,
           especially
           the
           12.
           and
           8.
           which
           ,
           by
           possessing
           foure
           places
           ,
           make
           up
           halfe
           the
           number
           of
           the
           proportionals
           .
           First
           ,
           we
           see
           that
           twelve
           in
           the
           first
           row
           ,
           is
           nothing
           else
           but
           the
           result
           of
           the
           product
           of
           6.
           in
           8.
           divided
           by
           4.
           
           And
           secondly
           ,
           that
           8.
           in
           the
           second
           row
           ,
           casteth
           downe
           ,
           by
           its
           division
           ,
           whatsoever
           by
           its
           multiplication
           it
           builded
           up
           in
           the
           first
           ;
           upon
           which
           observations
           we
           may
           ground
           these
           Sequels
           ,
           that
           12.
           may
           be
           safely
           left
           out
           ,
           both
           in
           the
           fourth
           ,
           and
           sixth
           place
           ,
           taking
           instead
           of
           it
           the
           number
           of
           4.
           6.
           and
           8.
           in
           whose
           potentia
           it
           is
           :
           and
           next
           8.
           undoing
           in
           one
           place
           ,
           what
           it
           doth
           in
           another
           ,
           may
           with
           greater
           ease
           void
           them
           both
           .
           So
           that
           by
           this
           abbreviated
           way
           of
           Analogising
           ,
           4.
           and
           6.
           alone
           in
           their
           due
           order
           before
           14.
           which
           is
           the
           third
           terme
           of
           the
           second
           row
           ,
           conduce
           as
           much
           to
           the
           obtaining
           of
           the
           fourth
           ,
           or
           if
           you
           will
           eighth
           proportionall
           21.
           as
           if
           the
           other
           foure
           termes
           of
           the
           two
           eights
           ,
           and
           twelves
           ,
           were
           concurrent
           with
           it
           .
           How
           plaine
           all
           this
           is
           ,
           no
           question
           needs
           to
           be
           made
           ,
           and
           therefore
           ,
           to
           returne
           to
           our
           Resolvers
           (
           for
           the
           explicating
           whereof
           ,
           we
           thought
           good
           to
           make
           this
           digression
           )
           we
           must
           understand
           that
           the
           finall
           Resolver
           ,
           (
           in
           its
           generall
           expression
           )
           made
           out
           of
           them
           (
           they
           being
           as
           they
           are
           materially
           displayed
           ,
           
             Nag-Mu-Torp
             ☞
             Myr
          
           ,
           &
           
             Torp-Me-Nag
             ☞
             Mur
          
           )
           is
           no
           other
           then
           
             Nag-Mu-Na
             ☞
             Mur
          
           ,
           that
           is
           ,
           
             As
             the
             Sine
             complement
             of
             one
             verticall
             is
             to
             the
             Tangent
             complement
             of
             a
             Subtendent
             :
             So
             the
             Sine
             complement
             of
             another
             verticall
             ,
             to
             the
             Tangent
             complement
             of
             another
             Subtendent
             :
          
           and
           Analytically
           to
           trace
           the
           running
           of
           this
           operation
           ,
           even
           to
           the
           source
           from
           whence
           it
           flowes
           ,
           by
           foysting
           in
           the
           Perpendicular
           ,
           and
           Radius
           ,
           we
           may
           bring
           it
           to
           the
           consistence
           of
           the
           former
           two
           subordinate
           Resolvers
           ,
           whereof
           the
           first
           is
           ,
           
             As
             the
             Sine
             complement
             of
             a
             first
             ,
             or
             a
             double
             verticall
             ,
             to
             the
             Tangent
             complement
             of
             a
             first
             Subtendent
             :
             so
             the
             Radius
             to
             the
             Tangent
             complement
             of
             the
             Perpendicular
          
           ;
           and
           the
           second
           ,
           
             As
             the
          
           
           
             Radius
             ,
             to
             the
             Tangent
             complement
             of
             the
             Perpendicular
             :
             so
             the
             Sine
             complement
             of
             a
             second
             ,
             or
             a
             double
             verticall
             ,
             to
             the
             Tangent
             complement
             of
             a
             second
             Subtendent
             ,
          
           which
           is
           the
           side
           required
           ,
           and
           the
           fourth
           proportionall
           of
           
             Nag
             —
             Mu
             —
             Na
             ☞
             Mur.
          
           Whose
           generality
           is
           to
           be
           contracted
           to
           every
           one
           of
           the
           three
           Cases
           of
           this
           Mood
           thus
           :
           If
           both
           the
           Angles
           given
           be
           of
           the
           same
           nature
           ,
           they
           being
           the
           first
           verticals
           ,
           from
           which
           the
           Cathetus
           fals
           on
           either
           side
           ,
           increased
           according
           to
           the
           demand
           of
           the
           side
           ,
           as
           in
           the
           first
           Case
           ,
           Dasimforauxy
           ,
           we
           must
           particularise
           the
           common
           Resolver
           ,
           in
           this
           manner
           ,
           
             Nat
             —
             Mut
             —
             Neud
             ☞
             Nwd
             *
             Yr
             ,
          
           that
           is
           ,
           
             As
             the
             Antisine
             of
             the
             double
             verticall
             ,
             is
             to
             the
             Antitangent
             of
             the
             first
             ,
             and
             great
             Subtendent
             :
             so
             the
             Antisine
             of
             the
             second
             verticall
             in
             the
             lesser
             rectangle
             ,
             to
             the
             Antitangent
             of
             the
             second
             Subtendent
             in
             the
             same
             little
             rectangle
             ,
          
           which
           Subtendent
           is
           the
           side
           required
           .
           For
           the
           second
           Case
           of
           this
           Mood
           ,
           viz.
           Dadiscracforeug
           ,
           we
           must
           say
           ,
           
             Naud
             —
             Mud
             —
             Nat
             ☞
             Mwt
             *
             Yr
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             complement
             of
             the
             first
             and
             little
             verticall
             to
             the
             Tangent
             complement
             of
             the
             first
             ,
             and
             little
             Subtendent
             :
             so
             the
             Sine
             complement
             of
             the
             double
             verticall
             ,
             to
             the
             Tangent
             complement
             of
             the
             second
             and
             great
             subtendent
             .
          
           And
           lastly
           ,
           for
           the
           third
           Case
           Dadiscramgatin
           ,
           the
           finall
           Resolver
           is
           determinated
           thus
           ,
           
             Nauth
             —
             Muth
             —
             Neuth
             ☞
             Mwth
             *
             Yr
             ,
          
           that
           is
           ,
           
             As
             the
             Co-sine
             of
             the
             first
             Co-verticall
             ,
             is
             to
             the
             Co-tangent
             of
             the
             first
             Co-subtendent
             :
             so
             the
             Co-sine
             of
             the
             second
             Co-verticall
             ,
             to
             the
             Co-tangent
             of
             the
             second
             Co-subtendent
             ,
          
           which
           is
           the
           side
           in
           this
           third
           Case
           required
           .
        
         
           The
           truth
           of
           all
           these
           operations
           is
           grounded
           on
           the
           Axiome
           of
           Naverprortes
           ,
           as
           we
           are
           certified
           by
           the
           first
           syllable
           of
           its
           Directory
           Allera
           ,
           which
           we
           may
           perceive
           by
           the
           direct
           Analogy
           that
           is
           betweene
           the
           Sines
           complements
           of
           the
           verticall
           Angles
           ,
           and
           the
           Tangents
           complements
           ,
           (
           and
           consequently
           reciprocall
           'twixt
           them
           and
           the
           Tangents
           )
           of
           the
           verticall
           sides
           ,
           which
           in
           this
           Mood
           are
           alwayes
           second
           Subtendents
           .
        
         
           The
           second
           Disergetick
           ,
           and
           Angulary
           Figure
           ,
           is
           Ahamepnare
           ,
           which
           embraceth
           all
           those
           Obliquangularie
           Sphericals
           ,
           wherein
           
             two
             Angles
          
           being
           given
           with
           an
           
             opposite
             side
          
           ,
           another
           Angle
           ,
           or
           the
           
             side
             interjacent
          
           ,
           is
           demanded
           :
           this
           Figure
           ,
           conforme
           to
           the
           two
           severall
           Quaesitas
           ,
           hath
           two
           Moods
           ,
           viz.
           Amanepra
           ,
           and
           Ammanepreb
           .
        
         
         
           The
           first
           Mood
           hereof
           ,
           which
           is
           Amanepra
           ,
           belongeth
           to
           all
           those
           Loxogonosphericall
           questions
           ,
           wherein
           ,
           
             two
             Angles
             with
             an
             opposite
             side
             being
             proposed
             ,
             the
             third
             Angle
             is
             required
             ,
          
           which
           is
           alwayes
           a
           
             first
             verticall
          
           ,
           a
           
             second
             verticall
          
           ,
           or
           a
           
             first
             co-verticall
          
           :
           to
           the
           notice
           of
           all
           which
           ,
           that
           we
           may
           with
           ease
           attaine
           ,
           the
           generall
           Maxim
           of
           the
           Cathetothesis
           of
           this
           Mood
           is
           to
           be
           considered
           ,
           which
           is
           Cafriq
           that
           is
           to
           say
           ,
           that
           in
           all
           the
           Cases
           of
           Amanepra
           ,
           the
           Perpendicular
           falleth
           from
           the
           Angle
           required
           upon
           the
           side
           opposite
           to
           that
           Angle
           ,
           and
           terminated
           by
           the
           other
           two
           Angles
           ,
           which
           side
           is
           to
           be
           increased
           ,
           if
           need
           be
           .
        
         
           Now
           in
           regard
           ,
           that
           besides
           the
           Cathetothesis
           of
           this
           Mood
           ,
           and
           some
           three
           moe
           ,
           to
           wit
           ,
           all
           those
           Loxogonosphericals
           wherein
           the
           quaesitum
           is
           either
           a
           
             partiall
             verticall
          
           ,
           or
           
             segment
             at
             the
             Base
          
           ,
           there
           is
           a
           peculiar
           Mensurator
           ,
           pertaining
           to
           every
           one
           of
           the
           foure
           ,
           called
           in
           my
           Trissotetras
           the
           
             plus
             minus
          
           ,
           because
           it
           sheweth
           by
           the
           specieses
           thereof
           to
           the
           Moods
           appropriated
           ,
           whether
           the
           summe
           ,
           or
           difference
           of
           the
           verticall
           Angles
           ,
           and
           segments
           at
           the
           Base
           ,
           be
           the
           Angle
           ,
           or
           side
           required
           ,
           and
           so
           clearly
           leadeth
           us
           thorough
           all
           the
           Cases
           of
           each
           of
           the
           Moods
           ,
           that
           either
           by
           abstracting
           the
           fourth
           proportionall
           from
           an
           Angle
           or
           a
           segment
           ,
           or
           by
           abstracting
           an
           Angle
           ,
           or
           a
           segment
           from
           it
           ,
           or
           lastly
           ,
           by
           joyning
           it
           to
           an
           Angle
           ,
           or
           a
           segment
           ,
           with
           an
           incredible
           facility
           we
           attaine
           to
           the
           knowledge
           of
           the
           
             maine
             quaesitum
          
           ,
           whether
           Angulary
           ,
           or
           laterall
           .
           Let
           the
           Reader
           then
           be
           pleased
           to
           know
           ,
           that
           the
           Mensurator
           ,
           or
           
             Plus
             minus
          
           of
           this
           Mood
           ,
           is
           Sindifora
           ,
           which
           evidently
           declareth
           (
           as
           by
           its
           representatives
           in
           the
           explanation
           of
           the
           Table
           is
           apparent
           )
           that
           ,
           if
           the
           demission
           of
           the
           Perpendicular
           be
           internall
           ,
           the
           summe
           ;
           if
           exterior
           ,
           the
           difference
           of
           the
           
             verticall
             Angles
          
           ,
           is
           the
           Angle
           required
           .
        
         
           Seeing
           thus
           the
           notice
           of
           the
           manner
           of
           the
           Perpendiculars
           falling
           is
           so
           necessary
           ,
           it
           is
           expedient
           ,
           for
           our
           better
           information
           therein
           ,
           that
           we
           severally
           perpend
           the
           three
           especiall
           Tenets
           of
           the
           generall
           Maxim
           of
           the
           Cathetothesis
           of
           this
           Mood
           ,
           which
           are
           
             Dadissepamforaur
             Dadissexamforeur
          
           ,
           and
           Dasimatin
           .
        
         
           Dadissepamforaur
           ,
           which
           is
           the
           Tenet
           of
           the
           first
           Case
           ,
           sheweth
           ,
           that
           when
           the
           Angles
           given
           are
           of
           a
           different
           nature
           ,
           and
           that
           the
           proposed
           side
           is
           opposite
           to
           an
           obtuse
           Angle
           ,
           the
           Perpendicular
           falleth
           outwardly
           ,
           and
           the
           first
           verticall
           is
           the
           Angle
           required
           .
        
         
         
           The
           second
           Tenet
           belonging
           to
           the
           second
           Case
           of
           this
           Mood
           ,
           viz.
           Dadissexamforeur
           ,
           sheweth
           ,
           that
           if
           the
           proposed
           Angles
           be
           of
           discrepant
           affections
           ,
           and
           that
           the
           side
           given
           be
           conterminat
           with
           an
           
             obtuse
             Angle
          
           ,
           the
           Perpendicular
           is
           demitted
           externally
           ,
           and
           the
           demanded
           Angle
           is
           a
           second
           verticall
           .
        
         
           The
           third
           Tenet
           pertaining
           to
           the
           last
           Case
           of
           this
           Mood
           ,
           to
           wit
           ,
           Dasimatin
           ,
           evidenceth
           ,
           that
           if
           the
           Angles
           proposed
           be
           of
           the
           same
           quality
           ,
           the
           Perpendicular
           falleth
           interiourly
           ,
           and
           the
           double
           verticall
           is
           the
           Angle
           required
           .
        
         
           Having
           thus
           (
           as
           I
           suppose
           )
           hereby
           evinced
           every
           difficulty
           of
           the
           Perpendiculars
           demission
           in
           all
           the
           Cases
           of
           this
           Mood
           ,
           I
           may
           the
           more
           boldly
           in
           the
           interim
           proceed
           to
           the
           three
           rectangular
           works
           thereto
           belonging
           .
           Now
           ,
           it
           being
           manifest
           that
           the
           Praenoscendas
           of
           this
           Mood
           ,
           or
           the
           Quaesitas
           of
           the
           first
           operation
           thereof
           ,
           are
           the
           same
           with
           those
           of
           the
           two
           Moods
           of
           the
           first
           Disergetick
           Figure
           ,
           to
           wit
           ,
           the
           
             double
             verticall
             ,
             the
             first
             verticaline
             ,
          
           and
           the
           
             first
             co-verticall
          
           ;
           and
           that
           ,
           without
           any
           alteration
           at
           all
           ,
           they
           are
           to
           be
           obtained
           by
           the
           same
           Datas
           ,
           both
           of
           side
           ,
           and
           Angle
           in
           this
           Mood
           of
           Amanepra
           ,
           that
           ,
           they
           were
           in
           the
           former
           Moods
           of
           Alamebna
           ,
           and
           Allamebne
           ,
           without
           any
           further
           specifying
           what
           these
           given
           sides
           ,
           and
           Angles
           are
           (
           which
           are
           to
           the
           full
           expressed
           in
           the
           last
           two
           forementioned
           Moods
           )
           I
           must
           make
           bold
           thither
           to
           direct
           you
           ,
           where
           you
           shall
           be
           sure
           also
           to
           learne
           all
           that
           is
           necessary
           to
           know
           of
           the
           Subservient
           and
           Resolver
           of
           the
           first
           operation
           of
           this
           Mood
           ,
           both
           which
           ,
           to
           wit
           ,
           Upalam
           and
           
             To
             —
             Tag
             —
             Nu
             ☞
             Mir
             ,
          
           are
           inseparable
           dependents
           on
           all
           the
           Angularie
           Praenoscendas
           of
           the
           Loxogonosphericall
           Disergeticks
           :
           And
           though
           within
           the
           generality
           of
           this
           Subservient
           be
           compreded
           the
           peculiar
           Problemets
           of
           
             Vtopat
             ,
             Udobaud
          
           ,
           and
           Uthophauth
           ,
           which
           are
           all
           three
           at
           large
           couched
           in
           the
           Trissotetras
           of
           this
           Mood
           ;
           yet
           ,
           because
           what
           hath
           beene
           already
           said
           thereof
           in
           the
           foresaid
           Figure
           ,
           may
           very
           well
           suffice
           for
           this
           place
           ,
           the
           Readers
           diligence
           (
           I
           hope
           )
           in
           the
           turning
           of
           a
           leaf
           ,
           will
           save
           me
           the
           labour
           of
           any
           further
           recapitulation
           .
        
         
           The
           Praenoscendas
           ,
           or
           the
           verticall
           Angles
           ,
           according
           to
           the
           nature
           of
           the
           Case
           ,
           being
           by
           the
           foresaid
           Datas
           thus
           found
           out
           ,
           must
           needs
           joyne
           with
           each
           its
           correspondent
           opposite
           ,
           specified
           by
           the
           characteristicks
           of
           π.
           β.
           φ.
           for
           the
           obtaining
           of
           the
           Perpendicular
           ,
           
           which
           in
           all
           the
           rest
           of
           the
           Disergetick
           Moods
           ,
           as
           well
           as
           this
           ,
           is
           alwayes
           the
           quaesitum
           of
           the
           second
           operation
           ,
           thorough
           all
           the
           Cases
           thereof
           .
           Of
           this
           work
           Amaner
           is
           the
           subservient
           ,
           by
           whose
           Resolver
           ,
           
             Say
             —
             Nag
             —
             To
             ☞
             Nyr
             ,
          
           the
           three
           sub-problems
           ,
           
             Opatca
             ,
             Obaudca
          
           ,
           and
           Ophauthca
           ,
           are
           made
           known
           ,
           and
           the
           same
           quaesitum
           attained
           unto
           by
           the
           Datas
           of
           three
           several
           both
           cathetopposites
           ,
           and
           verticals
           ,
           it
           being
           the
           only
           Mood
           which
           with
           Alamebna
           ,
           hath
           a
           cathetopposite
           and
           verticall
           catheteuretick
           identity
           .
           The
           Perpendicular
           being
           thus
           obtained
           ,
           is
           ,
           for
           the
           effecting
           of
           the
           third
           and
           last
           operation
           ,
           to
           concurre
           with
           the
           next
           cathetopposite
           ,
           the
           prime
           cathetopposite
           ,
           and
           the
           second
           cocathetopposite
           ,
           as
           the
           Case
           requires
           it
           ,
           thereby
           to
           find
           out
           the
           main
           quaesitum
           ;
           which
           in
           the
           first
           Case
           by
           abstracting
           the
           fourth
           proportionall
           ,
           in
           the
           second
           by
           abstracting
           from
           the
           fourth
           proportionall
           ,
           and
           in
           the
           third
           by
           adding
           the
           fourth
           proportionall
           to
           another
           verticall
           ,
           is
           easily
           obtained
           by
           those
           that
           have
           the
           skill
           to
           discerne
           which
           be
           the
           greater
           ,
           or
           lesser
           of
           two
           verticals
           proposed
           .
           To
           the
           perfecting
           of
           this
           third
           work
           ,
           Exoman
           is
           the
           Subservient
           ,
           whose
           Resolver
           
             Ne-To-Nag
             ☞
             Sir
          
           ,
           instructeth
           us
           ,
           how
           to
           unfold
           the
           peculiar
           Problems
           of
           
             Cathobeud
             Cathopat
          
           ,
           &
           Cathopseuth
           .
        
         
           Now
           ,
           the
           nature
           of
           proportion
           requiring
           that
           of
           two
           rowes
           of
           proportionals
           ,
           when
           the
           fourth
           in
           the
           first
           order
           is
           first
           in
           the
           second
           ,
           that
           then
           the
           multiplyers
           become
           dividers
           ,
           and
           the
           dividers
           multiplyers
           :
           as
           by
           these
           numbers
           following
           you
           may
           perceive
           ,
           viz.
           2
           —
           4
           —
           6
           ☞
           12.
           for
           the
           first
           row
           ,
           and
           12
           —
           4
           —
           15
           ☞
           5
           ,
           for
           the
           second
           ;
           of
           which
           proportionals
           ,
           because
           of
           the
           fourth
           terme
           in
           the
           first
           rowes
           being
           first
           in
           the
           second
           ,
           if
           you
           turne
           as
           many
           multiplyers
           into
           dividers
           as
           you
           can
           ,
           and
           (
           where
           the
           identity
           of
           a
           Figure
           requires
           it
           )
           dash
           out
           a
           multiplyer
           against
           a
           divider
           ,
           you
           will
           find
           ,
           the
           two
           foures
           by
           this
           reason
           being
           raced
           out
           ,
           and
           the
           two
           twelves
           (
           because
           of
           their
           being
           in
           the
           power
           of
           the
           three
           first
           proportionals
           of
           the
           first
           row
           )
           likewise
           left
           out
           ,
           that
           this
           Analogy
           of
           6
           —
           2
           —
           15
           doth
           the
           same
           effect
           ,
           that
           the
           former
           seven
           proportionals
           ,
           for
           obtaining
           of
           the
           quaesitum
           ,
           viz.
           5.
           the
           reason
           whereof
           is
           altogether
           grounded
           upon
           the
           inversion
           of
           a
           permutat
           proportion
           ,
           or
           the
           Retrograd
           Analogy
           of
           the
           alternat
           termes
           ,
           whereby
           the
           Consequents
           are
           compared
           to
           Consequents
           ,
           and
           Antecedents
           to
           Antecedents
           ,
           in
           the
           preposterous
           method
           of
           
           beginning
           at
           the
           second
           of
           both
           the
           Consequents
           and
           Antecedents
           ,
           and
           ending
           at
           the
           first
           :
           therefore
           (
           as
           I
           was
           telling
           you
           )
           the
           nature
           of
           proportion
           requiring
           that
           in
           such
           a
           Case
           the
           multiplyers
           and
           dividers
           be
           bound
           to
           interchange
           their
           places
           ,
           the
           Resolvers
           of
           the
           last
           two
           operations
           ,
           viz.
           
             Say
             —
             Nag
             —
             To
             ☞
             Nyr
             ,
          
           and
           
             Ne
             —
             To
             —
             Nag
             ☞
             Sir
             ,
          
           the
           first
           whereof
           being
           ,
           
             As
             the
             Sine
             of
             a
             verticall
             Angle
             ,
             to
             the
             Sine
             complement
             of
             an
             Angle
             at
             the
             Base
             ,
             or
             one
             of
             the
             Cathetopposites
             :
             so
             the
             Radius
             to
             the
             Sine
             complement
             of
             the
             Perpendicular
             :
          
           and
           the
           second
           ,
           
             As
             the
             Sine
             complement
             of
             the
             Perpendicular
             ,
             to
             the
             Radius
             :
             so
             the
             Sine
             complement
             of
             one
             of
             the
             Cathetopposite
             Angles
             ,
             to
             one
             of
             the
             verticals
             ,
          
           may
           both
           of
           them
           (
           according
           to
           the
           former
           rule
           )
           be
           handsomely
           compacted
           in
           this
           one
           Analogy
           ,
           
             Na
             —
             Say
             —
             Nag
             ☞
             Sir
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             complement
             of
             an
             opposite
             is
             to
             the
             Sine
             of
             a
             verticall
             :
             so
             the
             Sine
             complement
             of
             another
             opposite
             ,
             to
             the
             Sine
             of
             another
             verticall
             .
          
        
         
           This
           foresaid
           generall
           Resolver
           ,
           according
           to
           the
           three
           severall
           cases
           of
           this
           Mood
           ,
           is
           to
           be
           specialised
           into
           so
           many
           finall
           Resolvers
           ;
           the
           first
           whereof
           for
           
             Dadissepamforaur
             ,
             Nop
             —
             Sat
             —
             Nob
             —
             ☞
             Seudfr*
             At*
             Aut*
             ir
             ,
          
           that
           is
           ,
           
             As
             the
             sine
             complement
             of
             the
             prime
             cathetopposite
             ,
             to
             the
             sine
             of
             the
             double
             verticall
             :
             so
             the
             sine
             complement
             of
             the
             nearest
             cathetopposite
             ,
             to
             the
             sine
             of
             the
             second
             verticalin
             ;
             the
             which
             subtracted
             from
             the
             double
             verticall
             ,
             leaveth
             the
             first
             and
             great
             verticall
             ,
             which
             is
             the
             Angle
             required
             .
          
        
         
           Next
           ,
           for
           the
           second
           Case
           of
           this
           
             Mood
             ,
             Dadissexamforeur
          
           ,
           we
           must
           make
           use
           of
           ,
           
             Nob
             —
             Saud
             —
             Nop
             ☞
             Satfr
             ,
          
           *
           
             Aud*
             Eut*
             ir
          
           ,
           that
           is
           ,
           
             As
             the
             sine
             complement
             of
             the
             next
             opposite
             ,
             to
             the
             sine
             of
             the
             first
             verticallet
             :
             so
             the
             sine
             complement
             of
             the
             prime
             opposite
             ,
             to
             the
             sine
             of
             the
             double
             verticall
             ,
          
           from
           which
           ,
           
             if
             you
             deduce
             the
             first
             verticalm
             ,
             there
             will
             remaine
             the
             second
             and
             great
             verticall
             for
             the
             Angle
             demanded
             .
          
        
         
           Lastly
           ,
           for
           the
           third
           
             Case
             ,
             Dasimatin
          
           ,
           we
           must
           ,
           say
           
             Noph
             —
             Sauth
             —
             Nop●
             ☞
             Seuth*
             jauth*
             ir
             ,
          
           that
           is
           ,
           
             As
             the
             sine
             complement
             of
             the
             first
             co-opposite
             ,
             to
             the
             sine
             of
             the
             first
             co-verticall
             :
             so
             the
             sine
             complement
             of
             the
             second
             co-opposite
             ,
             to
             the
             sine
             of
             the
             second
             co
             verticall
             ,
             which
             added
             to
             the
             first
             co-verticall
             ,
             maketh
             up
             the
             Angle
             we
             desire
             .
          
        
         
           The
           veritie
           of
           all
           these
           operations
           is
           grounded
           on
           the
           Axiome
           Nabadprosver
           ,
           as
           the
           second
           syllable
           of
           its
           directorie
           Alama
           ,
           giveth
           us
           understand
           ,
           and
           as
           we
           may
           discerne
           more
           easily
           by
           the
           samenesse
           in
           species
           amongst
           the
           proportionall
           termes
           ;
           for
           they
           
           are
           all
           Angles
           ,
           the
           first
           ,
           and
           third
           being
           Angles
           at
           the
           Base
           (
           for
           these
           are
           alwaies
           of
           the
           opposits
           )
           and
           the
           second
           ,
           and
           fourth
           termes
           of
           the
           verticall
           Angles
           ,
           which
           verticall
           Angles
           in
           the
           finall
           resolvers
           of
           this
           Mood
           ,
           are
           according
           to
           the
           foresaid
           Axiome
           ,
           
             to
             the
             Angles
             of
             the
             Base
             directly
             proportionall
             ,
             and
             contrarily
             .
          
        
         
           The
           second
           Mood
           of
           the
           second
           Angularie
           figure
           of
           the
           Loxogonosphericall
           Disergeticks
           ,
           named
           Ahamepnare
           is
           Ammanepreb
           ,
           which
           is
           said
           of
           all
           those
           obliquangularie
           problems
           ,
           wherein
           
             two
             Angles
             ,
             and
             an
             opposite
             side
             being
             given
             ,
             the
             side
             between
             is
             required
             ,
          
           and
           is
           alwaies
           one
           of
           the
           basal-segments
           :
           to
           the
           knowledge
           whereof
           ,
           that
           we
           may
           the
           more
           easily
           attaine
           ,
           we
           must
           consider
           the
           generall
           maxime
           of
           the
           Cathetothesis
           of
           this
           Mood
           ,
           which
           is
           Cafregpagyq
           that
           is
           ,
           that
           the
           perpendicular
           falleth
           still
           from
           the
           given
           side
           ,
           opposite
           to
           both
           the
           Angles
           given
           ,
           and
           upon
           the
           side
           required
           ,
           continued
           ,
           if
           need
           be
           ,
           in
           all
           and
           every
           one
           of
           the
           cases
           of
           Ammanepreb
           .
        
         
           The
           Plusminus
           of
           this
           Mood
           ,
           is
           Sindiforiu
           ,
           that
           is
           to
           say
           ,
           the
           
             summe
             of
             the
             segments
             of
             the
             Base
             ,
             if
             the
             perpendicular
             fall
             inwardly
             ,
             and
             the
             difference
             of
             the
             Bases
             ,
             if
             exteriorly
             ,
             is
             the
             side
             demanded
             .
          
        
         
           The
           perpendiculars
           demission
           ,
           being
           a
           
             Sine
             quo
             non
          
           in
           all
           disergetick
           operations
           ,
           it
           will
           not
           be
           amisse
           ,
           that
           we
           ponder
           what
           the
           three
           
             severall
             tenets
          
           are
           of
           the
           Cathetothesis
           of
           this
           Mood
           ,
           and
           what
           is
           meaned
           by
           
             Dadissepamfor
             ,
             Dadissexamfor
          
           ,
           and
           Dasimin
           .
        
         
           Dadissepamfor
           ,
           the
           tenet
           of
           the
           first
           Case
           of
           this
           Mood
           ,
           sheweth
           ,
           that
           
             if
             the
             given
             Angles
             be
             of
             severall
             natures
             ,
             and
             that
             the
             proposed
             side
             be
             opposite
             to
             an
             obtuse
             Angle
             ,
             the
             perpendicular
             falleth
             externally
             .
          
        
         
           The
           second
           tenet
           ,
           Dadissexamfor
           expresseth
           ,
           that
           
             if
             the
             proposed
             Angles
             be
             different
             ,
             and
             that
             the
             side
             given
             be
             conterminat
             with
             the
             obtuse
             Angle
             ,
             it
             falleth
             likewise
             outwardly
             .
          
        
         
           But
           Dasimin
           ,
           which
           is
           the
           third
           tenet
           signifieth
           ,
           that
           if
           
             the
             given
             Angles
             be
             of
             the
             same
             affection
             ,
             the
             falling
             of
             the
             perpendicular
             is
             internall
             .
          
        
         
           This
           much
           being
           premised
           of
           the
           perpendicular
           ,
           we
           may
           securely
           goe
           on
           to
           the
           orthogonosphericall
           works
           of
           the
           Mood
           ;
           and
           so
           beginning
           with
           the
           first
           operation
           ,
           consider
           what
           the
           praenoscendas
           are
           ,
           which
           are
           alwaies
           the
           quaesitas
           by
           the
           first
           operation
           obtainable
           ,
           and
           in
           this
           Mood
           the
           Bases
           of
           the
           Triangle
           ;
           but
           more
           particularly
           to
           descend
           to
           the
           illustration
           of
           the
           Cases
           of
           Ammanepreb
           ,
           
           the
           praenoscendum
           of
           the
           first
           Case
           ,
           is
           the
           first
           and
           
             great
             Base
          
           ,
           of
           the
           second
           ,
           the
           first
           but
           little
           Base
           ,
           and
           of
           the
           third
           ,
           the
           first
           co-base
           .
           Now
           ,
           though
           these
           three
           praenoscendas
           ,
           be
           totally
           different
           from
           those
           of
           the
           three
           former
           Moods
           ,
           yet
           are
           they
           to
           be
           acquired
           by
           the
           same
           ,
           and
           no
           other
           Datas
           ;
           because
           none
           of
           the
           Angularie
           figures
           must
           differ
           from
           one
           another
           in
           the
           Datas
           of
           their
           praenoscendas
           ,
           as
           out
           of
           the
           definition
           of
           an
           Angularie
           figure
           in
           the
           entrie
           ,
           of
           the
           second
           Mood
           set
           downe
           ,
           is
           easie
           to
           be
           collected
           :
           these
           Datas
           being
           tendred
           to
           us
           of
           intermixed
           circularie
           parts
           ,
           that
           is
           to
           say
           ,
           of
           both
           sides
           and
           Angles
           ,
           the
           side
           being
           the
           first
           subtendentall
           ,
           or
           great
           subtendent
           ,
           the
           first
           subtendentine
           ,
           or
           little
           subtendent
           ,
           and
           the
           first
           co-subtendent
           :
           and
           the
           Angles
           the
           
             prime
             cathetopposite
          
           ,
           the
           next
           cathetopposite
           ,
           and
           the
           first
           co-catheopposite
           ;
           so
           that
           considering
           what
           is
           demanded
           ,
           and
           that
           the
           Datoquaere
           thereof
           must
           be
           expressed
           thus
           ,
           
             the
             hypotenusa
             ,
             and
             an
             oblique
             being
             given
             ,
             to
             finde
             the
             Ambient
             conterminate
             with
             the
             proposed
             Angle
             ,
          
           we
           are
           ,
           for
           the
           calculation
           of
           this
           work
           ,
           necessitated
           to
           have
           recourse
           to
           Vbamen
           ,
           which
           ,
           in
           the
           Table
           of
           my
           Trissotetras
           obtaineth
           the
           roome
           of
           its
           subservient
           ,
           to
           the
           end
           ,
           that
           by
           its
           Resolver
           
             Torp
             —
             Mu
             —
             Lag
             ☞
             Myr
             ,
          
           being
           instructed
           how
           by
           cutting
           off
           the
           Logarithm
           of
           the
           Radius
           ,
           from
           the
           summe
           of
           the
           Logarithms
           of
           the
           M.
           of
           one
           of
           the
           
             first
             subtendents
          
           ,
           and
           secant
           complement
           of
           one
           of
           the
           cathetopposits
           ,
           or
           Angles
           at
           the
           Base
           ,
           residuats
           the
           Logarithm
           of
           the
           Tangent
           complement
           of
           the
           Base
           required
           ,
           we
           may
           deliveredly
           extract
           ,
           out
           of
           the
           generality
           of
           that
           proposition
           ,
           the
           peculiar
           Subordinate
           resolutions
           of
           these
           three
           Problemets
           of
           Ubamen
           ,
           viz.
           
             Utopaet
             ,
             Vdobaed
          
           ,
           and
           Uthophaeth
           ,
           varied
           (
           as
           you
           see
           )
           according
           to
           the
           diversity
           of
           the
           Praenoscendas
           ,
           which
           being
           (
           as
           you
           were
           told
           already
           )
           the
           
             first
             Basal
          
           ,
           or
           great
           Base
           ,
           the
           
             first
             Baset
          
           or
           little
           Base
           ,
           and
           the
           
             first
             Co-base
          
           ;
           I
           will
           not
           detain
           you
           any
           longer
           upon
           this
           matter
           ,
           but
           the
           rather
           hasten
           my
           transition
           to
           the
           other
           work
           ,
           that
           in
           the
           Praenoscendall
           partition
           of
           Ammanepreb
           ,
           there
           is
           enough
           thereof
           set
           downe
           in
           the
           Table
           of
           my
           Trissotetras
           .
        
         
           The
           Praenoscendas
           of
           Ammanepreb
           ,
           or
           the
           three
           severall
           
             first
             Bases
          
           ,
           conforme
           to
           the
           various
           nature
           of
           the
           Cases
           thereof
           ,
           being
           by
           the
           foresaid
           Datas
           happily
           obtained
           ,
           must
           concurre
           with
           each
           its
           correspondent
           Cathetopposite
           (
           discernable
           ,
           in
           their
           severall
           
           qualities
           ,
           by
           the
           Characteristicks
           of
           π.
           β.
           φ.
           )
           for
           finding
           out
           of
           the
           perpendicular
           ,
           which
           is
           the
           perpetuall
           quaesitum
           of
           the
           second
           operation
           .
           The
           subservient
           of
           this
           work
           is
           Ethaner
           ,
           by
           whose
           Resolver
           ,
           
             To
             —
             Tag
             —
             Se
             ☞
             Tyr
             ,
          
           we
           come
           to
           the
           knowledge
           of
           Ethaners
           three
           Subdatoquaeres
           ,
           viz.
           
             Aetopca
             ,
             Aedobca
          
           ,
           and
           Aethophca
           ,
           whereby
           we
           may
           perceive
           ,
           that
           the
           same
           quaesitum
           ,
           to
           wit
           ,
           the
           perpendicular
           is
           obtained
           by
           the
           Datas
           of
           the
           three
           severall
           both
           Bases
           ,
           and
           
             Cathet
             opposite
          
           Angles
           .
        
         
           This
           so
           often
           mentioned
           perpendicular
           being
           thus
           made
           known
           ,
           must
           ,
           for
           the
           performance
           of
           the
           last
           and
           third
           work
           ,
           joyne
           with
           the
           nixt
           Cathetopposite
           ,
           the
           prime
           Cathetopposite
           ,
           and
           the
           second
           Co-cathetopposite
           ,
           as
           the
           Case
           will
           beare
           it
           ,
           the
           Datas
           being
           the
           same
           in
           every
           point
           here
           ,
           that
           in
           the
           last
           operation
           of
           the
           foregoing
           Mood
           (
           as
           by
           the
           subservients
           ,
           Exoman
           and
           Epsoner
           ,
           is
           obvious
           to
           any
           judicious
           Reader
           )
           thereby
           to
           obtaine
           the
           maine
           quaesitum
           ,
           which
           in
           the
           first
           Case
           ,
           by
           abstracting
           the
           fourth
           proportionall
           from
           the
           first
           great
           Base
           ,
           in
           the
           second
           by
           abstracting
           from
           the
           fourth
           proportionall
           ,
           the
           first
           little
           Base
           ,
           and
           in
           the
           third
           by
           adding
           the
           fourth
           proportionall
           to
           another
           segment
           of
           the
           Base
           ,
           is
           findable
           by
           any
           ,
           that
           will
           undergoe
           the
           labour
           of
           adding
           ,
           and
           substracting
           .
           For
           the
           acomplishment
           of
           this
           last
           operation
           Epsoner
           is
           the
           Subservient
           ,
           by
           whose
           Resolver
           
             Tag
             —
             Tolb
             —
             Te
             ☞
             Syr
             ,
          
           we
           are
           taught
           how
           to
           deale
           with
           its
           three
           Subproblems
           ,
           
             Cathoboed
             ,
             Cathopoet
          
           ,
           and
           Cathopsoeth
           .
        
         
           These
           last
           two
           operations
           being
           thus
           to
           the
           full
           extended
           ,
           it
           remaineth
           now
           to
           treat
           how
           they
           ought
           to
           be
           in
           one
           compacted
           ,
           or
           rather
           ,
           for
           brevitie
           of
           computation
           ,
           we
           should
           compact
           them
           both
           in
           one
           ,
           before
           we
           take
           the
           paines
           to
           extend
           them
           :
           yet
           ,
           because
           practice
           requireth
           one
           method
           ,
           &
           the
           order
           of
           Doctrine
           another
           ,
           we
           will
           ,
           that
           we
           may
           be
           the
           lesse
           troublesome
           to
           the
           Readers
           memory
           ,
           goe
           on
           (
           by
           ejecting
           some
           ,
           and
           reserving
           other
           proportional
           termes
           )
           in
           our
           usuall
           course
           of
           conflating
           two
           Resolvers
           together
           .
           These
           Resolvers
           are
           in
           this
           Mood
           ,
           
             To
             —
             Tag
             —
             Se
             ☞
             Tyr
             ,
          
           and
           
             Tag
             —
             To
             —
             To
             ☞
             Syr
             ,
          
           the
           first
           thereof
           ,
           sounding
           ,
           
             As
             the
             Radius
             ,
             to
             the
             Tangent
             of
             one
             of
             the
             Cathetopposite
             Angles
             ,
             or
             Angles
             at
             the
             Base
             :
             so
             the
             Sine
             of
             one
             of
             the
             first
             Bases
             ,
             to
             the
             Tangent
             of
             the
             perpendicular
             :
          
           and
           the
           second
           ,
           
             As
             the
             Tangent
             of
             one
             of
             the
             other
             Cathetopposite
             Angles
             to
             the
             Radius
             :
             so
             the
             Tangent
             of
             the
             perpendicular
             ,
             to
             the
             sine
             of
             the
             side
             required
             .
          
        
         
         
           Here
           may
           the
           Reader
           be
           pleased
           to
           consider
           ,
           that
           in
           all
           the
           glosse
           upon
           the
           posterior
           operations
           of
           my
           Disergeticks
           ,
           I
           have
           beene
           contented
           to
           set
           downe
           (
           as
           he
           may
           see
           in
           the
           last
           two
           propositions
           )
           the
           bare
           Theorems
           of
           the
           Resolvers
           ,
           conforme
           to
           the
           nature
           of
           their
           Analogy
           ,
           without
           troubling
           my selfe
           ,
           or
           any
           body
           else
           ,
           with
           repeating
           ,
           or
           reiterating
           the
           way
           ,
           how
           the
           Logarithms
           of
           the
           middle
           ,
           and
           initiall
           termes
           are
           to
           be
           handled
           ,
           for
           the
           obtaining
           of
           a
           fourth
           Logarithm
           ;
           all
           that
           can
           be
           desired
           therein
           ,
           being
           to
           the
           full
           expressed
           already
           in
           my
           ample
           comments
           upon
           the
           Orthogonosphericall
           Problems
           ;
           to
           the
           which
           the
           industrious
           Reader
           ,
           in
           case
           of
           doubting
           ,
           may
           (
           if
           he
           please
           )
           have
           recourse
           ,
           without
           any
           great
           losse
           of
           time
           ,
           or
           labour
           :
           however
           ,
           for
           his
           better
           encouragement
           ,
           I
           give
           another
           hint
           thereof
           in
           the
           closure
           of
           this
           Treatise
           .
        
         
           But
           to
           returne
           where
           we
           left
           ,
           seeing
           out
           of
           these
           two
           Resolvers
           ,
           
             To
             —
             Tag
             —
             Se
             ☞
             Tyr
             ,
          
           and
           
             Tag
             —
             To
             —
             Te
             ☞
             Syr
             ,
          
           according
           to
           the
           rules
           of
           coalescency
           ,
           mentioned
           in
           both
           the
           Moods
           of
           Ahalebmane
           ,
           both
           the
           Perpendicular
           and
           Radius
           may
           be
           ejected
           without
           any
           danger
           of
           losing
           our
           aime
           of
           the
           maine
           quaesitum
           ,
           it
           is
           evident
           ,
           that
           the
           proportion
           of
           the
           Remanent
           termes
           ,
           is
           ,
           
             Ta
             —
             Tag
             —
             Se
             ☞
             Syr
             ,
          
           which
           comprehendeth
           both
           the
           last
           two
           Resolvers
           ,
           and
           the
           three
           foresaid
           Problemets
           thereto
           belonging
           ,
           and
           being
           interpreted
           ,
           
             As
             the
             Tangent
             of
             one
             Cathetopposite
             Angle
             ,
             to
             the
             Tangent
             of
             another
             Cathetopposite
             :
             so
             the
             sine
             of
             one
             of
             the
             first
             Bases
             ,
             to
             the
             sine
             of
             a
             side
             ,
             which
             ushers
             in
             the
             side
             required
             .
          
        
         
           This
           generall
           Resolver
           ,
           according
           to
           the
           three
           severall
           Cases
           of
           this
           Mood
           ,
           is
           to
           be
           particularised
           into
           so
           many
           finall
           Resolvers
           ;
           the
           first
           whereof
           ,
           for
           Dadissepanefor
           ,
           is
           
             Tob-Top-Saet
             ☞
             Soedfr
             *
             ,
             Aet*
             Dyr
             ,
          
           that
           is
           ,
           
             As
             the
             Tangent
             of
             the
             next
             opposite
             ,
             to
             the
             Tangent
             of
             the
             prime
             opposite
             :
             so
             the
             Sine
             of
             the
             first
             great
             Base
             ,
             to
             the
             Sine
             of
             the
             second
             little
             Base
             ;
             which
             subducted
             from
             the
             foresaid
             first
             great
             Base
             ,
             will
             for
             the
             remainder
             afford
             us
             that
             segment
             of
             the
             Base
             ,
             which
             is
             the
             side
             in
             the
             first
             Case
             required
             .
          
        
         
           Then
           for
           the
           second
           Case
           ,
           Dadissexamfor
           ,
           the
           finall
           Resolver
           is
           
             Top
             —
             Tob
             —
             Saed
             ☞
             Soetfr
             *
             Aed*
             Dyr
             ,
          
           that
           is
           ,
           
             As
             the
             Tangent
             of
             the
             prime
             Cathetopposite
             to
             the
             Tangent
             of
             the
             next
             opposite
             :
             so
             the
             Sine
             of
             the
             first
             Baset
             ,
             or
             little
             Base
             ,
             to
             the
             Sine
             of
             the
             second
             and
          
           
           
             great
             Base
          
           ;
           from
           which
           if
           we
           abstract
           the
           foresaid
           first
           little
           Base
           ,
           the
           difference
           or
           remainer
           will
           be
           that
           Segment
           of
           the
           Base
           ,
           which
           is
           the
           side
           demanded
           .
        
         
           Lastly
           ,
           for
           the
           Case
           Dasimin
           ,
           the
           finall
           Resolver
           is
           
             Tops
             —
             Toph
             —
             Saeth
             ☞
             Soethj*
             Aeth*
             Syr
             ,
          
           that
           is
           ,
           
             As
             the
             Tangent
             of
             the
             second
             co-opposite
             ,
             to
             the
             Tangent
             of
             the
             first
             co-opposite
             :
             so
             the
             Sine
             of
             the
             first
             co-base
             ,
             to
             the
             Sine
             of
             the
             second
             co-base
          
           ;
           the
           summe
           of
           which
           two
           co-bases
           is
           the
           totall
           Base
           or
           side
           in
           the
           third
           Case
           required
           .
        
         
           The
           reason
           of
           all
           this
           is
           proved
           by
           the
           third
           Disergetick
           Axiome
           ,
           which
           is
           Siubprortab
           ,
           as
           is
           pointed
           at
           by
           the
           first
           syllable
           of
           its
           Directory
           Ammena
           ,
           and
           manifested
           to
           us
           in
           all
           the
           Analogies
           of
           this
           Mood
           ,
           every
           one
           whereof
           runneth
           upon
           Tangents
           of
           Angles
           ,
           and
           Sines
           of
           Segments
           ,
           both
           to
           the
           Base
           belonging
           :
           nor
           can
           any
           doubt
           ,
           that
           heares
           the
           resolution
           of
           the
           Cases
           of
           Ammanepreb
           ,
           but
           that
           the
           habitude
           ,
           which
           all
           the
           termes
           thereof
           have
           to
           one
           another
           ,
           proceedeth
           meerly
           from
           
             the
             reciprocall
             proportion
             ,
             which
             the
             Tangents
             of
             the
             opposite
             Angles
             have
             to
             the
             Basal-segments
             ,
             and
             contrariwise
             .
          
        
         
           The
           third
           Loxogonosphericall
           Disergetick
           Figure
           ,
           and
           first
           of
           the
           Laterals
           (
           that
           is
           ,
           the
           first
           two
           termes
           of
           whose
           Datas
           are
           sides
           ,
           what
           ere
           the
           quaesitum
           be
           )
           is
           Ehenabrole
           ,
           which
           comprehendeth
           all
           those
           Problems
           ,
           wherein
           two
           sides
           being
           given
           ,
           and
           an
           
             Angle
             betweene
          
           ,
           either
           a
           
             cathetopposite
             Angle
          
           ,
           or
           the
           third
           side
           is
           demanded
           .
           This
           Figure
           ,
           conforme
           to
           the
           two
           severall
           Quaesitas
           ,
           hath
           two
           Moods
           ,
           to
           wit
           ,
           Enerablo
           ,
           and
           Ennerable
           .
        
         
           The
           first
           Mood
           hereof
           ,
           Enerablo
           ,
           containeth
           all
           those
           obliquangularie
           questions
           ,
           wherein
           
             two
             sides
             with
             the
             Angle
             comprehended
             within
             them
             ,
             being
             proposed
             ,
             another
             Angle
             is
             required
             ,
          
           which
           Angle
           is
           alwayes
           one
           of
           the
           Cathetopposites
           or
           Angles
           at
           the
           Base
           ,
           that
           is
           ,
           either
           the
           complement
           to
           a
           Semicircle
           of
           the
           next
           
             Cathetopposite
             ,
             the
             prime
             Cathetopposite
             ,
             or
             the
             second
             Cocathetopposite
             :
          
           to
           the
           knowledge
           of
           all
           which
           ,
           that
           we
           may
           with
           facility
           attaine
           ,
           let
           us
           consider
           the
           generall
           Maxim
           of
           the
           Cathetothesis
           of
           this
           Mood
           ,
           which
           is
           Cafregpigeq
           that
           is
           to
           say
           ,
           that
           the
           Perpendicular
           in
           all
           the
           Cases
           of
           Enerablo
           falleth
           from
           that
           given
           side
           ,
           which
           is
           opposite
           to
           the
           Angle
           required
           ,
           upon
           the
           other
           given
           side
           ,
           continued
           ,
           if
           need
           be
           ;
           and
           according
           to
           the
           variety
           of
           the
           
           Angle
           at
           the
           Base
           which
           is
           the
           Angle
           sought
           for
           ,
           there
           be
           these
           three
           especiall
           Tenets
           of
           the
           generall
           Maxim
           of
           this
           Mood
           ,
           viz.
           
             Dacramfor
             ,
             Damracfor
          
           ,
           and
           Dasimquaein
           .
        
         
           Dacramfor
           ,
           which
           is
           the
           Tenet
           of
           the
           first
           Case
           ,
           sheweth
           ,
           that
           
             if
             the
             proposed
             Angle
             be
             sharp
             ,
             and
             the
             required
             flat
             ,
             the
             Perpendicular
             must
             fall
             outwardly
             .
          
        
         
           Damracfor
           ,
           the
           Tenet
           of
           the
           second
           Case
           ,
           signifieth
           ,
           that
           if
           a
           blunt
           ,
           or
           obtuse
           Angle
           be
           given
           ,
           and
           an
           acute
           or
           sharp
           demanded
           ,
           the
           demission
           of
           the
           Perpendicular
           must
           (
           as
           in
           the
           last
           )
           be
           externall
           .
        
         
           Lastly
           ,
           Dasimquaein
           ,
           the
           Tenet
           of
           the
           third
           Case
           ,
           sheweth
           ,
           that
           if
           the
           given
           ,
           and
           required
           Angles
           be
           of
           the
           same
           nature
           ,
           the
           Perpendicular
           must
           fall
           inwardly
           .
        
         
           Having
           thus
           unfolded
           all
           the
           intricacies
           in
           my
           Trissotetras
           of
           the
           Cathetothetick
           partition
           of
           this
           Mood
           ,
           I
           may
           ,
           without
           breaking
           order
           ,
           step
           back
           ,
           to
           explicate
           what
           is
           contained
           in
           the
           preceding
           partition
           ,
           and
           for
           the
           accomplishing
           of
           the
           first
           Orthogonosphericall
           work
           of
           this
           Mood
           ,
           consider
           what
           its
           Praenoscendas
           are
           ,
           and
           by
           what
           Datas
           they
           are
           to
           be
           obtained
           :
           but
           ,
           seeing
           both
           the
           Praenoscendas
           ,
           and
           the
           Datas
           ,
           together
           with
           the
           subservient
           ,
           and
           its
           Resolver
           ,
           with
           all
           the
           three
           Subdatoquaeres
           ;
           and
           in
           a
           word
           ,
           the
           whole
           contents
           of
           the
           first
           partition
           of
           this
           Mood
           of
           Enerablo
           ,
           is
           the
           same
           in
           all
           and
           every
           jot
           with
           the
           
             Praenoscendas
             ,
             Datas
             ,
             Subservient
             ,
             Resolver
             ,
          
           and
           Problemets
           ,
           contained
           in
           the
           first
           partition
           of
           the
           last
           
             Mood
             Ammanepreb
          
           ;
           I
           will
           not
           need
           to
           tell
           you
           any
           more
           ,
           then
           that
           (
           the
           Trissotetras
           it selfe
           (
           though
           otherwise
           short
           enough
           )
           shewing
           that
           Ubamen
           is
           the
           subservient
           to
           the
           
             Praenoscendas
             :
             Torp
             —
             Mu
             —
             Lag
             ☞
             Myr
             ,
          
           its
           Resolver
           :
           and
           
             Vtopaet
             ,
             Vdobaed
          
           ,
           and
           Vthophaeth
           ,
           the
           three
           Subproblems
           both
           of
           this
           and
           the
           next
           preceding
           Mood
           )
           you
           be
           pleased
           to
           have
           recourse
           to
           the
           glosse
           upon
           the
           last
           Mood
           ,
           where
           this
           matter
           is
           treated
           of
           at
           large
           ;
           to
           the
           which
           ,
           for
           avoyding
           of
           repetition
           ,
           I
           doe
           heartily
           recommend
           you
           .
        
         
           The
           first
           work
           being
           thus
           expedited
           ,
           we
           are
           to
           find
           out
           the
           Perpendicular
           by
           the
           second
           ,
           but
           so
           as
           that
           my
           direction
           to
           the
           Reader
           for
           the
           performance
           thereof
           shall
           detaine
           me
           no
           longer
           here
           ,
           then
           the
           time
           I
           am
           willing
           to
           bestow
           ,
           in
           telling
           him
           ,
           that
           the
           whole
           progresse
           of
           this
           operation
           ,
           as
           well
           as
           of
           the
           preceding
           ,
           
           is
           amply
           expressed
           in
           my
           comment
           on
           the
           last
           Mood
           ,
           from
           which
           ,
           what
           ere
           is
           written
           of
           the
           Subservient
           ,
           Ethaner
           ,
           its
           Resolver
           ,
           
             To
             —
             Tag
             —
             Se
             ☞
             Tyr
             ,
          
           or
           the
           under-problems
           ,
           
             Aetopca
             ,
             Aedobca
          
           ,
           and
           Aethophca
           ,
           thereby
           resolved
           ,
           may
           conveniently
           be
           transplaced
           hither
           ,
           and
           reseated
           there
           againe
           ,
           without
           any
           prejudice
           to
           either
           ;
           Ammanepreb
           being
           the
           onely
           Mood
           ,
           which
           with
           this
           of
           Enerablo
           hath
           a
           basal
           and
           opposite
           catheteuretick
           identity
           .
        
         
           The
           Perpendicular
           ,
           by
           these
           meanes
           being
           found
           out
           ,
           must
           be
           employed
           in
           the
           last
           work
           of
           this
           Mood
           ,
           to
           concurre
           with
           the
           second
           Basidion
           ,
           or
           little
           Base
           ,
           the
           second
           great
           Base
           ,
           and
           the
           second
           Co-base
           ,
           for
           obtaining
           of
           such
           Cathetopposites
           as
           are
           ,
           or
           usher
           the
           maine
           quaesitas
           ,
           which
           in
           the
           first
           Case
           is
           the
           complement
           of
           the
           fourth
           proportionall
           
             (
             viz.
          
           the
           next
           Cathetopposite
           )
           to
           a
           Semicircle
           ;
           in
           the
           second
           Case
           the
           prime
           Cathetopposite
           ,
           and
           in
           the
           third
           ,
           the
           second
           Cocathetopposite
           .
           For
           the
           perfecting
           of
           this
           operation
           ,
           Erelam
           is
           the
           Subservient
           ,
           by
           whose
           Resolver
           ,
           
             Sei
             —
             Teg
             —
             To
             ☞
             Tir
             ,
          
           we
           are
           instructed
           how
           to
           unfold
           its
           peculiar
           Problemets
           ,
           
             oedcathob
             ,
             oetcathop
          
           ,
           and
           oethcathops
           .
        
         
           All
           the
           three
           operations
           being
           thus
           singly
           accomplished
           ,
           according
           to
           our
           wonted
           manner
           ,
           the
           last
           two
           must
           be
           inchaced
           into
           one
           ,
           and
           therefore
           their
           Resolvers
           ,
           
             To
             —
             Tag
             —
             Se
             ☞
             Tyr
             ,
          
           and
           
             Sei
             —
             Teg
             —
             To
             ☞
             Tir
             ,
          
           must
           be
           untermed
           of
           some
           of
           their
           proportionals
           :
           the
           which
           ,
           that
           we
           may
           performe
           the
           more
           judicionsly
           ,
           let
           us
           consider
           what
           they
           signifie
           apart
           ;
           the
           first
           importeth
           (
           as
           in
           the
           last
           Mood
           I
           told
           you
           )
           that
           ,
           
             As
             the
             Radius
             is
             to
             the
             Tangent
             of
             one
             of
             the
             opposite
             Angles
             :
             so
             the
             Sine
             of
             one
             of
             the
             first
             Bases
             ,
             to
             the
             Tangent
             of
             the
             Perpendicular
             :
          
           the
           second
           soundeth
           ,
           
             As
             the
             Sine
             of
             one
             of
             the
             second
             Bases
             ,
             to
             the
             Tangent
             of
             the
             Perpendicular
             :
             so
             the
             Radius
             ,
             to
             the
             Tangent
             of
             an
             Angle
             ,
             which
             either
             ushers
             ,
             of
             is
             the
             Angle
             required
             .
          
        
         
           Hereby
           it
           is
           evident
           ,
           how
           the
           Radius
           is
           a
           multiplyer
           in
           the
           one
           ,
           and
           a
           divider
           in
           the
           other
           ,
           and
           that
           the
           Perpendicular
           ,
           which
           with
           the
           Radius
           is
           a
           multiplyer
           in
           the
           second
           row
           ,
           is
           in
           the
           power
           of
           the
           three
           first
           termes
           of
           the
           first
           row
           ,
           whereof
           the
           Radius
           is
           one
           ,
           by
           vertue
           of
           all
           which
           ,
           we
           must
           proceed
           just
           so
           with
           these
           last
           two
           operations
           here
           ,
           as
           we
           have
           already
           done
           with
           the
           two
           last
           of
           the
           Moods
           of
           
             Alamebna
             ,
             Allamebne
          
           ,
           and
           Ammanepreb
           ,
           and
           ejecting
           the
           Radius
           and
           Perpendicular
           out
           of
           both
           ,
           instead
           of
           
           
             To
             —
             Tag
             —
             Se
             ☞
             Tyr
          
           and
           
             Sei
             —
             Teg
             —
             To
             ☞
             Tir
             ,
          
           set
           downe
           
             Sei
             —
             Tag
             —
             Se
             ☞
             Tir
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             of
             one
             of
             the
             second
             Bases
             to
             the
             Tangent
             of
             one
             of
             the
             Cathetopposites
             :
             so
             is
             the
             Sine
             of
             one
             of
             the
             first
             Bases
             ,
             to
             the
             Tangent
             of
             one
             of
             the
             other
             Cathetopposites
             :
          
           which
           proposition
           comprehendeth
           to
           the
           full
           the
           last
           two
           operations
           ,
           and
           according
           to
           the
           three
           severall
           Cases
           of
           this
           Mood
           is
           to
           be
           individuated
           into
           so
           many
           finall
           Resolvers
           .
        
         
           The
           first
           thereof
           ,
           for
           Dacramfor
           ,
           is
           
             Soed
             —
             Top
             —
             Saet
             ☞
             Tob
             *
             Kir
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             of
             the
             second
             Basidion
             ,
             or
             little
             Base
             ,
             is
             to
             the
             Tangent
             of
             the
             prime
             Cathetopposite
             :
             so
             the
             Sine
             of
             the
             first
             ,
             and
             great
             Base
             ,
             to
             the
             Tangent
             of
             the
             next
             Cathetopposite
             ,
          
           whose
           complement
           to
           a
           Semicircle
           is
           the
           Angle
           required
           .
        
         
           The
           second
           finall
           Resolver
           ,
           is
           for
           Damracfor
           ,
           the
           Tenet
           of
           the
           second
           Case
           ,
           and
           is
           
             Soet
             —
             Tob
             —
             Saed
             ☞
             Top
             *
             Ir
             ,
          
           that
           is
           to
           say
           ,
           
             As
             the
             Sine
             of
             the
             second
             ,
             and
             great
             Base
             ,
             to
             the
             Tangent
             of
             the
             next
             Cathetopposite
             :
             so
             the
             Sine
             of
             the
             first
             Basidion
             ,
             to
             the
             Tangent
             of
             the
             prime
             opposite
             ,
          
           which
           is
           the
           Angle
           required
           .
        
         
           The
           third
           and
           last
           finall
           Resolver
           ,
           is
           for
           the
           third
           
             Case
             Dasimquaein
          
           ,
           and
           is
           couched
           thus
           ,
           
             Soeth
             —
             Toph
             —
             Saeth
             ☞
             Tops
             *
             Ir
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             of
             the
             second
             Co-base
             is
             to
             the
             Tangent
             of
             the
             first
             Cocathetopposite
             :
             so
             is
             the
             Sine
             of
             the
             first
             Co-base
             to
             the
             Tangent
             of
             the
             second
             Co-cathetopposite
             ,
          
           which
           is
           the
           Angle
           required
           .
        
         
           The
           fundamentall
           reason
           of
           all
           this
           ,
           is
           from
           the
           third
           Disergetick
           Axiome
           Siubprortab
           ,
           the
           second
           Determinater
           of
           whose
           Directory
           ,
           Ammena
           ,
           sheweth
           that
           the
           Mood
           of
           Enerablo
           ,
           in
           all
           the
           finall
           Resolvers
           thereof
           ,
           oweth
           the
           truth
           of
           its
           Analogy
           to
           the
           Maxim
           of
           Siubprortab
           ;
           because
           of
           the
           
             reciprocall
             proportion
             tha●
             is
             amongst
             its
             termes
             ,
             to
             be
             found
             betwixt
             the
             Sines
             of
             the
             basall
             segments
             and
             the
             Tangents
             of
             the
             Cathetopposite
             Angles
             .
          
        
         
           The
           second
           Mood
           of
           Ehenabrole
           is
           Ennerable
           ,
           which
           comprehendeth
           all
           those
           Obliquangulary
           Problems
           ,
           wherein
           two
           sides
           being
           given
           ,
           with
           an
           Angle
           intercepted
           therein
           ,
           the
           third
           side
           ▪
           demanded
           ,
           which
           side
           is
           alwayes
           one
           of
           the
           
             second
             Subtendent●
             ▪
          
           that
           is
           either
           the
           
             second
             Subtendentine
          
           ,
           the
           
             second
             Subtendentall
          
           ,
           〈◊〉
           the
           
             second
             Co-subtendent
          
           :
           to
           the
           notice
           of
           all
           which
           ,
           that
           we
           may
           the
           more
           easily
           attaine
           ,
           let
           us
           perpend
           the
           generall
           Maxim
           of
           the
           Cathetothesis
           of
           this
           
             Mood
             ,
             Cafregpaq
          
           the
           meaning
           whereof
           is
           ,
           that
           in
           this
           Mood
           ,
           whatever
           the
           Case
           be
           ,
           
             the
             Perpendicular
             may
             fall
          
           
           
             from
             the
             extremity
             of
             either
             of
             the
             given
             sides
             ,
             but
             must
             fall
             from
             one
             of
             them
             ,
             opposite
             to
             the
             Angle
             proposed
             ,
             and
             upon
             the
             other
             given
             side
             ,
             continued
             ,
             if
             need
             be
             .
          
        
         
           Here
           may
           the
           Reader
           be
           pleased
           to
           observe
           ,
           that
           the
           clause
           of
           the
           Perpendiculars
           falling
           opposite
           to
           the
           proposed
           Angle
           ,
           though
           it
           be
           onely
           mentioned
           in
           this
           place
           ,
           might
           have
           as
           well
           beene
           spoke
           of
           in
           any
           one
           of
           the
           rest
           of
           the
           Cathetothetick
           comments
           ;
           because
           it
           is
           a
           generall
           tie
           incumbent
           on
           the
           demission
           of
           Perpendiculars
           in
           all
           Loxogonosphericall
           Disergetick
           Figures
           ,
           whether
           Amblygonian
           or
           Oxygonian
           ,
           that
           
             it
             fall
             alwayes
             opposite
             to
             a
             knowne
             Angle
             ,
             and
             from
             the
             extremity
             of
             a
             knowne
             side
             .
          
        
         
           Of
           this
           generall
           Maxim
           ,
           Cafregpaq
           according
           to
           the
           variety
           of
           the
           second
           Subtendent
           ,
           which
           is
           the
           side
           required
           ,
           there
           be
           these
           three
           especiall
           Tenets
           ,
           
             Dacforamb
             ,
             Damforac
          
           ,
           and
           Dakinatam
           .
        
         
           Dacforamb
           ,
           the
           Tenet
           of
           the
           first
           Case
           ,
           giveth
           us
           to
           understand
           ,
           that
           if
           the
           given
           Angle
           be
           acute
           ,
           and
           that
           one
           onely
           of
           the
           other
           two
           be
           an
           obtuse
           Angle
           ,
           the
           Perpendicular
           falleth
           outwardly
           .
        
         
           Damforac
           ,
           the
           Tenet
           of
           the
           second
           Case
           ,
           signifieth
           ,
           that
           if
           the
           given
           Angle
           be
           obtuse
           ,
           and
           the
           other
           two
           acute
           ,
           that
           the
           demission
           of
           the
           Perpendicular
           is
           externall
           ,
           as
           in
           the
           first
           .
        
         
           Thirdly
           ,
           Dakinatam
           ,
           the
           Tenet
           of
           the
           third
           Case
           ,
           and
           variator
           of
           the
           first
           ,
           sheweth
           ,
           that
           if
           the
           proposed
           Angle
           be
           of
           the
           same
           affection
           with
           one
           of
           the
           other
           Angles
           of
           the
           Triangle
           ,
           as
           in
           the
           first
           Case
           ,
           the
           Perpendicular
           may
           fall
           inwardly
           .
        
         
           The
           Cathetology
           of
           this
           Mood
           being
           thus
           expeded
           ,
           the
           Pranoscendas
           thereof
           come
           next
           in
           hand
           to
           be
           discussed
           ,
           which
           are
           the
           first
           Bases
           ,
           whose
           subservient
           is
           Vbamen
           ,
           and
           its
           Resolver
           ,
           
             Torp
             —
             Mu
             —
             Lag
             ☞
             Myr
             ,
          
           upon
           which
           depend
           the
           three
           Subdatoquaeras
           of
           
             Vtopaet
             ,
             Vdobaed
          
           ,
           and
           Vthophaeth
           .
        
         
           Thus
           much
           I
           beleeve
           is
           expressed
           in
           the
           very
           Table
           of
           my
           Trissotetras
           ;
           and
           though
           a
           large
           explication
           might
           be
           with
           reason
           expected
           in
           this
           place
           ,
           of
           what
           is
           but
           summarily
           mentioned
           there
           ,
           yet
           because
           what
           concerneth
           this
           matter
           ,
           hath
           beene
           already
           treated
           of
           in
           the
           last
           two
           Moods
           of
           Enerablo
           ,
           and
           〈◊〉
           ,
           the
           whole
           discourse
           whereof
           may
           be
           as
           conveniently
           perused
           ,
           as
           if
           it
           were
           couched
           here
           ,
           I
           will
           not
           dull
           the
           Reader
           with
           tedious
           rehearsals
           of
           one
           and
           the
           same
           thing
           ,
           but
           ,
           letting
           passe
           the
           progresse
           
           of
           this
           first
           work
           ,
           with
           the
           manner
           of
           which
           (
           by
           my
           former
           instructions
           ,
           I
           suppose
           him
           sufficiently
           well
           acquainted
           )
           will
           proceed
           to
           the
           Cathetouretick
           operation
           of
           this
           Mood
           ,
           and
           perpend
           by
           what
           Datas
           the
           perpendicular
           is
           to
           be
           found
           out
           .
        
         
           To
           this
           effect
           ,
           the
           Praenoscendas
           of
           Ennerable
           ,
           to
           wit
           ,
           the
           
             first
             Basal
          
           ,
           the
           
             first
             Basidion
          
           ,
           and
           the
           
             first
             co-Base
          
           ,
           being
           by
           the
           last
           work
           already
           obtained
           ,
           must
           concurre
           with
           each
           its
           correspondent
           first
           subtendent
           ,
           viz.
           the
           first
           Subtendentall
           ,
           the
           first
           Subtendentine
           ,
           and
           the
           first
           co-Subtendent
           ,
           discernable
           in
           their
           severall
           natures
           ,
           by
           the
           figuratives
           of
           τ
           δ
           θ.
           for
           the
           perfecting
           of
           this
           second
           operation
           .
           The
           subservient
           of
           this
           work
           ,
           is
           Uch●ner
           ,
           by
           whose
           Resolver
           
             Neg
             —
             To
             —
             Nu
             ☞
             Nyr
             ,
          
           the
           three
           subproblems
           
             Utaeta
             ,
             Vtadca
          
           ,
           and
           Vthaethca
           ,
           are
           made
           manifest
           :
           by
           vertue
           whereof
           it
           is
           perceivable
           ,
           how
           the
           same
           quaesitum
           is
           attained
           unto
           by
           the
           Datas
           of
           three
           severall
           ,
           both
           first
           Subtendents
           ,
           and
           first
           Bases
           .
        
         
           The
           perpendicular
           being
           thus
           obtained
           ,
           must
           assist
           some
           other
           terme
           in
           the
           third
           operation
           ,
           for
           the
           finding
           out
           of
           the
           maine
           quaesitum
           ;
           which
           quaesitum
           ,
           though
           it
           be
           different
           from
           the
           finall
           one
           of
           the
           last
           Mood
           ,
           yet
           is
           the
           knowledge
           of
           them
           both
           attained
           unto
           ,
           by
           meanes
           of
           the
           same
           Datas
           ;
           the
           perpendicular
           ,
           and
           the
           three
           second
           Bases
           ,
           being
           ingredients
           in
           both
           .
        
         
           It
           being
           certaine
           then
           ,
           that
           the
           perpendicular
           must
           concurre
           in
           the
           last
           work
           of
           this
           Mood
           with
           the
           second
           Basidion
           ,
           the
           second
           Basal
           ,
           and
           second
           co-Base
           ,
           for
           obtaining
           the
           second
           Subtendentine
           ,
           the
           second
           Subtendentall
           ,
           and
           second
           
             co-Subtendent
             ;
             Enerul
          
           ,
           is
           made
           use
           of
           for
           their
           subservient
           ,
           by
           whose
           Resolver
           ,
           
             To
             —
             Neg
             —
             Ne
             ☞
             Nur
             ,
          
           we
           are
           raught
           how
           to
           deale
           with
           its
           subordinat
           Problems
           ,
           
             Catheudwd
             ,
             Cathatwt
          
           ,
           and
           Catheuthwth
           .
        
         
           All
           the
           three
           works
           being
           thus
           specified
           apart
           ,
           according
           to
           our
           accustomed
           Method
           ,
           we
           will
           declare
           what
           way
           the
           last
           two
           are
           to
           be
           joyned
           into
           one
           ;
           for
           the
           better
           effectuating
           whereof
           ,
           their
           Resolvers
           ,
           
             Neg
             —
             To
             —
             Nu
             —
             ☞
             Nyr
             :
          
           and
           
             To
             —
             Neg
             —
             Ne
             ☞
             Nur
             ,
          
           must
           be
           interpreted
           ;
           the
           first
           being
           ,
           
             As
             the
             sine
             complement
             of
             a
             first
             Base
             to
             the
             Radius
             :
             so
             the
             sine
             complement
             of
             a
             first
             subtendent
             ,
             to
             the
             sine
             complement
             of
             the
             perpendicular
             .
          
           And
           the
           second
           .
           
             As
             the
             Radius
             ,
             to
             the
             sine
             complement
             of
             a
             second
             Base
             :
             so
             the
             sine
             complement
             of
             the
             perpendicular
             to
             the
             sine
             complement
             of
             a
             second
             subtendent
             ,
          
           which
           is
           the
           side
           required
           .
        
         
         
           Now
           ,
           seeing
           a
           multiplier
           must
           be
           dashed
           against
           a
           divider
           ,
           being
           both
           quantified
           alike
           ,
           and
           that
           all
           unnecessary
           pestring
           of
           a
           work
           with
           superfluous
           ingredients
           is
           to
           be
           avoided
           ;
           we
           are
           to
           deale
           with
           the
           Radius
           ,
           and
           perpendicular
           in
           this
           place
           ,
           as
           formerly
           we
           have
           done
           in
           the
           Moods
           of
           
             Alamebna
             ,
             Allamebne
             ,
             Ammanepreb
          
           ,
           and
           Enerablo
           ,
           where
           we
           did
           eject
           them
           forth
           of
           both
           the
           orders
           of
           proportionalls
           ;
           and
           when
           we
           have
           done
           the
           like
           here
           ,
           instead
           of
           
             Neg
             —
             To
             —
             Nu
             ☞
             Nyr
             ,
          
           and
           
             To
             —
             Neg
             —
             Ne
             ☞
             Nur
             ,
          
           we
           may
           with
           the
           same
           efficacie
           say
           ,
           
             Neg
             —
             Nu
             —
             Ne
             ☞
             Nur
             ,
          
           that
           is
           ,
           
             As
             the
             sine
             complement
             of
             one
             side
             ,
             is
             to
             the
             sine
             complement
             of
             a
             subtendent
             :
             so
             the
             sine
             complement
             of
             another
             side
             ,
             to
             the
             sine
             complement
             of
             another
             subtendent
          
           ;
           or
           more
           determinatly
           ,
           
             As
             the
             sine
             complement
             of
             a
             first
             Base
             ,
             to
             the
             sine
             complement
             of
             a
             first
             subtendent
             :
             so
             the
             sine
             complement
             of
             a
             second
             Base
             ,
             to
             the
             sine
             complement
             of
             a
             second
             subtendent
             .
          
        
         
           This
           theorem
           comprehendeth
           to
           the
           full
           both
           the
           last
           operations
           ,
           and
           according
           to
           the
           number
           of
           the
           Cases
           of
           this
           Mood
           ,
           is
           particularized
           into
           three
           finall
           Resolvers
           ,
           the
           first
           whereof
           for
           the
           first
           
             Case
             ,
             Dacforamb
          
           ,
           is
           
             Naet
             —
             Nut
             —
             Noed
             ☞
             Nwd*yr
             ,
          
           that
           is
           ,
           
             As
             the
             sine
             complement
             of
             the
             first
             Basal
             ,
             or
             great
             Base
             to
             the
             sine
             complement
             of
             the
             first
             Subtendentall
             ,
             or
             great
             subtendent
             :
             so
             the
             sine
             complement
             of
             the
             second
             Basidion
             ,
             or
             little
             Base
             ,
             to
             the
             sine
             complement
             of
             the
             second
             subtendentine
             ,
             or
             little
             subtendent
             ,
          
           which
           is
           the
           side
           required
           .
        
         
           The
           second
           finall
           Resolver
           ,
           is
           for
           Damforac
           ,
           the
           second
           Case
           ,
           and
           is
           set
           downe
           thus
           ,
           
             Naed
             —
             Nud
             —
             Noet
             ☞
             Nwt*yr
             ,
          
           that
           is
           ,
           
             As
             the
             sine
             complement
             of
             the
             first
             Basidion
             ,
             to
             the
             sine
             complement
             of
             the
             first
             subtendentine
             :
             so
             the
             sine
             complement
             of
             the
             second
             Basal
             ,
             to
             the
             sine
             complement
             of
             the
             second
             subtendentall
             ,
          
           which
           is
           the
           side
           in
           this
           Case
           required
           .
        
         
           The
           third
           ,
           and
           last
           finall
           Resolver
           is
           for
           Dakinatamb
           ,
           and
           is
           expressed
           thus
           ,
           
             Naeth
             —
             Nuth
             —
             Noeth
             ☞
             Nwth*yr
             ,
          
           that
           is
           to
           say
           ,
           
             As
             the
             sine
             complement
             of
             the
             first
             co-base
             ,
             to
             the
             sine
             complement
             of
             the
             first
             co-subtendent
             :
             so
             the
             sine
             complement
             of
             the
             second
             co-base
             ,
             to
             the
             sine
             complement
             of
             the
             second
             co-subtendent
             ,
          
           which
           in
           the
           third
           Case
           is
           alwayes
           the
           side
           required
           .
        
         
           The
           reason
           of
           all
           this
           is
           proved
           out
           of
           the
           fourth
           ,
           and
           last
           disergetick
           Axiom
           ,
           Niubprodnesver
           ,
           whose
           directer
           Ennerra
           ,
           sheweth
           by
           its
           Determinater
           ,
           the
           syllable
           Enn
           ,
           that
           the
           Datoquaere
           of
           Ennerable
           ,
           
           is
           bound
           for
           the
           veritie
           of
           its
           proportion
           ,
           in
           all
           the
           finall
           Resolvers
           thereof
           ,
           to
           the
           maxime
           of
           Niubprodnesver
           ,
           because
           off
           the
           direct
           analogie
           that
           ,
           amongst
           its
           termes
           ,
           is
           to
           be
           seen
           betwixt
           
             the
             sines
             complements
             of
             the
             segments
             of
             the
             Base
             &
             the
             sines
             complements
             of
             the
             sides
             of
             the
             verticall
             Angles
          
           ;
           which
           in
           all
           this
           Treatise
           ,
           both
           for
           plainesse
           ,
           and
           brevity
           sake
           ,
           I
           have
           thought
           fit
           to
           call
           by
           the
           names
           of
           first
           and
           
             second
             Subtendents
          
           .
        
         
           The
           fourth
           and
           last
           Loxogonosphericall
           Disergetick
           figure
           ,
           and
           second
           of
           the
           Lateralls
           ,
           is
           Eherolabme
           ,
           which
           is
           of
           all
           those
           obliquangularie
           problems
           ,
           wherin
           two
           sides
           being
           given
           ,
           and
           an
           opposite
           Angle
           ,
           the
           interjacent
           Angle
           ,
           or
           one
           of
           the
           other
           sides
           is
           demanded
           ;
           and
           ,
           conforme
           to
           its
           two
           severall
           quaesitas
           ,
           hath
           two
           Moods
           ,
           viz.
           Erelomab
           ,
           and
           Errelome
           .
        
         
           The
           first
           Mood
           hereof
           Erelomab
           comprehendeth
           all
           those
           Loxogonosphericall
           Problems
           ,
           wherein
           two
           sides
           with
           an
           opposite
           Angle
           being
           proposed
           ,
           the
           Angle
           between
           is
           demanded
           ,
           which
           Angle
           is
           still
           one
           of
           the
           verticals
           ,
           that
           is
           ,
           the
           first
           verticall
           ,
           the
           second
           verticall
           ,
           or
           the
           double
           vertical
           :
           to
           the
           notice
           of
           all
           which
           ,
           that
           we
           may
           the
           more
           easily
           attain
           ,
           we
           must
           consider
           the
           general
           Maxim
           of
           the
           Cathetothesis
           of
           this
           Mood
           ,
           which
           is
           Cafriq
           the
           very
           same
           in
           name
           with
           the
           generall
           
             Cathetothetick
             Maxim
          
           of
           Amanepra
           ,
           and
           thus
           far
           agreeing
           with
           it
           ,
           that
           the
           Perpendicular
           in
           both
           must
           fal
           from
           the
           Angle
           required
           ,
           and
           upon
           the
           side
           opposite
           to
           that
           Angle
           ,
           increased
           if
           need
           be
           :
           but
           in
           this
           point
           different
           ,
           that
           in
           Amanepra
           ,
           the
           Perpendiculars
           demission
           is
           from
           the
           Angle
           required
           upon
           the
           opposite
           side
           ,
           conterminat
           with
           the
           two
           proposed
           Angles
           ,
           and
           in
           Erelomab
           ,
           it
           falleth
           from
           the
           required
           Angle
           ,
           upon
           the
           opposite
           side
           conterminat
           with
           the
           two
           proposed
           sides
           :
           and
           ,
           according
           to
           the
           variety
           of
           the
           fourth
           proportionall
           ,
           which
           ,
           in
           the
           Analogies
           to
           this
           Mood
           belonging
           ,
           ushers
           in
           the
           verticall
           required
           ,
           there
           be
           those
           three
           especiall
           Tenets
           of
           the
           generall
           Maxim
           of
           this
           Mood
           ,
           viz.
           
             Dacracforaur
             ,
             Damraeforeur
          
           ,
           and
           Dacrambatin
           .
        
         
           Dacracforaur
           ,
           which
           is
           the
           Tenet
           of
           the
           first
           Case
           ,
           sheweth
           ,
           that
           if
           the
           given
           and
           demanded
           Angles
           be
           acute
           ,
           and
           the
           third
           an
           obtuse
           Angle
           ,
           the
           Perpendicular
           falleth
           outwardly
           upon
           the
           third
           side
           ,
           and
           the
           required
           Angle
           is
           a
           first
           verticall
           .
        
         
           Dambracforeur
           ,
           the
           Tenet
           of
           the
           second
           Case
           ,
           importeth
           ,
           that
           if
           the
           proposed
           Angle
           be
           obtuse
           ,
           and
           an
           acute
           Angle
           required
           ,
           
           the
           third
           Angle
           being
           acute
           ,
           the
           Perpendicular
           must
           likewise
           in
           this
           Case
           fall
           outwardly
           upon
           the
           third
           side
           ,
           and
           the
           Angle
           demanded
           be
           a
           second
           verticall
           .
        
         
           Dacrambatin
           ,
           the
           Tenet
           of
           the
           third
           Case
           ,
           signifieth
           ,
           that
           if
           the
           proposed
           Angle
           be
           acute
           ,
           and
           an
           obtuse
           Angle
           required
           ,
           the
           Perpendicular
           falleth
           inwardly
           ,
           and
           the
           demanded
           Angle
           is
           a
           double
           verticall
           .
        
         
           I
           had
           almost
           forgot
           to
           tell
           you
           ,
           that
           Sindifora
           is
           the
           Plus-minus
           of
           this
           Mood
           ,
           whereby
           we
           are
           given
           to
           understand
           ,
           
             that
             the
             summe
             of
             the
             top
             Angles
             ,
             if
             the
             Perpendicular
             fall
             within
             ,
             and
             their
             difference
             ,
             if
             it
             fall
             without
             ,
             is
             the
             Angle
             required
             :
          
           and
           ,
           seeing
           it
           varieth
           neither
           in
           name
           ,
           nor
           interpretation
           from
           the
           mensurator
           of
           Amanepra
           (
           the
           diversity
           betwixt
           them
           being
           onely
           in
           this
           ,
           that
           the
           verticals
           there
           are
           invested
           with
           Sines
           ,
           and
           here
           with
           Sine
           complements
           )
           I
           must
           make
           bold
           to
           desire
           the
           Reader
           to
           look
           back
           to
           that
           place
           ,
           if
           he
           know
           not
           why
           it
           is
           that
           some
           Moods
           are
           Plus-minused
           ,
           and
           not
           others
           ;
           for
           there
           he
           will
           find
           that
           Sindiforation
           is
           meerly
           proper
           to
           those
           Cases
           ,
           in
           the
           Analogies
           whereof
           the
           fourth
           proportionall
           is
           not
           the
           maine
           quaesitum
           it selfe
           ,
           but
           the
           illaticious
           terme
           that
           brings
           it
           in
           .
        
         
           The
           Praenoscendas
           of
           the
           Mood
           ,
           or
           Quaesitas
           of
           the
           first
           operation
           ,
           falling
           next
           in
           order
           to
           be
           treated
           of
           ,
           it
           is
           fitting
           we
           perpend
           of
           what
           nature
           they
           be
           in
           this
           Mood
           of
           Erelomab
           ,
           that
           if
           they
           be
           different
           from
           those
           of
           other
           Moods
           ,
           we
           may
           ,
           according
           to
           our
           accustomed
           diligence
           ,
           formerly
           observed
           in
           the
           like
           occasions
           ,
           appropriate
           ,
           in
           this
           parcell
           of
           the
           comment
           to
           their
           explication
           ,
           for
           the
           Readers
           instruction
           ,
           the
           greater
           share
           of
           discourse
           ,
           the
           lesse
           that
           before
           in
           any
           part
           of
           this
           Tractar
           ,
           they
           have
           beene
           mentioned
           :
           But
           if
           it
           be
           so
           farre
           otherwise
           ,
           that
           for
           their
           coincidence
           with
           other
           proturgetick
           Quaesitas
           ,
           there
           can
           no
           materiall
           document
           concerning
           them
           be
           delivered
           here
           ,
           which
           hath
           not
           beene
           spoke
           of
           already
           in
           some
           one
           or
           other
           of
           our
           foregoing
           Datoquaeres
           ,
           it
           were
           but
           an
           unnecessary
           wasting
           of
           both
           time
           and
           paper
           to
           make
           repetition
           of
           that
           ,
           which
           in
           other
           places
           we
           have
           handled
           to
           the
           full
           ;
           and
           therefore
           ,
           seeing
           the
           Praenoscendas
           of
           this
           Mood
           ,
           to
           wit
           ,
           the
           double
           
             top
             Angle
          
           or
           verticall
           ,
           the
           first
           
             top
             Anglet
          
           or
           verticalin
           ,
           and
           the
           first
           Co-top-Angle
           ,
           or
           co-verticall
           ,
           together
           with
           the
           Datas
           ,
           whereby
           these
           are
           obtained
           ,
           viz.
           for
           the
           side
           ,
           the
           
           first
           subtendentall
           ,
           the
           first
           subtendentine
           ,
           and
           the
           first
           co-subtendent
           ,
           and
           for
           the
           Angle
           ,
           the
           prime
           Cathetopposite
           ,
           the
           next
           Cathetopposite
           ,
           and
           the
           first
           Co-cathetopposite
           ,
           and
           consequently
           the
           subservient
           Upalam
           ,
           its
           Resolver
           
             To
             —
             Tag
             —
             Nu
             ☞
             Mir
             ,
          
           and
           their
           three
           peculiar
           Problemets
           ,
           
             Vtopat
             ,
             Udobaud
          
           ,
           and
           Uthophauth
           ,
           are
           all
           and
           every
           one
           of
           them
           the
           same
           in
           this
           Mood
           of
           Erelomab
           ,
           that
           they
           were
           in
           the
           three
           preceding
           Moods
           of
           
             Alamebna
             ,
             Allamebne
          
           ,
           and
           Amanepra
           (
           for
           these
           are
           the
           foure
           Moods
           ,
           which
           have
           an
           Angulary
           praenoscendall
           identity
           )
           we
           will
           not
           need
           (
           I
           hope
           )
           to
           talk
           any
           more
           thereof
           in
           this
           place
           ,
           seeing
           what
           hath
           beene
           already
           said
           concerning
           that
           purpose
           ,
           will
           undoubtedly
           satisfie
           the
           desire
           of
           any
           industrious
           civill
           Reader
           .
        
         
           The
           praenoscendas
           of
           the
           Mood
           ,
           or
           the
           verticall
           Angle
           ,
           according
           to
           the
           nature
           of
           the
           Case
           ,
           being
           by
           the
           foresaid
           Datas
           thus
           obtained
           ,
           must
           needs
           concurre
           with
           each
           its
           correspondent
           first
           subtendent
           ,
           determined
           by
           the
           figuratives
           of
           τ.
           δ.
           θ
           for
           finding
           out
           of
           the
           Perpendicular
           ,
           of
           which
           work
           ,
           Ubamen
           being
           the
           subservient
           ,
           by
           whose
           
             Resolver
             Nag
             —
             Mu
             —
             Torp
             ☞
             Myr
             ,
          
           the
           sub-problems
           of
           
             Utatca
             ,
             Vdaudca
          
           ,
           and
           Vthauthca
           ,
           are
           made
           known
           ,
           if
           I
           utter
           any
           more
           of
           this
           purpose
           ,
           I
           must
           intrench
           upon
           what
           I
           spoke
           before
           in
           the
           second
           operation
           of
           Allaemebne
           ,
           it
           being
           the
           onely
           Mood
           which
           ,
           with
           this
           of
           Erelomab
           ,
           hath
           a
           verticall
           ,
           and
           subtendentine
           
             Catheteuretick
             identity
          
           .
        
         
           The
           second
           operation
           being
           thus
           accomplished
           ,
           the
           perpendicular
           ,
           which
           is
           alwayes
           an
           ingredient
           in
           the
           third
           work
           ,
           must
           joyne
           with
           one
           of
           the
           rere
           subtendents
           for
           obtaining
           of
           the
           illatitious
           terme
           of
           the
           maine
           quaesitum
           :
           or
           ,
           more
           particularly
           ,
           by
           the
           concurrence
           of
           the
           Perpendicular
           with
           the
           second
           subtendentine
           ,
           the
           second
           subtendentall
           ,
           and
           second
           Co-subtendent
           ,
           according
           to
           the
           variety
           of
           the
           Case
           ,
           we
           are
           to
           find
           out
           three
           verticals
           ,
           which
           ,
           by
           abstracting
           the
           first
           from
           another
           verticall
           ,
           then
           by
           abstracting
           another
           verticall
           from
           the
           second
           ,
           and
           lastly
           by
           adding
           the
           third
           verticall
           to
           another
           ,
           afford
           the
           summe
           ,
           and
           differences
           ,
           which
           are
           the
           required
           verticals
           .
        
         
           All
           this
           being
           fully
           set
           downe
           in
           my
           comment
           upon
           the
           Resolutory
           partition
           of
           Amanepra
           ,
           in
           which
           Mood
           the
           maine
           quaesitum
           is
           the
           same
           as
           here
           (
           though
           otherwise
           endowed
           )
           I
           need
           not
           any
           longer
           insist
           thereon
           .
        
         
         
           For
           the
           performance
           of
           this
           work
           ,
           Ukelamb
           is
           the
           subservient
           ,
           by
           whose
           Resolver
           
             Meg
             —
             To
             —
             Mu
             ☞
             Nir
             ,
          
           we
           are
           taught
           how
           to
           unfold
           the
           peculiar
           problemets
           of
           
             Wdcathaud
             ,
             Wicatha●
          
           ,
           and
           Wthcatheuth
           .
        
         
           All
           the
           three
           works
           being
           in
           this
           manner
           perfected
           ,
           according
           to
           our
           accustomed
           method
           ,
           we
           will
           shew
           unto
           you
           what
           way
           the
           last
           two
           are
           to
           be
           compacted
           in
           one
           :
           for
           the
           better
           expediting
           whereof
           ,
           their
           Resolvers
           
             Nag
             —
             Mu
             —
             To
             ☞
             Myr
             ,
          
           and
           
             Meg
             —
             To
             —
             Mu
             ☞
             Nir
             ,
          
           must
           be
           explained
           ,
           the
           first
           being
           ,
           
             As
             the
             Sine
             complement
             of
             an
             Angle
             ,
             to
             the
             Tangent
             complement
             of
             a
             subtendent
             :
             so
             the
             Radius
             ,
             to
             the
             Tangent
             complement
             of
             the
             side
             required
             :
          
           Or
           ,
           more
           particularly
           ,
           
             As
             the
             Sine
             complement
             of
             a
             verticall
             ,
             to
             the
             Tangent
             complement
             of
             a
             first
             subtendent
             :
             so
             the
             Radius
             ,
             to
             the
             Tangent
             complement
             of
             the
             Perpendicular
             :
          
           And
           the
           second
           Resolver
           being
           ,
           
             As
             the
             Tangent
             complement
             of
             a
             given
             side
             ,
             to
             the
             Radius
             :
             so
             the
             Tangent
             complement
             of
             a
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             a
             required
             Angle
             :
          
           Or
           ,
           more
           particularly
           ,
           
             As
             the
             Tangent
             complement
             of
             the
             Perpendicular
             ,
             to
             the
             Radius
             :
             so
             the
             Tangent
             complement
             of
             a
             first
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             a
             verticall
             ,
          
           which
           ushers
           the
           quaesitum
           .
        
         
           Now
           ,
           seeing
           it
           falleth
           forth
           ,
           that
           the
           Perpendicular
           ,
           which
           is
           the
           fourth
           terme
           in
           the
           first
           order
           of
           proportionals
           ,
           becometh
           first
           in
           the
           second
           row
           ;
           and
           that
           in
           such
           an
           exigent
           (
           as
           I
           proved
           already
           for
           illustration
           of
           the
           same
           point
           in
           the
           Mood
           of
           Amanepra
           )
           the
           multiplyers
           and
           dividers
           of
           the
           first
           row
           must
           interchange
           their
           roomes
           ,
           and
           consequently
           make
           the
           Radius
           ejectable
           ,
           without
           any
           prejudice
           or
           hindrance
           to
           the
           progresse
           of
           the
           Analogy
           ;
           and
           a
           place
           being
           left
           for
           the
           Perpendicular
           in
           both
           the
           rowes
           ,
           without
           taking
           the
           paines
           to
           find
           our
           its
           value
           ,
           because
           it
           is
           but
           a
           subordinate
           quaesitum
           for
           obtaining
           of
           the
           maine
           ,
           and
           lieth
           hid
           in
           the
           power
           of
           the
           three
           first
           proportionals
           ,
           instead
           of
           
             Nag
             —
             Mu
             —
             To
             ☞
             Myr
             ,
          
           and
           
             Meg
             —
             To
             —
             Mu
             ☞
             Nir
             ,
          
           we
           may
           ,
           with
           as
           much
           truth
           and
           energy
           ,
           say
           ,
           
             Mu
             —
             Nag
             —
             Mu
             ☞
             Nir
             ,
          
           that
           is
           ,
           
             As
             the
             Tangent
             complement
             of
             a
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             an
             Angle
             :
             so
             the
             Tangent
             complement
             of
             another
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             another
             Angle
             :
          
           Or
           ,
           more
           particularly
           ,
           
             As
             the
             Tangent
             complement
             of
             a
             first
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             a
             verticall
             :
             so
             the
             Tangent
             complement
             of
             a
             second
             subtendent
             ,
          
           
           
             to
             the
             Sine
             complement
             of
             a
             verticall
             illative
             to
             the
             quesitum
             .
          
        
         
           This
           proposition
           to
           the
           full
           containeth
           all
           that
           is
           in
           both
           the
           last
           operations
           ,
           and
           ,
           according
           to
           the
           number
           of
           the
           Cases
           of
           this
           Mood
           ,
           is
           specialized
           into
           so
           many
           
             finall
             Resolvers
          
           ;
           the
           first
           whereof
           ,
           for
           the
           first
           
             Case
             Dacracforaur
          
           ,
           is
           
             Mutnat
             —
             Mwd
             ☞
             Neud-fr*At*Aut*ir
          
           ,
           that
           is
           ,
           
             As
             the
             Tangent
             complement
             of
             the
             first
             subtendentall
             ,
             to
             the
             sine
             complement
             of
             the
             double
             verticall
             :
             so
             the
             tangent
             complement
             of
             the
             second
             Subtendentine
             ,
             to
             the
             sine
             complement
             of
             the
             second
             verticalin
             ,
          
           which
           subtracted
           from
           the
           
             double
             verticall
          
           ,
           leaves
           the
           
             first
             verticall
          
           for
           the
           Angle
           required
           .
        
         
           The
           second
           finall
           Resolver
           ,
           is
           for
           Damracforeur
           ,
           the
           second
           Case
           ,
           and
           is
           expressed
           thus
           ,
           
             Mud
             —
             Naud
             —
             Mwt
             ☞
             Natfr*Aud*Eut*ir
             ,
          
           that
           is
           ,
           
             As
             the
             tangent
             complement
             of
             the
             first
             Subtendentine
             ,
             to
             the
             sine
             complement
             of
             the
             first
             verticalin
             :
             so
             the
             tangent
             complement
             of
             the
             second
             Subtendentall
             ,
             to
             the
             sine
             complement
             of
             the
             double
             verticall
          
           ;
           from
           which
           if
           you
           deduce
           the
           first
           verticalin
           ,
           there
           will
           remaine
           the
           second
           verticall
           for
           the
           Angle
           required
           .
        
         
           The
           last
           
             finall
             Resolver
          
           is
           for
           the
           third
           
             Case
             ,
             Dacrambatin
          
           ,
           and
           is
           couched
           thus
           ,
           
             Muth
             —
             Nauth
             —
             Mwth
             ☞
             Neuth*jauth*ir
             ,
          
           that
           is
           ,
           
             As
             the
             tangent
             complement
             of
             the
             first
             co-subtendent
             ,
             to
             the
             sine
             complement
             of
             the
             first
             co-verticall
             :
             so
             the
             Tangent
             complement
             of
             the
             second
             co-subtendent
             ,
             to
             the
             sine
             complement
             of
             the
             second
             co-verticall
             ,
          
           which
           ,
           joyned
           to
           the
           first
           co-verticall
           ,
           affordeth
           the
           Angle
           required
           .
        
         
           The
           proofe
           of
           the
           veritie
           of
           all
           these
           Analogies
           ,
           is
           taken
           out
           of
           the
           second
           Disergetick
           Amblygonosphericall
           Axiome
           ,
           Naverprortes
           ,
           the
           second
           Determinater
           of
           whose
           Directorie
           sheweth
           ,
           that
           this
           Mood
           is
           one
           of
           its
           dependents
           ;
           and
           with
           reason
           ,
           because
           of
           the
           reciprocall
           Analogie
           ,
           that
           amongst
           its
           termes
           is
           perceivable
           betwixt
           the
           Tangents
           of
           the
           verticall
           sides
           ,
           which
           in
           this
           Mood
           are
           alwayes
           first
           subtendents
           ,
           and
           the
           sine-complements
           of
           the
           verticall
           Angles
           ;
           that
           is
           tosay
           (
           according
           to
           the
           literal
           meaning
           of
           my
           finall
           Resolvers
           of
           this
           Mood
           )
           the
           
             direct
             proportion
             that
             is
             betwixt
             the
             tangent-complements
             of
             the
             verticall
             sides
             ,
             or
             rere
             subtendents
             ,
             &
             the
             sine-complements
             of
             the
             vertical
             Angles
             ,
          
           for
           the
           proportion
           is
           the
           same
           with
           that
           ,
           wherof
           I
           have
           told
           you
           somewhat
           already
           in
           the
           Mood
           of
           Allamebme
           ,
           the
           fellow
           dependent
           of
           Erelomab
           .
        
         
           The
           second
           Mood
           of
           Eherolabme
           ,
           fourth
           of
           the
           Laterals
           ,
           
           eighth
           of
           the
           
             Sphericobliquangularie
             Disergeticks
          
           ,
           twelfth
           of
           the
           Loxogonosphericalls
           ,
           eight
           and
           twentieth
           of
           the
           Sphericals
           ,
           and
           one
           and
           fourtieth
           or
           last
           of
           the
           Triangulars
           ,
           is
           Errelome
           ,
           which
           comprehendeth
           all
           those
           obliquangularie
           Problems
           ,
           wherein
           two
           sides
           being
           given
           with
           an
           opposite
           Angle
           ,
           the
           third
           side
           is
           required
           ,
           which
           side
           is
           alwayes
           either
           one
           of
           the
           segments
           of
           the
           Base
           ,
           or
           the
           Base
           it selfe
           :
           to
           the
           knowledge
           of
           all
           which
           ,
           that
           we
           may
           reach
           with
           ease
           ,
           we
           must
           perpend
           the
           generall
           Maxim
           of
           the
           Cathetothesis
           of
           this
           Mood
           ,
           which
           is
           Cacurgyq
           that
           is
           to
           say
           ,
           
             the
             Perpendiculars
             demission
             ,
             in
             all
             the
             Cases
             of
             Errelome
             ,
             must
             be
             from
             the
             concurse
             of
             the
             given
             sides
             ,
             upon
             the
             side
             required
             ,
             continued
             ,
             if
             need
             be
             .
          
        
         
           The
           Plus-minus
           of
           this
           Mood
           is
           Sindiforiu
           ,
           which
           importeth
           ,
           that
           if
           the
           Perpendicular
           fall
           internally
           ,
           the
           summe
           of
           the
           segments
           of
           the
           Base
           ,
           or
           the
           totall
           Base
           ,
           is
           the
           side
           demanded
           :
           and
           if
           it
           fall
           without
           ,
           the
           difference
           of
           the
           Bases
           (
           the
           little
           Base
           ,
           being
           alwayes
           but
           a
           segment
           of
           the
           greater
           )
           is
           the
           maine
           quaesitum
           .
        
         
           The
           Mood
           of
           Ammanepreb
           is
           sindiforated
           in
           the
           same
           manner
           as
           this
           is
           ;
           because
           the
           maine
           Quaesitas
           ,
           and
           fourth
           proportionals
           of
           both
           doe
           in
           nothing
           differ
           ,
           but
           that
           those
           are
           sinused
           ,
           and
           these
           run
           upon
           sine-complements
           .
        
         
           The
           prosiliencie
           of
           the
           Perpendicular
           in
           all
           sphericall
           Disergeticks
           ,
           being
           so
           necessary
           to
           be
           knowne
           (
           as
           I
           have
           often
           told
           you
           )
           because
           of
           the
           facility
           thereby
           to
           reduce
           them
           to
           Rectangulary
           operations
           ,
           it
           falleth
           out
           most
           conveniently
           here
           ,
           according
           to
           the
           method
           proposed
           to
           my selfe
           ,
           to
           speak
           somewhat
           of
           the
           three
           severall
           Tenets
           of
           the
           Cathetothesis
           of
           this
           Mood
           ,
           and
           what
           is
           understood
           by
           
             Dakyxamfor
             ,
             Dambyxamfor
          
           ,
           and
           Dakypambin
           .
        
         
           Dakyxamfor
           ,
           which
           is
           the
           Tenet
           of
           the
           first
           Case
           ,
           declareth
           ,
           that
           if
           the
           proposed
           Angle
           be
           acute
           ,
           and
           the
           side
           required
           conterminate
           with
           an
           obtuse
           Angle
           ,
           the
           demission
           of
           the
           Perpendicular
           is
           extrinsecall
           .
        
         
           Dambyxamfor
           ,
           the
           Tenet
           of
           the
           second
           Case
           ,
           importeth
           ,
           that
           if
           the
           given
           Angle
           be
           obtuse
           ,
           and
           that
           the
           side
           required
           be
           annexed
           thereto
           ,
           the
           Perpendicular
           must
           ,
           as
           in
           the
           last
           ,
           fall
           outwardly
           .
        
         
           Thirdly
           ,
           Dakypambin
           ,
           the
           Tenet
           of
           the
           last
           Case
           ,
           signifieth
           ,
           that
           if
           the
           angle
           proposed
           be
           sharp
           ,
           &
           that
           the
           demanded
           side
           be
           subjacent
           
           to
           an
           obtuse
           or
           blunt
           Angle
           ,
           the
           Perpendicular
           falleth
           inwardly
           .
        
         
           Having
           thus
           proceeded
           in
           the
           enumeration
           of
           the
           Cathetothetick
           Tenets
           of
           this
           Mood
           ,
           according
           to
           the
           manner
           by
           me
           observed
           in
           those
           of
           all
           the
           former
           Disergeticks
           ,
           save
           the
           first
           ,
           I
           am
           confident
           the
           Reader
           (
           if
           he
           hath
           perused
           all
           the
           Tractat
           untill
           this
           place
           )
           will
           not
           think
           strange
           why
           ,
           Dakypambin
           being
           but
           the
           third
           ,
           I
           should
           call
           it
           the
           Tenet
           of
           the
           last
           Case
           of
           this
           Mood
           ;
           for
           though
           in
           Alamebna
           I
           spoke
           somewhat
           of
           every
           Amblygonosphericall
           Disergetick
           Moods
           generall
           Cathetothetick
           maximes
           division
           into
           foure
           especiall
           Tenets
           ,
           appropriable
           to
           so
           many
           severall
           Cases
           :
           yet
           the
           fourth
           Case
           ,
           viz.
           that
           wherein
           all
           the
           Angles
           are
           homogeneall
           ,
           whether
           blunt
           or
           sharp
           ,
           not
           being
           limited
           to
           any
           one
           Mood
           ,
           but
           adaequatly
           extended
           to
           all
           the
           eight
           ,
           it
           seemed
           to
           me
           more
           expedient
           to
           let
           its
           generality
           be
           known
           by
           mentioning
           it
           once
           or
           twice
           ,
           then
           (
           by
           doing
           no
           more
           in
           effect
           )
           to
           make
           superfluous
           repetitions
           ;
           and
           ,
           as
           in
           the
           first
           Disergetick
           Case
           ,
           for
           the
           Readers
           instruction
           ,
           I
           did
           under
           the
           name
           of
           Simomatin
           ,
           explicate
           the
           nature
           thereof
           :
           so
           ,
           for
           his
           better
           remembrance
           ,
           have
           I
           choosed
           rather
           to
           shut
           up
           my
           Cathetothetick
           comment
           with
           the
           same
           discourse
           wherewith
           I
           did
           begin
           it
           ,
           then
           unnecessarily
           to
           weary
           him
           with
           frequent
           reiterations
           ,
           and
           a
           tedious
           rehearsall
           of
           one
           and
           the
           same
           thing
           in
           all
           the
           six
           severall
           intermediat
           Moods
           .
        
         
           It
           is
           not
           amisse
           now
           ,
           that
           the
           perpendicularity
           of
           this
           Mood
           is
           discussed
           ,
           to
           consider
           what
           the
           praenoscendas
           thereof
           are
           ,
           or
           the
           Quaesitas
           of
           the
           first
           operation
           :
           but
           ,
           as
           I
           said
           in
           the
           last
           Mood
           ,
           that
           there
           is
           no
           need
           to
           insist
           so
           long
           upon
           the
           explication
           of
           those
           praenoscendas
           ,
           whereof
           ample
           relation
           hath
           beene
           already
           made
           in
           some
           of
           my
           Proturgetick
           comments
           ,
           as
           upon
           those
           others
           ,
           which
           ,
           for
           being
           altogether
           different
           from
           such
           as
           have
           beene
           formerly
           mentioned
           ,
           claim
           (
           by
           the
           law
           of
           parity
           ,
           in
           their
           imparity
           )
           right
           to
           a
           large
           discourse
           apart
           ,
           I
           will
           confine
           my
           pen
           upon
           this
           subject
           ,
           within
           those
           prescribed
           bounds
           ,
           and
           seeing
           the
           
             first
             Basal
          
           ,
           the
           
             first
             Basidion
          
           ,
           and
           
             first
             Co-base
          
           ,
           together
           with
           the
           Datas
           ,
           whereby
           they
           are
           found
           out
           ,
           viz.
           for
           the
           side
           ,
           the
           
             first
             subtendentall
          
           ,
           the
           
             first
             subtendentine
          
           ,
           and
           the
           
             first
             co
             subtendent
          
           ;
           and
           for
           the
           Angle
           the
           prime
           Cathetopposite
           ,
           the
           
             next
             Cathetopposite
          
           ,
           and
           the
           
             first
             Co-cathetopposite
          
           (
           the
           Datas
           being
           both
           for
           side
           and
           Angle
           
           the
           same
           here
           ,
           that
           they
           were
           in
           the
           former
           Mood
           )
           then
           the
           Subservient
           Ubamen
           ,
           and
           its
           Resolver
           
             Torp
             —
             Mu
             —
             Lag
          
           ☞
           Myr
           ,
           with
           the
           three
           peculiar
           Problemets
           thereto
           belonging
           ,
           
             Utopat
             ,
             Vdobaed
          
           ,
           and
           Vthophaeth
           ,
           are
           all
           and
           every
           one
           of
           them
           the
           same
           in
           this
           Mood
           of
           Errelome
           ,
           that
           they
           were
           in
           the
           three
           foregoing
           Moods
           of
           
             Ammanepreb
             ,
             Enerablo
          
           ,
           and
           Ennerable
           ,
           these
           being
           the
           onely
           foure
           Moods
           which
           have
           a
           laterall
           praenoscendal
           identity
           ,
           the
           Reader
           will
           not
           (
           in
           my
           opinion
           )
           be
           so
           prodigall
           of
           his
           owne
           labour
           ,
           nor
           covetous
           of
           mine
           ,
           that
           either
           he
           would
           put
           himselfe
           ,
           or
           me
           to
           any
           further
           paines
           ,
           then
           have
           beene
           already
           bestowed
           upon
           this
           matter
           by
           my selfe
           for
           his
           instruction
           ;
           and
           therefore
           ,
           leaving
           it
           for
           a
           supposed
           certainty
           ,
           that
           the
           Praenoscendas
           ,
           or
           first
           Bases
           (
           according
           to
           the
           nature
           of
           the
           Case
           )
           cannot
           escape
           the
           Readers
           knowledge
           ,
           by
           what
           hath
           beene
           by
           me
           delivered
           of
           them
           ;
           I
           purpose
           here
           to
           give
           him
           notice
           ,
           that
           these
           foresaid
           
             first
             Bases
          
           must
           concurre
           with
           each
           its
           correspondent
           
             first
             Subtendent
          
           ,
           to
           wit
           ,
           the
           
             first
             subtendentall
          
           ,
           the
           
             first
             subtendentine
          
           ,
           and
           
             first
             co-subtendent
          
           ,
           dignoscible
           by
           the
           Characteristicks
           of
           τ.
           δ.
           θ
           for
           obtaining
           of
           the
           Perpendicular
           ,
           of
           which
           operation
           ,
           Vchener
           being
           the
           Subservient
           ,
           by
           whose
           Resolver
           
             Neg
             —
             To
             —
             Nu
          
           ☞
           Nyr
           ,
           the
           Problemets
           of
           
             Utaetca
             ,
             Udaedca
          
           ,
           and
           Uthaethca
           ,
           are
           made
           manifest
           ,
           as
           to
           the
           same
           effect
           it
           remaines
           couched
           in
           my
           comment
           upon
           Ennerable
           ,
           which
           is
           the
           onely
           Mood
           ,
           that
           ,
           with
           this
           of
           Errelome
           ,
           hath
           a
           subtendentine
           and
           
             Basal
             Catheteuretick
          
           identity
           .
        
         
           The
           second
           work
           being
           thus
           perfected
           ,
           the
           perpendicular
           ,
           thereby
           found
           out
           ,
           is
           to
           assist
           one
           of
           the
           rere
           subtendents
           ,
           in
           obtaining
           the
           illatitious
           terme
           of
           the
           maine
           quasitum
           ,
           correspondent
           thereto
           ,
           discernable
           by
           the
           Characteristicks
           or
           Figuratives
           of
           δ.
           τ.
           θ
           or
           ,
           more
           plainly
           to
           expresse
           it
           ,
           the
           Perpendicular
           must
           concurre
           (
           according
           as
           the
           Case
           requires
           it
           )
           with
           the
           
             second
             subtendentine
          
           ,
           the
           
             second
             subtendentall
          
           ,
           and
           
             second
             co-subtendent
          
           ,
           (
           as
           you
           may
           see
           in
           the
           last
           Mood
           ,
           the
           Datas
           of
           the
           Resolutory
           partition
           whereof
           are
           the
           same
           as
           here
           )
           to
           find
           out
           three
           Bases
           ,
           which
           ,
           by
           abstracting
           the
           first
           from
           another
           Base
           ,
           then
           by
           abstracting
           another
           Base
           from
           the
           second
           ;
           and
           lastly
           ,
           by
           adding
           the
           third
           Base
           to
           another
           ,
           afford
           the
           summe
           and
           differences
           ,
           which
           are
           the
           required
           Bases
           .
        
         
           For
           the
           performance
           of
           this
           operation
           ,
           the
           same
           Subservient
           
           and
           Resolver
           suffice
           ,
           which
           served
           for
           the
           last
           :
           so
           that
           
             Uchener
             subserveth
          
           it
           ,
           by
           whose
           Resolver
           
             Neg
             —
             To
             —
             Nu
          
           ☞
           Nyr
           ,
           we
           are
           instructed
           how
           to
           explicate
           the
           Subdatoquaeres
           of
           
             Wdcathoed
             ,
             Wtcathoet
          
           ,
           and
           Wthcathoeth
           ,
           or
           more
           orderly
           
             Cathwdoed
             ,
             Cathwtoet
          
           ,
           and
           Cathwthoeth
           .
        
         
           All
           the
           three
           works
           being
           thus
           accomplished
           ,
           the
           manner
           of
           conflating
           the
           last
           two
           in
           one
           rests
           to
           be
           treated
           of
           ;
           for
           the
           better
           perfecting
           of
           which
           designe
           ,
           the
           two
           Resolvers
           ,
           or
           the
           same
           in
           its
           greatest
           generality
           doubled
           ,
           
             viz.
             Neg
             —
             To
             —
             Nu
          
           ☞
           Nyr
           ,
           and
           
             Neg
             —
             To
             —
             Nu
          
           ☞
           Nyr
           ,
           must
           be
           interpreted
           :
           The
           truth
           is
           ,
           both
           of
           them
           ,
           as
           they
           sound
           in
           their
           vastest
           extent
           of
           signification
           ,
           expresse
           the
           same
           Analogy
           ,
           without
           any
           difference
           ,
           which
           is
           ,
           
             As
             the
             Sine
             complement
             of
             a
             given
             side
             ,
             to
             the
             Radius
             :
             so
             the
             Sine
             complement
             of
             a
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             another
             side
             :
          
           but
           when
           more
           contractedly
           ,
           according
           to
           the
           specification
           of
           the
           side
           ,
           they
           doe
           suppone
           severally
           ,
           they
           should
           be
           thus
           expounded
           ;
           the
           first
           ,
           
             As
             the
             Sine
             complement
             of
             a
             first
             Base
             ,
             to
             the
             totall
             Sine
             :
             so
             the
             Sine
             complement
             of
             a
             first
             subtendent
             to
             the
             Sine
             complement
             of
             the
             Perpendicular
             :
          
           and
           the
           second
           ,
           
             As
             the
             Sine
             complement
             of
             the
             Perpendicular
             ,
             to
             the
             totall
             Sine
             :
             so
             the
             Sine
             complement
             of
             a
             second
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             a
             second
             Base
             ,
          
           which
           ushers
           the
           main
           quaesitum
           .
        
         
           Now
           ,
           the
           Perpendicular
           ,
           and
           Radius
           ,
           being
           both
           to
           be
           expelled
           these
           two
           foresaid
           orders
           of
           proportionall
           termes
           ,
           for
           the
           reasons
           which
           ,
           in
           the
           last
           preceding
           Mood
           ,
           and
           some
           others
           before
           it
           ,
           I
           have
           already
           mentioned
           ,
           and
           which
           to
           repeat
           (
           further
           then
           that
           the
           sympathy
           of
           this
           place
           with
           that
           may
           be
           manifested
           in
           the
           tranf-seating
           of
           multiplyers
           and
           dividers
           ,
           occasioned
           by
           the
           fourth
           terme
           in
           the
           first
           rowes
           ,
           being
           first
           in
           the
           second
           )
           is
           altogether
           unnecessary
           :
           in
           lieu
           of
           
             Neg
             —
             To
             —
             Nu
          
           ☞
           Nyr
           ,
           and
           
             Neg
             —
             To
             —
             Nu
          
           ☞
           Nyr
           ,
           we
           may
           say
           ,
           with
           as
           much
           truth
           ,
           power
           ,
           and
           efficacie
           ,
           and
           farre
           more
           compendiously
           ,
           
             Nu
             —
             Ne
             —
             Nu
          
           ☞
           Nyr
           ,
           that
           is
           ,
           
             As
             the
             Sine
             complement
             of
             a
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             a
             side
             :
             so
             the
             sine
             complement
             of
             another
             Subtendent
             ,
             to
             the
             Sine
             complement
             of
             another
             side
             :
          
           Or
           ,
           more
           particularly
           ,
           and
           appliably
           to
           the
           present
           Analogy
           ,
           
             As
             the
             Sine
             complement
             of
             a
             first
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             a
             first
             Base
             :
             so
             the
             Sine
             complement
             of
             a
             second
             subtendent
             ,
             to
             the
             Sine
             complement
          
           
           
             of
             a
             second
             Base
          
           ,
           illative
           to
           the
           quaesitum
           .
        
         
           This
           theorem
           ,
           or
           proposition
           ,
           comprehendeth
           in
           every
           point
           all
           that
           is
           in
           the
           two
           last
           operations
           ,
           and
           ,
           not
           transcending
           the
           number
           of
           the
           Cases
           of
           this
           Mood
           ,
           is
           divided
           into
           so
           many
           finall
           Resolvers
           ;
           the
           first
           whereof
           for
           the
           first
           
             Case
             ,
             Dakyxamfor
          
           is
           ,
           
             Nut
             —
             Naet
             —
             Nwd
             ☞
             Noedfr*Aet*
             Dyr
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             complement
             of
             the
             first
             subtendent
             all
             ,
             to
             the
             Sine
             complement
             of
             the
             first
             Basall
             ▪
             so
             the
             Sine
             complement
             of
             the
             second
             subtendentine
             ,
             to
             the
             Sine
             complement
             of
             the
             second
             Base
          
           ;
           which
           subducted
           from
           the
           first
           Basal
           ,
           residuats
           the
           segment
           that
           is
           the
           side
           required
           .
        
         
           The
           second
           finall
           Resolver
           of
           this
           Mood
           ,
           and
           that
           which
           is
           for
           the
           second
           Case
           thereof
           ,
           Dambyxamfor
           ,
           is
           
             Nud
             —
             Naed
             —
             Nwt
             ☞
             Noetfr*Aed*
             Dyr
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             complement
             of
             the
             first
             subtendentine
             ,
             to
             the
             Sine
             complement
             of
             the
             first
             Basidion
             :
             so
             the
             Sine
             complement
             of
             the
             second
             subtendentall
             ,
             to
             the
             Sine
             complement
             of
             the
             second
             Basall
          
           ;
           which
           ,
           the
           first
           Basidion
           being
           subtracted
           from
           it
           ,
           leaves
           ,
           for
           Remainder
           ,
           or
           difference
           ,
           that
           segment
           of
           the
           Base
           ,
           which
           is
           the
           side
           demanded
           .
        
         
           The
           last
           finall
           Resolver
           of
           this
           Mood
           (
           belonging
           to
           the
           third
           
             Case
             ,
             Dakypambin
          
           ,
           as
           also
           to
           the
           fourth
           ,
           Simomatin
           ,
           (
           if
           what
           we
           have
           already
           spoke
           of
           that
           matter
           will
           permit
           us
           to
           call
           it
           the
           fourth
           )
           for
           Simomatin
           ,
           together
           with
           the
           third
           Case
           of
           every
           Mood
           ,
           is
           still
           resolved
           by
           the
           last
           finall
           Resolver
           thereof
           )
           is
           
             Nuth
             —
             Naeth
             —
             Nwth
             ☞
             Noethj*Aeth*
             Syr
             ,
          
           that
           is
           ,
           
             As
             the
             Sine
             complement
             of
             the
             first
             co-subtendent
             ,
             to
             the
             Sine
             complement
             of
             the
             first
             Co-base
             :
             so
             the
             Sine
             complement
             ,
             of
             the
             second
             Co-subtendent
             ,
             or
             alterne
             subtendent
             ,
             to
             the
             Sine
             complement
             of
             the
             second
             Co-base
             or
             alterne
             Base
          
           ;
           which
           added
           to
           the
           first
           Co-base
           ,
           summes
           an
           Aggregat
           of
           subjacent
           sides
           ,
           which
           is
           the
           totall
           Base
           ,
           or
           side
           required
           .
        
         
           The
           fundamentall
           ground
           of
           the
           truth
           of
           these
           Analogies
           ,
           is
           in
           the
           fourth
           and
           last
           Amblygonosphericall
           Axiome
           ,
           Niubprodnesver
           ;
           (
           as
           we
           are
           made
           to
           understand
           by
           the
           second
           determinater
           of
           its
           Directory
           Ennerra
           )
           for
           by
           the
           direct
           proportion
           that
           ,
           amongst
           the
           terms
           thereof
           ,
           is
           visible
           ,
           
             (
             viz.
          
           betwixt
           the
           Sines
           complements
           of
           the
           subtendents
           ,
           or
           Sides
           of
           the
           verticall
           Angles
           ,
           and
           the
           segments
           of
           the
           Bases
           ,
           and
           inversedly
           )
           it
           is
           apparent
           ,
           that
           this
           Mood
           doth
           no
           lesse
           firmely
           depend
           upon
           it
           ,
           then
           that
           of
           Ennerable
           formerly
           explained
           .
        
         
         
           Now
           ,
           with
           reason
           doe
           I
           conjecture
           ,
           that
           ,
           without
           disappointing
           the
           Reader
           of
           his
           expectation
           ,
           I
           may
           here
           securely
           make
           an
           end
           of
           this
           Trigonometricall
           Treatise
           ;
           because
           of
           that
           Trissotetrall
           Table
           ,
           which
           comprehendeth
           all
           the
           Mysteries
           ,
           Axiomes
           ,
           Principles
           ,
           Analogies
           ,
           and
           Precepts
           of
           the
           Science
           of
           Triangular
           Calculations
           ,
           I
           have
           omitted
           no
           materiall
           point
           unexplained
           :
           yet
           seeing
           ,
           for
           avoyding
           of
           prolixity
           ,
           I
           was
           pleased
           in
           my
           comment
           upon
           the
           eighth
           Loxogonospherical
           Disergeticks
           ,
           barely
           to
           expresse
           in
           their
           finall
           Resolvers
           ,
           the
           Analogie
           of
           the
           termes
           ,
           without
           putting
           my selfe
           to
           the
           paines
           I
           took
           in
           my
           Sphericorectangulars
           ,
           how
           to
           order
           the
           Logarithms
           ,
           ,
           and
           Antilogarithms
           of
           the
           proportionalls
           ,
           ,
           for
           obtaining
           of
           the
           maine
           Quaesitas
           ,
           and
           that
           by
           having
           to
           the
           full
           explicated
           the
           variety
           of
           the
           proportions
           of
           the
           foresaid
           Moods
           ,
           and
           upon
           what
           severall
           Axiomes
           they
           doe
           depend
           ,
           thereby
           making
           the
           way
           more
           pervious
           ,
           thorough
           Logarithmicall
           difficulties
           ,
           for
           the
           Readers
           understanding
           ,
           I
           deliberatly
           proposed
           to
           my selfe
           this
           method
           at
           first
           ,
           and
           chose
           ,
           rather
           then
           dispersedly
           to
           treat
           of
           those
           things
           in
           the
           glosse
           (
           where
           ,
           by
           reason
           of
           the
           disturbed
           order
           ,
           the
           correspondencie
           or
           reference
           to
           one
           another
           of
           these
           Sphericobliquangulary
           Datoquaeres
           ,
           could
           not
           by
           any
           meanes
           have
           beene
           so
           conceivable
           )
           to
           summon
           their
           appearance
           to
           the
           Catastrophe
           of
           this
           Tractat
           ,
           that
           ,
           having
           them
           all
           in
           a
           front
           before
           us
           ,
           we
           may
           the
           more
           easily
           judge
           of
           the
           semblance
           ,
           or
           dissimilitude
           of
           their
           proportionalities
           ,
           ,
           and
           what
           affinity
           ,
           or
           relation
           ,
           whether
           of
           parity
           or
           imparity
           is
           amongst
           their
           respective
           proportionall
           terms
           :
           all
           which
           ,
           both
           for
           intelligibility
           and
           memory
           ,
           are
           quicklier
           apprehended
           ,
           and
           longer
           retained
           ,
           by
           being
           accumulatively
           reserved
           to
           this
           place
           ,
           then
           if
           they
           had
           beene
           each
           in
           its
           proper
           cell
           (
           though
           never
           so
           amply
           )
           discoursed
           upon
           apart
           .
        
         
           Here
           therfore
           ,
           that
           the
           Reader
           may
           take
           a
           generall
           view
           at
           once
           of
           all
           the
           Disergetick
           Amblygonosphericall
           analogised
           ingredients
           ,
           ready
           for
           Logarithmication
           ,
           I
           have
           thought
           fit
           to
           set
           downe
           a
           List
           of
           all
           the
           eight
           forenamed
           Moods
           ,
           together
           with
           the
           Finall
           Resolvers
           ,
           in
           their
           amplest
           extent
           thereto
           belonging
           ,
           in
           the
           manner
           as
           followeth
           .
        
         
         
           
             
               
                 Alamebna
                 .
              
               
                 
                   Say-Nag-Sa
                   ☞
                   Nir
                
              
            
             
               
                 Allamebne
                 .
              
               
                 
                   Nag-Mu-Na
                   ☞
                   Mur
                
              
            
             
               
                 Amanepra
                 .
              
               
                 
                   Na-Say-Nag
                   ☞
                   Sir
                
              
            
             
               
                 Ammanepreb
                 .
              
               
                 
                   Ta-Tag-Se
                   ☞
                   Syr.
                
                 
              
            
             
               
                 Enerablo
                 .
              
               
                 
                   Sei-Tag-Seg
                   ☞
                   Tir
                
              
            
             
               
                 Ennerable
                 .
              
               
                 
                   Neg-Nu-Ne
                   ☞
                   Nur
                
              
            
             
               
                 Erelomab
                 .
              
               
                 
                   Mu-Nag-Mu
                   ☞
                   Nir
                
              
            
             
               
                 Errelome
                 .
              
               
                 
                   Nu-Ne-Nu
                   ☞
                   Nyr
                
                 .
              
            
          
        
         
           These
           being
           the
           eight
           Disergeticks
           ,
           attended
           by
           their
           Adaequat
           finall
           Resolvers
           ,
           it
           is
           not
           amisse
           ,
           that
           we
           examine
           them
           all
           one
           after
           another
           ,
           and
           shew
           the
           Reader
           how
           ,
           with
           the
           help
           of
           a
           convenient
           Logarithmicall
           Canon
           ,
           he
           may
           easily
           out
           of
           the
           Analogie
           of
           the
           three
           first
           termes
           of
           each
           of
           them
           ,
           frame
           a
           computation
           apt
           for
           the
           finding
           out
           of
           a
           fourth
           proportionall
           ,
           to
           every
           severall
           ternarie
           correspondent
           :
           and
           so
           in
           order
           ,
           beginning
           at
           the
           first
           ,
           we
           will
           deale
           with
           
             Say
             —
             Nag
             —
             Sa
             ☞
             Nir
          
           (
           which
           is
           the
           Adaequat
           finall
           Resolver
           of
           Alamebna
           ,
           and
           composed
           (
           as
           it
           is
           appropriated
           to
           the
           first
           Mood
           of
           the
           Disergeticks
           )
           of
           the
           Sines
           of
           verticals
           ,
           and
           the
           Anti-sines
           of
           Cathetopposites
           )
           and
           so
           proceed
           therein
           ,
           that
           by
           adding
           to
           the
           summe
           of
           the
           Sine
           of
           a
           verticall
           ,
           and
           Co-sine
           of
           a
           Cathetopposite
           ,
           the
           Arithmetical
           complement
           of
           the
           Sine
           of
           another
           verticall
           ,
           we
           will
           be
           sure
           (
           cutting
           off
           the
           supernumerary
           digit
           or
           digits
           towards
           the
           left
           )
           to
           obtaine
           the
           Co-sine
           of
           the
           Cathetopposite
           required
           ,
           which
           Cathetopposites
           and
           verticals
           are
           particularised
           according
           to
           the
           Cases
           of
           the
           Mood
           .
        
         
           The
           second
           is
           ,
           
             Nag
             —
             Mu
             —
             Na
             ☞
             Mur
             ,
          
           which
           ,
           running
           upon
           the
           Anti-sines
           of
           verticals
           ,
           and
           the
           Co-tangents
           of
           
             subtendent
             sides
          
           ,
           sheweth
           ,
           that
           if
           to
           the
           Aggregat
           of
           a
           first
           hypotenusall
           Co-tangent
           ,
           and
           verticall
           Anti-sine
           ,
           we
           joyne
           the
           Arithmeticall
           complement
           of
           the
           Anti-sine
           of
           another
           verticall
           ,
           (
           observing
           the
           usuall
           presection
           )
           we
           cannot
           misse
           of
           the
           Co-tangent
           of
           the
           second
           subtendent
           side
           required
           ,
           which
           both
           second
           ,
           and
           
             first
             subtendents
          
           have
           their
           peculiar
           denominations
           ,
           according
           to
           the
           Cases
           of
           the
           Mood
           .
        
         
           The
           third
           Resolver
           is
           ,
           
             Na
             —
             Say
             —
             Nag
             ☞
             Sir
             ,
          
           which
           ,
           being
           nothing
           else
           but
           the
           first
           inverted
           ,
           runneth
           the
           same
           very
           way
           upon
           the
           Anti-sines
           of
           Cathetopposites
           ,
           and
           sines
           of
           verticals
           :
           and
           therefore
           doth
           the
           unradiused
           summe
           of
           the
           Anti-sine
           of
           a
           Cathetopposite
           ,
           the
           sine
           of
           a
           verticall
           ,
           and
           the
           Arithmeticall
           complement
           of
           the
           Anti-sine
           of
           another
           Cathetopposite
           ,
           afford
           the
           sine
           of
           the
           verticall
           ,
           illatitious
           to
           the
           Angle
           required
           ;
           which
           
           verticals
           and
           Cathetopposites
           are
           particularised
           according
           to
           the
           variety
           of
           the
           Cases
           of
           this
           
             Sindiforating
             Mood
          
           .
        
         
           The
           fourth
           generall
           Resolver
           is
           
             Ta
             —
             Tag
             —
             Se
             ☞
             Syr
             ,
          
           which
           ,
           coursing
           on
           the
           Tangents
           of
           all
           the
           Cathetopposites
           ,
           and
           sines
           of
           all
           the
           Bases
           ,
           evidenceth
           ,
           that
           the
           summe
           of
           the
           Tangent
           of
           a
           Cathetopposite
           ,
           and
           sine
           of
           a
           first
           Base
           ,
           added
           to
           the
           Arithmeticall
           complement
           of
           the
           Tangent
           of
           another
           Cathetopposite
           (
           unradiated
           )
           is
           the
           sine
           of
           the
           second
           Base
           ,
           illative
           to
           the
           segment
           required
           ;
           which
           Bases
           (
           both
           first
           and
           second
           )
           and
           Cathetopposites
           ,
           are
           specialised
           conform
           to
           the
           Cases
           of
           this
           
             Sindiforiuting
             Mood
          
           .
        
         
           The
           fifth
           Resolver
           is
           ,
           
             Sei
             —
             Tag
             —
             Se
             ☞
             Tir
             ,
          
           which
           ,
           composed
           of
           the
           sines
           of
           the
           second
           and
           first
           Bases
           ,
           and
           the
           Tangents
           of
           Cathetopposites
           ,
           giveth
           us
           to
           know
           ,
           that
           if
           to
           the
           summe
           of
           the
           sine
           of
           a
           first
           Base
           ,
           and
           the
           Tangent
           of
           a
           verticall
           ,
           we
           adde
           the
           Arithmeticall
           complement
           of
           the
           sine
           of
           a
           second
           Base
           ,
           (
           not
           omitting
           the
           usuall
           presection
           )
           we
           cannot
           faile
           of
           the
           Tangent
           of
           the
           Cathetopposite
           required
           ,
           which
           Cathetopposites
           ,
           and
           Bases
           ,
           both
           first
           and
           second
           ,
           are
           particularised
           according
           to
           the
           Cases
           of
           the
           Mood
           .
        
         
           The
           sixth
           generall
           Resolver
           is
           ,
           
             Neg
             —
             Nu
             —
             Ne
             ☞
             Nur
             ,
          
           which
           ,
           running
           along
           the
           Co-sines
           of
           all
           the
           Bases
           and
           Subtendents
           ,
           sheweth
           ,
           that
           by
           the
           summe
           of
           the
           Co-sines
           of
           a
           second
           Base
           ,
           and
           first
           subtendent
           joyned
           with
           the
           Arithmeticall
           complement
           of
           the
           Co-sine
           of
           a
           first
           Base
           (
           if
           we
           observe
           the
           customary
           presection
           )
           we
           find
           the
           second
           Subtendent
           required
           ,
           which
           both
           first
           and
           
             second
             Subtendents
          
           ,
           together
           with
           the
           first
           and
           
             second
             Bases
          
           ,
           are
           all
           of
           them
           particularised
           conforme
           to
           the
           Cases
           of
           the
           Mood
           .
        
         
           The
           seventh
           Resolver
           is
           ,
           
             Mu
             —
             Nag
             —
             Mu
             ☞
             Nir
             ,
          
           which
           ,
           coursing
           along
           the
           Anti-tangents
           of
           first
           ,
           and
           second
           Subtendents
           ,
           and
           the
           Anti-sines
           of
           verticals
           ,
           sheweth
           ,
           that
           the
           summe
           of
           the
           Anti-tangent
           of
           a
           second
           Subtendent
           ,
           and
           Anti-sine
           of
           a
           verticall
           ,
           together
           with
           the
           Arithmeticall
           complement
           of
           the
           Antitangent
           of
           a
           first
           Subtendent
           (
           the
           usuall
           presection
           being
           observed
           )
           is
           the
           Anti-tangent
           of
           that
           verticall
           ,
           which
           ushers
           in
           the
           verticall
           required
           ;
           all
           which
           ,
           both
           Verticals
           ,
           and
           Subtendents
           ,
           both
           first
           ,
           and
           second
           ,
           have
           their
           peculiar
           denominations
           conforme
           to
           the
           Cases
           of
           this
           
             Sindiforating
             Mood
          
           .
        
         
         
           The
           eighth
           and
           last
           generall
           Resolver
           is
           ,
           
             Nu
             —
             Ne
             —
             Nu
             ☞
             Nyr
             ,
          
           which
           (
           running
           altogether
           upon
           Co-sines
           of
           Subtendents
           ,
           and
           Bases
           ,
           both
           first
           ,
           and
           second
           of
           either
           ,
           and
           is
           nothing
           else
           but
           the
           sixth
           inverted
           )
           sheweth
           ,
           that
           the
           summe
           of
           the
           Cosines
           of
           a
           second
           subtendent
           ,
           and
           first
           Base
           ,
           with
           the
           Arithmeticall
           complement
           of
           the
           co-sine
           of
           a
           first
           Subtendent
           ,
           (
           observing
           the
           usuall
           presection
           )
           affords
           the
           Co-sine
           of
           the
           second
           Base
           ,
           illatitious
           to
           the
           segment
           required
           ;
           which
           Bases
           and
           Subtendents
           ,
           both
           first
           ,
           and
           second
           ,
           are
           peculiarly
           denominated
           according
           to
           the
           severall
           Cases
           of
           this
           
             Sindiforiuting
             Mood
          
           .
        
         
           Thus
           have
           I
           finished
           the
           Logarithmication
           of
           the
           generall
           Resolvers
           of
           the
           Loxogonosphericall
           Disergeticks
           ,
           so
           farre
           as
           is
           requisite
           ,
           wherein
           I
           often
           times
           mentioned
           the
           Arithmeticall
           complement
           of
           Sines
           ,
           Co-sines
           ,
           Tangents
           ,
           and
           Co-tangents
           :
           and
           though
           I
           spoke
           of
           that
           purpose
           sufficiently
           in
           my
           Sphericorectangular
           comments
           ,
           yet
           ,
           for
           the
           Readers
           better
           remembrance
           thereof
           ,
           I
           will
           once
           more
           define
           them
           here
           .
        
         
           The
           Arithmeticall
           complements
           of
           Sines
           are
           Co-secants
           ;
           of
           Co-sines
           ,
           Secants
           ;
           of
           Tangents
           ,
           Co-tangents
           ;
           and
           of
           Co-tangents
           ,
           Tangents
           ;
           each
           being
           the
           others
           complement
           to
           the
           double
           Radius
           :
           but
           if
           such
           a
           Canon
           were
           framed
           ,
           wherein
           the
           single
           Radius
           is
           left
           out
           of
           all
           Secants
           ,
           and
           Tangents
           of
           major
           Arches
           ,
           then
           would
           each
           be
           the
           others
           complement
           to
           the
           single
           Radius
           ,
           and
           all
           Logarithmicall
           operations
           in
           questions
           of
           Trigonometry
           so
           easily
           performable
           by
           addition
           onely
           ,
           that
           seldome
           would
           the
           presectionall
           digit
           exceed
           an
           unit
           .
        
         
           Having
           already
           said
           so
           much
           of
           these
           eight
           Disergeticks
           ,
           I
           will
           conclude
           my
           discourse
           of
           them
           with
           a
           summary
           delineation
           of
           the
           eight
           severall
           Concordances
           which
           I
           observed
           amongst
           them
           ;
           for
           either
           they
           resemble
           one
           another
           in
           the
           Datas
           of
           their
           Moods
           ,
           or
           in
           their
           Proturgetick
           operations
           ,
           or
           in
           their
           dependance
           upon
           the
           same
           Axiome
           ,
           or
           in
           the
           work
           of
           perpendicular
           finding
           ,
           or
           in
           their
           Datas
           for
           the
           main
           demand
           ,
           or
           in
           their
           materiall
           Quaesitas
           (
           though
           diversly
           endowed
           )
           or
           in
           their
           inversion
           ,
           or
           lastly
           in
           their
           sindiforation
           ,
           which
           affinity
           is
           onely
           betwixt
           two
           paires
           of
           them
           ,
           as
           the
           first
           two
           amongst
           two
           quaternaries
           apeece
           ,
           and
           the
           next
           five
           between
           foure
           couples
           each
           one
           ,
           the
           brief
           hypotyposis
           of
           all
           which
           is
           here
           exposed
           to
           the
           view
           of
           the
           Reader
           .
        
         
           
             
             
               CONCORDANCES
               .
            
             
               
                 Datall
                 .
              
            
             
               
                 Datangulary
                 .
              
               
                 Datolaterall
                 .
              
            
             
               
                 1.
                 
                 Alamebna
                 .
              
               
                 3.
                 
                 Amanepra
                 .
              
               
                 1.
                 
                 Enerablo
                 .
              
               
                 3.
                 
                 Erelomab
                 .
              
            
             
               
                 2.
                 
                 Allamebne
                 .
              
               
                 4.
                 
                 Ammanepreb
                 .
              
               
                 2.
                 
                 Ennerable
                 .
              
               
                 4.
                 
                 Errelome
                 .
              
            
             
               
                 Praenoscendall
                 .
              
            
             
               
                 Verticall
                 .
              
               
                 Basall
                 .
              
            
             
               
                 1.
                 
                 Alamebna
                 .
              
               
                 3.
                 
                 Amanepra
                 .
              
               
                 1.
                 
                 Ammanepreb
                 .
              
               
                 3.
                 
                 Ennerable
                 .
              
            
             
               
                 2.
                 
                 Allamebne
                 .
              
               
                 4.
                 
                 Erelomab
                 .
              
               
                 2.
                 
                 Enerablo
                 .
              
               
                 4.
                 
                 Errelome
                 .
              
            
             
               
                 Theorematick
                 .
              
            
             
               
                 Nabadprosver
              
               
                 Naverprortes
                 .
              
               
                 Siubprortab
              
               
                 Niubprodnesver
                 .
              
            
             
               
                 1.
                 
                 Alamebna
                 .
              
               
                 1.
                 
                 Allamebne
                 .
              
               
                 1.
                 
                 Ammanepreb
                 .
              
               
                 1.
                 
                 Ennerable
                 .
              
            
             
               
                 2.
                 
                 Amanepra
                 .
              
               
                 2.
                 
                 Erelomab
                 .
              
               
                 2.
                 
                 Enerablo
                 .
              
               
                 2.
                 
                 Errelome
                 .
              
            
             
               
                 Catheteuretick
                 .
              
            
             
               
                 Oppoverticall
              
               
                 Hypoverticall
                 .
              
               
                 Oppobasall
              
               
                 Hypobasall
                 .
              
            
             
               
                 1.
                 
                 Alamebna
                 .
              
               
                 1.
                 
                 Allamebne
                 .
              
               
                 1.
                 
                 Ammanepreb
                 .
              
               
                 1.
                 
                 Ennerable
                 .
              
            
             
               
                 2.
                 
                 Amanepra
                 .
              
               
                 2.
                 
                 Erelomab
                 .
              
               
                 2.
                 
                 Enerablo
                 .
              
               
                 2.
                 
                 Errelome
                 .
              
            
             
               
                 Datysterurgetick
                 .
              
            
             
               
                 Cathetoverticall
              
               
                 Oppocathetall
                 .
              
               
                 Cathetobasall
              
               
                 Hypocathetal
                 .
              
            
             
               
                 1.
                 
                 Alamebna
                 .
              
               
                 1.
                 
                 Amanepra
                 .
              
               
                 1.
                 
                 Enerablo
                 .
              
               
                 1.
                 Erelomab
              
            
             
               
                 2.
                 
                 Allamebne
                 .
              
               
                 2.
                 
                 Ammanepreb
                 .
              
               
                 2.
                 
                 Ennerable
                 .
              
               
                 2.
                 
                 Errelome
                 .
              
            
             
               
                 Zetetick
                 .
              
            
             
               
                 Cathetopposite
              
               
                 Hypotenusall
                 .
              
               
                 Verticall
              
               
                 Basall
                 .
              
            
             
               
                 1.
                 
                 
                   Alamebna*
                   S.
                
                 
              
               
                 1.
                 
                 
                   Allamebne*
                   M.
                
                 
              
               
                 1.
                 
                 
                   Amanepra*
                   S.
                
                 
              
               
                 1.
                 
                   ammanepreb
                   .
                   *
                   S
                
              
            
             
               
                 2.
                 
                 
                   Enerablo*
                   T.
                
                 
              
               
                 2.
                 
                 
                   Ennerable*
                   N.
                
                 
              
               
                 2.
                 
                 
                   Erelomab*
                   N.
                
                 
              
               
                 2.
                 
                 
                   Errelome*
                   N.
                
                 
              
            
             
               
                 Inversionall
                 .
              
            
             
               
                 Sinocosinall
              
               
                 Sinocotangentall
                 .
              
               
                 Tangentosinall
              
               
                 Cosinocosinall
                 .
              
            
             
               
                 1.
                 
                 Alamebna
                 .
              
               
                 1.
                 
                 Allamebne
                 .
              
               
                 1.
                 
                 Ammanepreb
                 .
              
               
                 1.
                 
                 Ennerable
                 .
              
            
             
               
                 2.
                 
                 Amanepra
                 .
              
               
                 2.
                 
                 Erelomab
                 .
              
               
                 2.
                 
                 Enerablo
                 .
              
               
                 2.
                 
                 Errelome
                 .
              
            
             
               
                 Sindiforall
                 .
              
            
             
               
                 Sindiforatall
                 .
              
               
                 Sindiforiutall
                 .
              
            
             
               
                 1.
                 
                 Amanepra
                 .
              
               
                 2.
                 
                 Erelomab
                 .
              
               
                 1.
                 
                 Ammanepreb
                 .
              
               
                 2.
                 
                 Errelome
                 .
              
            
          
        
      
    
     
       
         
         
           THE
           EPILOGUE
           .
        
         
           WHat
           concerneth
           the
           resolving
           of
           all
           manner
           of
           Triangles
           ,
           whether
           plain
           ,
           or
           
             Sphericall
             ,
             Rectangular
          
           ,
           or
           Obliquangular
           ,
           being
           now
           (
           conform
           to
           my
           promise
           in
           the
           Title
           )
           to
           the
           ful
           explained
           ,
           commented
           on
           ,
           perfected
           ,
           and
           with
           all
           possible
           brevity
           ,
           and
           perspicuity
           ,
           in
           all
           its
           abstrusest
           and
           most
           difficult
           Secrets
           ,
           from
           the
           very
           first
           principles
           of
           the
           Science
           it selfe
           ,
           made
           manifest
           ,
           proved
           ,
           and
           convincingly
           demonstrated
           ;
           I
           will
           here
           shut
           up
           my
           discourse
           ,
           and
           bring
           this
           Tractat
           to
           a
           period
           :
           which
           I
           may
           do
           with
           the
           more
           alacrity
           ,
           in
           that
           I
           am
           confident
           ,
           there
           is
           no
           Precept
           belonging
           to
           that
           faculty
           which
           is
           not
           herein
           included
           ,
           or
           reducible
           thereto
           :
           and
           therefore
           (
           I
           beleeve
           )
           the
           judicious
           Reader
           will
           not
           be
           frustrate
           of
           his
           expectation
           ,
           though
           by
           cutting
           the
           threed
           of
           my
           Glosse
           ,
           I
           doe
           not
           illustrate
           what
           I
           have
           written
           with
           variety
           of
           examples
           ;
           seeing
           practically
           to
           treat
           of
           Triangulary
           calculations
           ,
           in
           applying
           their
           doctrine
           to
           use
           ,
           were
           to
           digresse
           from
           the
           purpose
           in
           hand
           ,
           and
           incroach
           upon
           the
           subject
           of
           other
           Sciences
           ;
           a
           priviledge
           ,
           which
           I
           must
           decline
           ,
           as
           repugnant
           to
           the
           scope
           proposed
           to
           my selfe
           ,
           in
           keeping
           this
           book
           within
           the
           speculative
           bounds
           of
           Trigonometry
           :
           for
           ,
           as
           
             Logica
             utens
          
           ,
           is
           the
           Science
           to
           the
           which
           it
           is
           applyed
           ,
           and
           not
           Logick
           :
           So
           doth
           not
           the
           matter
           of
           Trigonometry
           ,
           exceed
           the
           Theory
           of
           a
           Triangle
           :
           And
           as
           
             Arithmeticall
             ,
             Geometricall
             ,
             Astronomicall
             ,
             Physicall
             ,
          
           and
           Metaphysicall
           definitions
           ,
           divisions
           ,
           and
           argumentations
           ,
           are
           no
           part
           of
           the
           Art
           that
           instructeth
           how
           to
           define
           ,
           divide
           ,
           and
           argue
           ,
           nor
           matter
           incumbent
           to
           him
           that
           teacheth
           it
           :
           even
           
           so
           ,
           by
           divulging
           this
           Treatise
           ,
           doe
           I
           present
           the
           Reader
           with
           a
           Key
           ,
           by
           meanes
           whereof
           he
           may
           enter
           into
           the
           chiefest
           treasures
           of
           the
           Mathematicall
           Sciences
           ;
           for
           the
           which
           ,
           in
           some
           measure
           ,
           I
           deserve
           thanks
           ,
           although
           I
           help
           him
           not
           to
           unshut
           the
           Coffers
           wherein
           they
           lie
           inclosed
           :
           for
           ,
           if
           the
           Lord
           chamberlain
           of
           the
           Kings
           houshold
           should
           give
           me
           a
           Key
           ,
           made
           to
           open
           all
           the
           doores
           of
           the
           Court
           ,
           I
           could
           not
           but
           graciously
           accept
           of
           it
           ,
           though
           he
           did
           not
           goe
           along
           with
           me
           to
           try
           how
           it
           might
           fit
           every
           lock
           .
           The
           application
           is
           so
           palpable
           ,
           that
           ,
           not
           minding
           to
           insist
           therein
           ,
           I
           will
           here
           stop
           the
           current
           of
           my
           Pen
           ,
           and
           by
           a
           circulary
           conclusion
           ,
           ending
           where
           I
           begun
           ,
           certifie
           the
           Reader
           ,
           that
           if
           he
           intend
           to
           approve
           himselfe
           an
           Artist
           in
           matters
           of
           
             Pleusiotechnie
             ,
             Poliechyrologie
             ,
             Cosmography
             ,
             Geography
             ,
             Astronomy
             ,
             Geodesie
             ,
             Gnomonicks
             ,
             Sciography
             ,
             Catoptricks
             ,
             Dioptricks
             ,
          
           and
           many
           other
           most
           exquisite
           Arts
           and
           Sciences
           ,
           Practical
           and
           Theoretick
           ,
           his
           surest
           course
           ,
           for
           attaining
           to
           so
           much
           knowledge
           ,
           is
           to
           be
           well
           versed
           in
           Trigonometry
           ,
           to
           understand
           this
           Treatife
           aright
           ,
           revolve
           all
           the
           passages
           thereof
           ,
           ruminate
           on
           the
           Table
           ,
           and
           peruse
           the
           Trissotetras
           .
        
      
       
         
         
           A
           Lexicidion
           of
           some
           of
           the
           hardest
           words
           ,
           that
           occurre
           in
           the
           discourse
           of
           this
           institution
           Trigonometricall
           .
        
         
           
             BEing
             certainly
             perswaded
             ,
             that
             a
             great
             many
             good
             spirits
             ply
             Trigonometry
             ,
             that
             are
             not
             versed
             in
             the
             learned
             Tongues
             ,
             I
             thought
             fit
             ,
             for
             their
             encouragement
             ,
             to
             subjoyne
             here
             the
             explication
             of
             the
             most
             important
             of
             those
             Greek
             ,
             and
             Latin
             termes
             ,
             which
             ,
             for
             the
             more
             efficacy
             of
             expression
             ,
             I
             have
             made
             use
             of
             in
             this
             Treatise
             :
             in
             doing
             whereof
             ,
             that
             I
             might
             both
             instruct
             the
             Reader
             ,
             and
             not
             weary
             him
             ,
             I
             have
             endeavoured
             perspicuity
             with
             shortnesse
             :
             though
             (
             I
             speak
             it
             ingenuously
             )
             to
             have
             been
             more
             prolixe
             therin
             ,
             could
             have
             cost
             but
             very
             little
             labor
             to
             me
             ,
             who
             have
             already
             bin
             pretty
             well
             versed
             in
             the
             like
             ,
             as
             may
             appear
             by
             my
             Etymologicall
             dictionary
             of
             above
             twenty
             seven
             thousand
             proper
             names
             ,
             mentioned
             in
             the
             Lemmas
             of
             my
             severall
             Volums
             of
             Epigrams
             ,
             the
             words
             whereof
             are
             for
             the
             most
             part
             abstruser
             ,
             derived
             from
             moe
             Languages
             ,
             and
             more
             liable
             to
             large
             ,
             and
             ample
             interpretations
             .
             However
             
               (
               caeteris
               paribus
            
             )
             brevity
             is
             to
             be
             preferred
             ;
             therefore
             let
             us
             proceed
             to
             the
             Vocabulary
             in
             hand
             .
          
        
         
           
           
             THE
             LEXICIDION
             .
          
           
             
               A.
               
            
             
               ACute
               ,
            
             
               comes
               from
               
                 Acuo
                 ,
                 acuere
              
               ,
               to
               sharpen
               ,
               and
               is
               said
               of
               an
               Angle
               ,
               whose
               including
               sides
               ,
               the
               more
               that
               its
               measure
               is
               lesse
               then
               a
               Quadrant
               ,
               have
               their
               concursive
               ,
               and
               angulary
               point
               the
               more
               penetrative
               ,
               sharp
               ,
               keen
               and
               pierceing
               :
               Whence
               an
               acutangled
               triangle
               .
            
             
               Adaequat
               ,
            
             
               is
               that
               ,
               which
               comprehendeth
               to
               the
               full
               ,
               whatever
               is
               in
               the
               thing
               to
               the
               which
               it
               is
               compared
               ,
               and
               for
               the
               most
               part
               in
               my
               Trissotetras
               is
               said
               of
               the
               generall
               finall
               Resolvers
               ,
               in
               relation
               to
               the
               Moods
               resolved
               by
               them
               .
               It
               is
               compounded
               of
               Ad
               ,
               and
               
                 aequo
                 ,
                 aequare
                 ,
                 parem
                 facere
                 ,
              
               to
               make
               one
               thing
               altogether
               like
               ,
               or
               equall
               to
               another
               .
            
             
               Adjacent
               ,
            
             
               signifieth
               to
               lie
               neare
               ,
               and
               close
               ,
               and
               is
               applyed
               both
               to
               sides
               ,
               and
               Angles
               ,
               in
               which
               sense
               likewise
               I
               make
               use
               of
               the
               words
               adjoyning
               the
               ,
               conterminat
               ,
               or
               conterminall
               with
               ,
               annexed
               to
               ,
               intercepted
               in
               ,
               and
               other
               such
               like
               ,
               for
               the
               more
               variety
               ,
               as
               adherent
               ,
               bounding
               ,
               bordering
               ,
               and
               so
               forth
               :
               It
               comes
               from
               
                 Adjaceo
                 ,
                 Adjacere
              
               ,
               to
               lie
               neere
               unto
               ,
               as
               the
               words
               Ad
               and
               jaceo
               ,
               which
               are
               the
               parts
               whereof
               it
               is
               compounded
               ,
               most
               perspicuously
               declare
               .
            
             
               Additionall
               ,
            
             
               is
               said
               of
               the
               Line
               ,
               which
               ,
               in
               my
               comment
               ,
               is
               indifferently
               called
               the
               Line
               of
               Addition
               ,
               the
               Line
               of
               continuation
               ,
               the
               extrinsecall
               Line
               ,
               the
               excesse
               of
               the
               Secant
               above
               the
               Radius
               ,
               the
               Refiduum
               ,
               or
               the
               new
               Secant
               :
               it
               comes
               from
               
                 Addo
                 ,
                 Addere
              
               ,
               which
               is
               compounded
               of
               Ad
               ,
               and
               do
               ,
               to
               put
               to
               and
               augment
               .
            
             
               Affection
               ,
            
             
               is
               the
               nature
               ,
               passion
               ,
               and
               quality
               of
               an
               Angle
               ,
               and
               consisteth
               either
               in
               the
               
                 obtusity
                 ,
                 acutenesse
              
               ,
               or
               rectitude
               thereof
               :
               It
               is
               a
               verball
               from
               
                 Afficio
                 ,
                 affeci
                 ,
                 affectum
              
               ,
               compounded
               of
               ad
               ,
               and
               facio
               .
            
             
               
               Aggregat
               ,
            
             
               is
               the
               summe
               ,
               totall
               ,
               or
               result
               of
               an
               Addition
               ,
               and
               is
               compounded
               of
               Ad
               ,
               and
               grex
               ;
               for
               ,
               as
               the
               Shepheard
               gathers
               his
               Sheep
               into
               a
               flock
               ,
               so
               doth
               the
               Arithmetician
               compact
               his
               numbers
               to
               be
               added
               into
               a
               summe
               .
            
             
               Alternat
               ,
            
             
               is
               said
               of
               Angles
               ,
               made
               by
               a
               Line
               cutting
               two
               or
               more
               parallels
               ,
               which
               Angles
               may
               be
               properly
               called
               so
               ;
               because
               they
               differ
               in
               nothing
               else
               but
               their
               situation
               ;
               for
               if
               the
               sectionary
               Line
               ,
               to
               the
               which
               I
               suppose
               the
               parallels
               to
               be
               fixed
               ,
               have
               the
               highest
               and
               lowest
               points
               thereof
               to
               interchange
               their
               sites
               ,
               by
               a
               motion
               progressive
               towards
               the
               roome
               of
               the
               under
               Alternat
               ,
               and
               terminating
               in
               that
               of
               the
               upper
               one
               ,
               we
               will
               find
               ,
               that
               both
               the
               inclination
               of
               the
               Lines
               towards
               one
               another
               ,
               and
               the
               quality
               of
               the
               Angles
               ,
               will
               ,
               notwithstanding
               that
               alteration
               ,
               be
               the
               same
               as
               before
               ;
               hence
               it
               is
               that
               they
               are
               called
               alternat
               ,
               because
               there
               is
               no
               other
               difference
               betwixt
               them
               :
               or
               ,
               if
               alternat
               be
               taken
               (
               as
               arithmetically
               it
               is
               )
               for
               that
               proportion
               ,
               wherein
               the
               Antecedent
               is
               compared
               to
               the
               Antecedent
               ,
               and
               the
               Consequent
               to
               the
               Consequent
               ,
               the
               sense
               will
               likewise
               hold
               in
               the
               foresaid
               Angles
               ;
               for
               if
               by
               the
               parallelisme
               of
               two
               right
               Lines
               ,
               cut
               with
               a
               third
               ,
               two
               blunt
               ,
               and
               two
               keen
               Angles
               be
               produced
               (
               as
               must
               needs
               ,
               unlesse
               the
               Secant
               line
               be
               to
               the
               parallels
               a
               perpendicular
               )
               the
               keen
               or
               acute
               Angle
               will
               be
               to
               its
               complement
               ,
               or
               successively
               following
               obtuse
               Angle
               ,
               as
               the
               other
               acute
               unto
               its
               following
               obtuse
               ;
               therefore
               alternly
               ,
               as
               the
               Antecedents
               are
               to
               one
               another
               ,
               viz.
               the
               Acute
               to
               the
               Acute
               :
               so
               the
               Consequents
               ,
               the
               obtuse
               to
               the
               obtuse
               .
               And
               if
               the
               Angles
               be
               right
               ,
               the
               direct
               ,
               and
               alternat
               proportion
               is
               one
               and
               the
               same
               ;
               the
               third
               ,
               and
               fourth
               terms
               of
               the
               Analogy
               being
               in
               nothing
               different
               from
               the
               first
               ,
               and
               second
               .
            
             
               Ambient
               ,
            
             
               is
               taken
               for
               any
               of
               the
               legs
               of
               a
               rectangle
               ,
               or
               the
               including
               ,
               containing
               ,
               or
               comprehending
               sides
               of
               the
               right
               Angle
               :
               it
               comes
               from
               
                 Ambio
                 ,
                 Ambire
              
               ,
               which
               is
               compounded
               of
               Am
               and
               eo
               ,
               i.
               e.
               circumeo
               :
               and
               more
               properly
               applied
               to
               both
               ,
               then
               to
               any
               one
               of
               them
               ,
               though
               usually
               it
               be
               usurped
               for
               one
               alone
               ,
               
                 vide
                 Leg.
              
               
            
             
               Amblygonian
               ,
            
             
               is
               said
               of
               obtuse
               angled
               Triangles
               ,
               and
               Amblygonosphericall
               of
               obtuse
               sphericals
               :
               It
               is
               composed
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               angulus
               .
            
             
               Amfractuosities
               ,
            
             
               are
               taken
               here
               for
               the
               cranklings
               ,
               windings
               ,
               turnings
               ,
               and
               involutions
               belonging
               to
               the
               equisoleary
               Scheme
               ;
               of
               am
               and
               
                 frango
                 ,
                 quod
                 sit
                 quasi
                 via
                 crebris
                 maeandris
                 undequaque
                 interrupta
                 .
              
            
             
               
               Analogy
               ,
            
             
               signifieth
               an
               equality
               of
               proportion
               ,
               a
               likenesse
               of
               reasons
               ,
               a
               conveniencie
               ,
               or
               habitude
               betwixt
               termes
               :
               It
               is
               compounded
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               aequaliter
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               ratio
               .
            
             
               Analytick
               ,
            
             
               resolutory
               ,
               and
               is
               said
               of
               those
               things
               that
               are
               resolved
               into
               their
               first
               principles
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               re
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               solvo
               .
            
             
               Antilogarithm
               ,
            
             
               is
               the
               Logarithm
               of
               the
               complement
               ;
               as
               for
               example
               ,
               the
               Anti-logarithm
               of
               a
               Sine
               is
               the
               Logarithm
               of
               the
               Sine
               complement
               ,
               
                 vide
                 Logarithm
              
               .
            
             
               Anti-secant
               ,
            
             
               Anti-sine
               ,
               and
               Anti-tangent
               ,
               are
               the
               complements
               of
               the
               Secant
               ,
               Sine
               ,
               and
               Tangent
               ,
               and
               are
               called
               sometime
               Co-secant
               ,
               Co-sine
               ,
               and
               Co-tangent
               :
               they
               have
               anti
               prefixed
               ,
               because
               they
               are
               not
               in
               the
               same
               colume
               ,
               and
               co
               ,
               because
               they
               are
               in
               the
               next
               to
               it
               .
            
             
               Apodictick
               ,
            
             
               is
               that
               ,
               which
               is
               demonstrative
               ,
               and
               giveth
               evident
               proofs
               of
               the
               truth
               of
               a
               conclusion
               ;
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 monstro
                 ,
                 ostendo
                 ,
                 unde
              
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               demonstratio
               .
            
             
               Area
               ,
            
             
               is
               the
               capacity
               of
               a
               Figure
               ,
               and
               whole
               content
               thereof
               .
            
             
               Arch
               ,
            
             
               or
               Ark
               ,
               is
               the
               segment
               of
               a
               circumference
               lesse
               then
               a
               semicircle
               :
               major
               Arch
               is
               above
               45.
               degrees
               ,
               a
               minor
               Arch
               ,
               lesse
               then
               
                 45.
                 vide
                 Circle
              
               .
            
             
               Arithmeticall
            
             
               complement
               ,
               is
               the
               difference
               betweene
               the
               Logarithm
               to
               be
               substracted
               ,
               and
               that
               of
               the
               double
               ,
               or
               single
               Radius
               .
            
             
               Artificiall
            
             
               numbers
               ,
               are
               the
               Logarithms
               ,
               and
               artificiall
               Sine
               the
               Logarithm
               of
               the
               Sine
               .
            
             
               Axiome
               ,
            
             
               is
               a
               maxim
               ,
               tenet
               ,
               or
               necessary
               principle
               ,
               whereupon
               the
               Science
               of
               a
               thing
               is
               grounded
               :
               it
               cometh
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               dignus
               ;
               because
               such
               things
               are
               worthy
               our
               knowledge
               .
            
          
           
             
               B.
               
            
             
               BAsall
               ,
            
             
               adjectively
               is
               that
               ,
               which
               belongeth
               to
               the
               Base
               ,
               or
               the
               subjacent
               side
               ,
               but
               substantively
               the
               great
               Base
               .
            
             
               Basangulary
               ,
            
             
               is
               said
               of
               the
               Angles
               at
               the
               Base
               .
            
             
               Basidion
               ,
            
             
               or
               baset
               ,
               is
               the
               little
               Base
               ,
               all
               which
               come
               from
               the
               Greek
               word
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Basiradius
               ,
            
             
               is
               the
               totall
               Sine
               of
               that
               Arch
               ,
               a
               Segment
               whereof
               is
               the
               Base
               of
               the
               proposed
               sphericall
               Triangle
               .
            
             
               Bisected
               ,
            
             
               and
               Bisegment
               ,
               are
               said
               of
               lines
               cut
               into
               two
               equall
               parts
               :
               it
               comes
               from
               
                 biseco
                 ,
                 bisecare
                 ,
                 bisectum
                 ,
                 bisegmen
                 .
              
            
             
               Bluntnesse
               ,
            
             
               or
               flatnesse
               ,
               is
               the
               obtuse
               affection
               of
               Angles
               .
            
             
               
               Bucarnon
               ,
            
             
               by
               this
               name
               is
               entitled
               the
               seven
               and
               fortieth
               proposition
               of
               the
               first
               of
               the
               elements
               of
               Euclid
               ;
               because
               of
               the
               oxe
               ,
               or
               ,
               (
               as
               some
               say
               )
               the
               hecatomb
               which
               Pythagoras
               ,
               for
               gladnesse
               of
               the
               invention
               ,
               sacrificed
               unto
               the
               gods
               :
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               bos
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 vicissim
                 aliquid
                 capio
              
               ;
               they
               being
               (
               as
               it
               is
               supposed
               )
               well
               pleased
               with
               that
               acknowledgement
               of
               his
               thankfulnesse
               for
               so
               great
               a
               favour
               ,
               as
               that
               was
               ,
               which
               he
               received
               from
               them
               :
               you
               may
               see
               the
               proposition
               in
               the
               seventeenth
               of
               my
               Apodicticks
               .
            
          
           
             
               C.
               
            
             
               CAnon
               ,
            
             
               is
               taken
               here
               for
               the
               Table
               of
               Sines
               and
               Tangents
               ,
               or
               of
               their
               Logarithms
               :
               it
               properly
               signifieth
               the
               needle
               or
               tongue
               of
               a
               balance
               ,
               and
               metaphorically
               a
               rule
               ,
               whereby
               things
               are
               examined
               .
            
             
               Cases
               ,
            
             
               are
               the
               parts
               wherein
               a
               Mood
               is
               divided
               from
               cado
               .
            
             
               Cathetos
               ,
            
             
               is
               a
               Perpendicular
               line
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               demitto
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Catheteuretick
               ,
            
             
               is
               concerning
               the
               finding
               out
               of
               the
               Perpendicular
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               invenio
               .
            
             
               Cathetobasall
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               Loxogonosphericall
               Moods
               ,
               in
               the
               Datas
               of
               the
               Perpendicular
               ,
               and
               the
               Base
               ,
               for
               finding
               out
               of
               the
               maine
               quaesitum
               .
            
             
               Cathetopposite
               ,
            
             
               is
               the
               Angle
               opposite
               to
               the
               Perpendicular
               ;
               it
               is
               a
               hybrid
               or
               mungrell
               word
               ,
               composed
               of
               the
               Greek
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               Latin
               oppositus
               .
            
             
               Cathetorabdos
               ,
            
             
               or
               Cathetoradius
               ,
               is
               the
               totall
               Sine
               of
               that
               Arch
               ,
               a
               Segment
               whereof
               is
               the
               Cathetos
               ,
               or
               Perpendicular
               of
               the
               proposed
               Orthogonosphericall
               .
            
             
               Cathetothesis
               ,
            
             
               and
               cathetothetick
               are
               said
               of
               the
               determinat
               position
               of
               the
               Perpendicular
               ,
               which
               is
               sometimes
               expressed
               by
               cathetology
               ,
               instructing
               us
               how
               it
               should
               be
               demitted
               :
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               pono
               ,
               colloco
               .
            
             
               Cathetoverticall
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               Loxogonosphericall
               Moods
               in
               the
               Datas
               of
               the
               perpendicular
               ,
               and
               the
               verticall
               Angle
               in
               the
               last
               operation
               .
            
             
               Catoptrick
               ,
            
             
               the
               Science
               of
               perspective
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               perspicio
               ,
               cerno
               .
            
             
               Characteristick
               ,
            
             
               is
               said
               of
               the
               letters
               ,
               which
               are
               the
               notes
               and
               marks
               of
               distinction
               ,
               called
               sometimes
               figuratives
               ,
               or
               determinaters
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               sculpo
               ,
               imprimo
               .
            
             
               
               Circles
               ,
            
             
               great
               circles
               are
               those
               which
               bisect
               the
               Sphere
               ,
               lesser
               Circles
               those
               which
               not
               .
            
             
               Circular
            
             
               parts
               ,
               are
               in
               opposition
               to
               the
               reall
               and
               naturall
               parts
               of
               a
               Triangle
               .
            
             
               Circumjacent
               ,
            
             
               things
               which
               lie
               about
               ,
               of
               circum
               and
               jaceo
               .
            
             
               Coalescencie
               ,
            
             
               a
               growing
               together
               ,
               a
               compacting
               of
               two
               things
               in
               one
               ;
               it
               is
               said
               of
               the
               last
               two
               operations
               of
               the
               Loxogonosphericals
               conflated
               into
               one
               ,
               from
               coalesco
               or
               coalco
               ,
               of
               con
               and
               alo
               .
            
             
               Cobase
               ,
            
             
               a
               fellow
               Base
               ,
               or
               that
               which
               with
               another
               Base
               hath
               a
               common
               Perpendicular
               ,
               of
               con
               and
               basis
               .
            
             
               Cocathetopposite
               ,
            
             
               is
               said
               of
               two
               ,
               Angles
               at
               the
               Base
               ,
               opposite
               to
               one
               and
               the
               same
               Cathetos
               .
            
             
               Coincidence
               ,
            
             
               a
               falling
               together
               upon
               the
               same
               thing
               ,
               from
               coincido
               ,
               of
               con
               ,
               and
               incido
               ,
               ex
               in
               ,
               &
               cado
               .
            
             
               Comment
               ,
            
             
               is
               an
               interpretation
               ,
               or
               exposition
               of
               a
               thing
               ,
               and
               comes
               from
               
                 comminiscor
                 ,
                 comminisci
                 ,
                 mentionem
                 facere
                 .
              
            
             
               Compacted
               ,
            
             
               joyned
               ,
               and
               knit
               together
               ,
               put
               in
               one
               ;
               from
               
                 compingo
                 ,
                 compegi
                 ,
                 compactum
              
               ,
               vide
               Coalescencie
               .
            
             
               Complement
               ,
            
             
               signifieth
               the
               perfecting
               that
               which
               a
               thing
               wanteth
               ,
               and
               usually
               is
               that
               ,
               which
               an
               Angle
               or
               a
               Side
               wanteth
               of
               a
               Quadrant
               ,
               or
               90.
               degrees
               :
               and
               of
               a
               Semicircle
               ,
               or
               180.
               from
               
                 compleo
                 ,
                 complere
              
               ,
               to
               fill
               up
               .
            
             
               Concurse
               ,
            
             
               is
               the
               meeting
               of
               lines
               ,
               or
               of
               the
               sides
               of
               a
               Triangle
               ,
               from
               
                 concurro
                 ,
                 concursum
              
               .
            
             
               Conflated
               ,
            
             
               compacted
               ,
               joyned
               together
               ,
               from
               
                 conflo
                 ,
                 conflatum
                 ,
                 conflare
              
               ,
               to
               blow
               together
               ,
               
                 vide
                 Inchased
              
               .
            
             
               Consectary
               ,
            
             
               is
               taken
               here
               for
               a
               Corollary
               ,
               or
               rather
               a
               secondary
               Axiome
               ,
               which
               dependeth
               on
               a
               prime
               one
               ,
               &
               being
               deduced
               from
               it
               ,
               doth
               necessarily
               follow
               .
               From
               
                 consector
                 ,
                 consectaris
              
               ,
               the
               frequentative
               of
               consequor
               .
            
             
               Consound
               ,
            
             
               to
               sound
               with
               another
               thing
               ;
               it
               is
               said
               of
               consonants
               ,
               which
               have
               no
               vocality
               without
               the
               help
               of
               the
               vowell
               .
            
             
               Constitutive
               ,
            
             
               is
               said
               of
               those
               things
               ,
               which
               help
               to
               frame
               ,
               make
               ,
               and
               build
               up
               :
               From
               constituo
               ,
               of
               con
               and
               statuo
               .
            
             
               Constitutive
            
             
               sides
               ,
               the
               ingredient
               sides
               of
               a
               Triangle
               .
            
             
               Constructive
            
             
               parts
               ,
               are
               those
               ,
               whereof
               a
               thing
               is
               built
               ,
               and
               framed
               :
               From
               
                 construo
                 ,
                 constructum
              
               ,
               to
               heap
               together
               ,
               and
               build
               up
               ,
               of
               con
               and
               strues
               .
            
             
               Conterminall
               ,
            
             
               is
               that
               which
               bordereth
               with
               ,
               and
               joyneth
               to
               a
               thing
               ,
               of
               
               con
               and
               
                 terminus
                 ,
                 vide
                 Adjacent
              
               ,
               or
               Insident
               .
            
             
               Cordes
               ,
            
             
               and
               cordall
               ,
               are
               said
               of
               subtenses
               metaphorically
               ;
               because
               the
               Arches
               and
               subtenses
               are
               as
               the
               bow
               and
               string
               :
               chorda
               ,
               comes
               of
               the
               Greek
               word
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               intestinum
               ,
               ilia
               ,
               quia
               ex
               illis
               chordae
               conficiuntur
               .
            
             
               Correctangle
               ,
            
             
               that
               is
               one
               which
               ,
               with
               another
               rectangle
               ,
               hath
               a
               common
               Perpendicular
               .
            
             
               Correspondent
               ,
            
             
               that
               which
               answereth
               with
               ,
               and
               hath
               a
               reference
               to
               another
               thing
               ,
               of
               con
               and
               respondeo
               .
            
             
               Cosinocosinall
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               Loxogonosphericall
               Moods
               ,
               agreeing
               ,
               in
               that
               the
               termes
               of
               their
               finall
               Resolvers
               run
               upon
               Cosines
               .
            
             
               Cosmography
               ,
            
             
               is
               taken
               here
               for
               the
               Science
               whereby
               is
               described
               the
               celestiall
               Globe
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Cosubtendent
               ,
            
             
               is
               the
               subtendent
               of
               a
               correctangle
               ,
               or
               that
               which
               with
               another
               is
               substerned
               to
               two
               right
               Angles
               ,
               made
               by
               the
               demission
               of
               one
               and
               the
               same
               Perpendicular
               .
            
             
               Coverticall
               ,
            
             
               is
               the
               fellow
               top
               Angle
               ,
               from
               whence
               the
               Perpendicular
               falleth
               .
            
          
           
             
               D.
               
            
             
               DAta
               ,
            
             
               is
               said
               of
               the
               parts
               of
               a
               Triangle
               ,
               which
               are
               given
               us
               ,
               whether
               they
               be
               sides
               or
               Angles
               ,
               or
               both
               ,
               of
               
                 do
                 ,
                 datum
                 ,
                 dare
              
               .
            
             
               Datimista
               ,
            
             
               are
               those
               Datas
               ,
               which
               are
               neither
               Angles
               onely
               ,
               nor
               sides
               onely
               ,
               but
               Angles
               ,
               and
               sides
               intermixedly
               :
               of
               data
               ,
               and
               mista
               ,
               from
               misceo
               .
            
             
               Datangulary
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               those
               Moods
               ,
               for
               the
               obtaining
               of
               whose
               Praenoscendas
               ,
               we
               have
               no
               other
               Datas
               ,
               but
               Angles
               ,
               unto
               the
               foresaid
               Moods
               common
               .
            
             
               Datapurall
               ,
            
             
               comes
               from
               datapura
               ,
               which
               be
               those
               Datas
               ,
               that
               are
               either
               meerly
               Angles
               ,
               or
               meerly
               sides
               .
            
             
               Datolaterall
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               those
               Moods
               ,
               for
               the
               obtaining
               of
               whose
               Praenoscendas
               ,
               the
               same
               sides
               serve
               for
               Datas
               .
            
             
               Datoquaere
               ,
            
             
               is
               the
               very
               Problem
               it selfe
               ,
               wherein
               two
               or
               three
               things
               are
               given
               ,
               and
               a
               third
               or
               fourth
               required
               ,
               as
               by
               the
               composition
               of
               the
               word
               appears
               .
            
             
               Datisterurgetick
               ,
            
             
               is
               said
               of
               those
               Moods
               which
               agree
               in
               the
               Datas
               of
               the
               last
               work
               :
               of
               data
               ,
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               postremum
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               opus
               .
            
             
               
               Demission
               ,
            
             
               is
               a
               letting
               fall
               of
               the
               Perpendicular
               :
               from
               
                 demitto
                 ;
                 demissum
              
               .
            
             
               Determinater
               ,
            
             
               is
               the
               characteristick
               or
               figurative
               letter
               of
               a
               directory
               :
               from
               determinare
               ,
               to
               prescribe
               and
               limit
               .
            
             
               Diagonall
               ,
            
             
               taken
               substantively
               ,
               or
               diagonie
               ,
               is
               a
               line
               drawn
               from
               one
               Angle
               to
               another
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               what
               the
               diagonie
               is
               in
               surfaces
               ,
               the
               axle
               is
               in
               solids
               .
            
             
               Diagrammatise
               ,
            
             
               to
               make
               a
               Scheme
               or
               Diagram
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               delineo
               .
            
             
               Diatyposis
               ,
            
             
               is
               a
               briefe
               summary
               description
               ,
               and
               delineation
               of
               a
               thing
               :
               or
               the
               couching
               of
               a
               great
               deale
               of
               matter
               ,
               for
               the
               instruction
               of
               the
               Reader
               ,
               in
               very
               little
               bounds
               ,
               and
               in
               a
               most
               neat
               and
               convenient
               order
               :
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               instituo
               ,
               item
               melius
               dispono
               ,
               vide
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Diodot
               ,
            
             
               is
               
                 Pythagorases
                 Bucarnon
              
               ,
               or
               the
               gift
               bestowed
               on
               him
               by
               the
               gods
               :
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               the
               genitive
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               datus
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               do
               ,
               vide
               Bucarnon
               .
            
             
               Dioptrick
               ,
            
             
               the
               art
               of
               taking
               heights
               and
               distances
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               pervidendo
               ,
               altitudinem
               dimensionemque
               turrium
               &
               murorum
               exploro
               .
            
             
               Directly
               ,
            
             
               is
               said
               of
               two
               rowes
               of
               proportionals
               ,
               where
               the
               first
               terme
               of
               the
               first
               row
               ,
               is
               to
               the
               first
               of
               the
               second
               ,
               as
               the
               last
               of
               the
               first
               ,
               is
               to
               the
               last
               of
               the
               second
               .
            
             
               Directory
               ,
            
             
               is
               that
               which
               pointeth
               out
               the
               Moods
               dependent
               on
               an
               Axiome
               .
            
             
               Discrepant
               ,
            
             
               different
               ,
               dissonant
               ,
               id
               est
               ,
               
                 diverso
                 modo
                 crepare
              
               .
            
             
               Disergeticks
               ,
            
             
               of
               two
               operations
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Document
               ,
            
             
               instruction
               ,
               from
               doceo
               .
            
          
           
             
               E.
               
            
             
               ELucidation
               ,
            
             
               a
               clearing
               ,
               explaining
               ,
               resolving
               of
               a
               doubt
               ,
               and
               commenting
               on
               some
               obscure
               passage
               ,
               from
               
                 elucido
                 ,
                 elucidare
              
               .
            
             
               Energie
               ,
            
             
               efficacie
               ,
               power
               ,
               force
               ;
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 qui
                 in
                 opere
                 est
              
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               opus
               .
            
             
               Enodandum
               ,
            
             
               that
               which
               is
               to
               be
               resolved
               and
               explicated
               ,
               declared
               ,
               and
               made
               manifest
               ,
               from
               
                 enodo
                 ,
                 enodare
              
               ,
               to
               unknit
               ,
               or
               cut
               away
               the
               knot
               .
            
             
               
               Equation
               ,
            
             
               or
               rather
               aequation
               ,
               a
               making
               equall
               ,
               from
               
                 aequo
                 ,
                 aequare
              
               .
            
             
               Equiangularity
               ,
            
             
               is
               that
               affection
               of
               Triangles
               ,
               whereby
               their
               Angles
               are
               equall
               .
            
             
               Equicrurall
               ,
            
             
               is
               said
               of
               Triangles
               ,
               whose
               legs
               or
               shanks
               are
               equall
               ;
               of
               aequale
               ,
               and
               
                 crus
                 ,
                 cruris
              
               ;
               leg
               being
               taken
               here
               for
               the
               thigh
               and
               leg
               .
            
             
               Equilaterall
               ,
            
             
               is
               said
               of
               Triangles
               ,
               which
               have
               all
               their
               sides
               ,
               shanks
               ,
               or
               legs
               equall
               ,
               of
               aequale
               and
               
                 latus
                 ,
                 lateris
              
               .
            
             
               Equipollencie
               ,
            
             
               is
               a
               samenesse
               ,
               or
               at
               least
               an
               equality
               of
               efficacie
               ,
               power
               ,
               vertue
               ,
               and
               energie
               ;
               of
               aequus
               and
               polleo
               .
            
             
               Equisolea
               ,
            
             
               and
               Equisolearie
               ,
               are
               said
               of
               the
               grand
               Orthogono
               sphericall
               Scheme
               ;
               because
               of
               the
               resemblance
               it
               hath
               with
               a
               horse-shooe
               ,
               and
               may
               in
               that
               sense
               be
               to
               this
               purpose
               applied
               with
               the
               same
               metaphoricall
               congruencie
               ,
               whereby
               it
               is
               said
               ,
               that
               the
               royall
               army
               at
               Edge-hill
               was
               imbatteld
               in
               a
               half-moon
               .
            
             
               Equivalent
               ,
            
             
               of
               as
               much
               worth
               and
               vertue
               ,
               of
               aequus
               and
               valeo
               .
            
             
               Erected
               ,
            
             
               is
               said
               of
               Perpendiculars
               ,
               which
               are
               set
               or
               raised
               upright
               upon
               a
               Base
               ,
               from
               erigere
               ,
               to
               raise
               up
               ,
               or
               set
               aloft
               .
            
             
               Externall
               ,
            
             
               extrinsecall
               ,
               exteriour
               ,
               outward
               ,
               or
               outer
               ,
               are
               said
               oftest
               of
               Angles
               ,
               which
               being
               without
               the
               Area
               of
               a
               Triangle
               ,
               are
               comprehended
               by
               two
               of
               its
               shanks
               meeting
               or
               cutting
               one
               another
               ,
               accordingly
               as
               one
               or
               both
               of
               them
               are
               protracted
               beyond
               the
               extent
               of
               the
               figure
               .
            
          
           
             
               F.
               
            
             
               FAciendas
               ,
            
             
               are
               the
               things
               which
               are
               to
               be
               done
               :
               faciendum
               is
               the
               gerund
               of
               facio
               .
            
             
               Figurative
               ,
            
             
               is
               the
               same
               thing
               as
               Characteristick
               ,
               and
               is
               applied
               to
               those
               letters
               which
               doe
               figure
               and
               point
               us
               out
               a
               resemblance
               and
               distinction
               in
               the
               Moods
               .
            
             
               Figures
               ,
            
             
               are
               taken
               here
               for
               those
               partitions
               of
               Trigonometry
               ,
               which
               are
               divided
               into
               Moods
               .
            
             
               Flat
               ,
            
             
               is
               said
               of
               obtuse
               ,
               or
               blunt
               Angles
               .
            
             
               Forwardly
               ,
            
             
               is
               said
               of
               Analogies
               ,
               progressive
               from
               the
               first
               terme
               to
               the
               last
               .
            
             
               Fundamentall
               ,
            
             
               is
               said
               of
               reasons
               ,
               taken
               from
               the
               first
               grounds
               and
               principles
               of
               a
               Science
               .
            
          
           
             
             
               G.
               
            
             
               GEodesie
               ,
            
             
               the
               Art
               of
               Surveying
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               or
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               terra
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               divido
               ,
               partior
               .
            
             
               Geography
               ,
            
             
               the
               Science
               of
               the
               Terrestriall
               Globe
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               terra
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               describo
               .
            
             
               Glosse
               ,
            
             
               signifieth
               a
               Commentary
               ,
               or
               explication
               ,
               it
               cometh
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Gnomon
               ,
            
             
               is
               a
               Figure
               lesse
               then
               the
               totall
               square
               ,
               by
               the
               square
               of
               a
               Segment
               :
               or
               ,
               according
               to
               Ramus
               ,
               a
               Figure
               composed
               of
               the
               two
               supplements
               ,
               and
               one
               of
               the
               Diagonall
               squares
               of
               a
               Quadrat
               .
            
             
               Gnomonick
               ,
            
             
               the
               Art
               of
               Dyalling
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               the
               cock
               of
               a
               Dyall
               .
            
             
               Great
               Circles
               ,
            
             
               vide
               Circles
               .
            
          
           
             
               H.
               
            
             
               HOmogeneall
               ,
            
             
               and
               Homogeneity
               ,
               are
               said
               of
               Angles
               of
               the
               same
               kind
               ,
               nature
               ,
               quality
               ,
               or
               affection
               :
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               communio
               generis
               .
            
             
               Homologall
               ,
            
             
               is
               said
               of
               sides
               congruall
               ,
               correspondent
               ,
               and
               agreeable
               ,
               viz.
               such
               as
               have
               the
               same
               reason
               or
               proportion
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               similis
               ratio
               .
            
             
               Hypobasall
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               those
               Loxogonosphericall
               Moods
               ,
               which
               ,
               when
               the
               Perpendicular
               is
               demitted
               ,
               have
               for
               the
               Datas
               of
               their
               second
               operation
               the
               same
               Subtendent
               and
               Base
               .
            
             
               Hypocathetall
               ,
            
             
               is
               said
               of
               those
               which
               for
               the
               Datas
               of
               their
               third
               operation
               have
               the
               same
               Subtendent
               and
               Perpendicular
               .
            
             
               Hypotenusall
               ,
            
             
               is
               said
               of
               Subtendent
               sides
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Hypotyposis
               ,
            
             
               a
               laying
               downe
               of
               severall
               things
               before
               our
               eyes
               at
               one
               time
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 oculis
                 subjicio
                 ,
                 delineo
                 ,
                 &
                 repraesento
                 ,
              
               vide
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Hypoverticall
               ,
            
             
               is
               said
               of
               Moods
               ,
               agreeing
               in
               the
               same
               Catheteuretick
               Datas
               of
               subtendent
               and
               verticall
               ,
               as
               the
               Analysis
               of
               the
               word
               doth
               shew
               .
            
          
           
             
               I
            
             
               IDentity
               ,
            
             
               a
               samenesse
               ,
               from
               idem
               ,
               the
               same
               .
            
             
               Illatitious
               ,
            
             
               or
               illative
               ,
               is
               said
               of
               the
               terme
               which
               bringeth
               in
               the
               quaesitum
               ,
               from
               
                 infero
                 ,
                 illatum
              
               .
            
             
               
               Inchased
               ,
            
             
               coagulated
               ,
               fixed
               in
               ,
               compacted
               ,
               or
               conflated
               ,
               is
               said
               of
               the
               last
               two
               Loxogonosphericall
               operations
               put
               into
               one
               ,
               
                 vide
                 Compacted
                 ,
                 Conflated
              
               ,
               and
               Coalescencie
               .
            
             
               Including
               sides
               ,
            
             
               are
               the
               containing
               sides
               of
               an
               Angle
               of
               what
               affection
               soever
               it
               be
               ,
               
                 vide
                 ,
                 Ambients
                 ,
                 Legs
              
               ,
               &c.
               
            
             
               Individuated
               ,
            
             
               brought
               to
               the
               lowest
               division
               ,
               
                 vide
                 ,
                 Specialised
              
               ,
               and
               Specification
               .
            
             
               Indowed
               ,
            
             
               is
               said
               of
               the
               termes
               of
               an
               Analogie
               ,
               whether
               sides
               ,
               or
               Angles
               ,
               as
               they
               stand
               affected
               with
               Sines
               ,
               Tangents
               ,
               Secants
               ,
               or
               their
               complements
               ,
               
                 vide
                 Invested
              
               .
            
             
               Ingredient
               ,
            
             
               is
               that
               which
               entreth
               into
               the
               composition
               of
               a
               Triangle
               ,
               or
               the
               progresse
               of
               an
               operation
               ,
               from
               ingredior
               ,
               of
               in
               and
               gradior
               .
            
             
               Initiall
               ,
            
             
               that
               which
               belongeth
               to
               the
               beginning
               ,
               from
               
                 initium
                 ,
                 ab
                 ineo
                 ,
                 significante
                 incipio
                 .
              
            
             
               Insident
               ,
            
             
               is
               said
               of
               Angles
               ,
               from
               insideo
               ,
               vide
               Adjacent
               ,
               or
               Conterminall
               .
            
             
               Interjacent
               ,
            
             
               lying
               betwixt
               ,
               of
               inter
               and
               jaceo
               ;
               it
               is
               said
               of
               the
               Side
               or
               Angle
               betweene
               .
            
             
               Intermediat
               ,
            
             
               is
               said
               of
               the
               middle
               termes
               of
               a
               proportion
               .
            
             
               Inversionall
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               those
               Moods
               which
               agree
               in
               the
               manner
               of
               their
               inversion
               ;
               that
               is
               in
               placing
               the
               second
               and
               fourth
               termes
               of
               the
               Analogy
               ,
               together
               with
               their
               indowments
               ,
               in
               the
               roomes
               of
               the
               first
               and
               third
               ,
               and
               contrariwise
               .
            
             
               Invested
               ,
            
             
               is
               the
               same
               as
               indowed
               ,
               from
               
                 investio
                 ,
                 investire
              
               .
            
             
               Irrationall
               ,
            
             
               are
               those
               which
               are
               commonly
               called
               surd
               numbers
               ,
               and
               are
               inexplicable
               by
               any
               number
               whatsoever
               ,
               whether
               whole
               ,
               or
               broken
               .
            
             
               Isosceles
               ,
            
             
               is
               the
               Greek
               word
               of
               equicrurall
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               crus
               .
            
          
           
             
               L.
               
            
             
               LAterall
               ,
            
             
               belonging
               to
               the
               sides
               of
               a
               Triangle
               from
               
                 latus
                 ,
                 lateris
              
               .
            
             
               Leg
               ,
            
             
               is
               one
               of
               the
               including
               sides
               of
               an
               Angle
               ,
               two
               sides
               of
               every
               Triangle
               being
               called
               the
               Legs
               ,
               and
               the
               third
               ,
               the
               Base
               ;
               the
               Legs
               therefore
               or
               shankes
               of
               an
               Angle
               are
               the
               bounds
               insisting
               or
               standing
               upon
               the
               Base
               of
               the
               Angle
               .
            
             
               Line
            
             
               of
               interception
               ,
               is
               the
               difference
               betwixt
               the
               Secant
               ,
               and
               the
               Radius
               ,
               and
               is
               commonly
               called
               the
               residuum
               .
            
             
               Logarithms
               ,
            
             
               are
               those
               artificiall
               numbers
               ,
               by
               which
               ,
               with
               addition
               and
               
               subtraction
               onely
               ,
               we
               work
               the
               same
               effects
               ,
               as
               by
               other
               numbers
               ,
               with
               multiplication
               and
               division
               :
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 ratio
                 ,
                 proportio
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               numerus
               .
            
             
               Logarithmication
               ,
            
             
               is
               the
               working
               of
               an
               Analogy
               by
               Logarithms
               ,
               without
               having
               regard
               to
               the
               old
               laborious
               way
               of
               the
               naturall
               Sines
               ,
               and
               Tangents
               ;
               we
               say
               likewise
               Logarithmicall
               and
               Logarithmically
               ,
               for
               Logarithmeticall
               ,
               and
               Logarithmetically
               ;
               for
               by
               the
               syncopising
               of
               et
               ,
               the
               pronunciation
               of
               those
               words
               is
               made
               to
               the
               eare
               more
               pleasant
               :
               a
               priviledge
               warranted
               by
               all
               the
               dialects
               of
               the
               Greek
               ,
               and
               other
               the
               most
               refined
               Languages
               in
               the
               world
               .
            
             
               Loxogonosphericall
               ,
            
             
               is
               said
               of
               oblique
               sphericals
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               obliquus
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 ad
                 sphaeram
                 pertinens
              
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               globus
               .
            
          
           
             
               M.
               
            
             
               MAjor
            
             
               and
               Minor
               Arches
               ,
               
                 vide
                 Arch.
              
               
            
             
               Maxim
               ,
            
             
               an
               axiome
               ,
               or
               principle
               ,
               called
               so
               (
               from
               maximus
               )
               because
               it
               is
               of
               greatest
               account
               in
               an
               Art
               or
               Science
               ,
               and
               the
               principall
               thing
               we
               ought
               to
               know
               .
            
             
               Meane
               ,
            
             
               or
               middle
               proportion
               ,
               is
               that
               ,
               the
               square
               whereof
               is
               equall
               to
               the
               plane
               of
               the
               extremes
               :
               and
               called
               so
               because
               of
               its
               situation
               in
               the
               Analogy
               .
            
             
               Mensurator
               ,
            
             
               is
               that
               ,
               whereby
               the
               illatitious
               terme
               is
               compared
               ,
               or
               measured
               with
               the
               maine
               quaesitum
               .
            
             
               Monotropall
               ,
            
             
               is
               said
               of
               figures
               ,
               which
               have
               one
               onely
               Mood
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ▪
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Monurgeticks
               ,
            
             
               are
               said
               of
               those
               Moods
               ,
               the
               maine
               Quaesitas
               whereof
               are
               obtained
               by
               one
               operation
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Moods
               ,
            
             
               aetermine
               unto
               us
               the
               severall
               manners
               of
               Triangles
               ,
               from
               modus
               ,
               a
               way
               ,
               or
               manner
               .
            
          
           
             
               N.
               
            
             
               NAturall
               ,
            
             
               the
               naturall
               parts
               of
               a
               Triangle
               ,
               are
               those
               of
               which
               it
               is
               compounded
               ,
               and
               the
               circular
               ,
               those
               whereby
               the
               maine
               quaesitum
               is
               found
               out
               .
            
             
               Nearest
               ▪
            
             
               or
               next
               ,
               is
               said
               of
               that
               Cathetopposite
               Angle
               ,
               which
               is
               immediatly
               opposite
               to
               the
               perpendicular
               .
            
             
               
               Notandum
               ,
            
             
               is
               set
               downe
               for
               an
               admonition
               to
               the
               Reader
               ,
               of
               some
               remarkable
               thing
               to
               follow
               ,
               and
               is
               the
               Gerund
               of
               
                 Noto
                 ,
                 notare
              
               .
            
          
           
             
               O.
               
            
             
               OBlique
               ,
            
             
               and
               obliquangulary
               ,
               are
               said
               of
               all
               Angles
               that
               are
               not
               right
               .
            
             
               Oblong
               ,
            
             
               is
               a
               parallelogram
               ,
               or
               square
               more
               long
               them
               large
               :
               from
               oblongus
               ,
               very
               long
               .
            
             
               Obtuse
               ,
            
             
               and
               obtuse
               angled
               ,
               are
               said
               of
               flat
               ,
               and
               blunt
               Angles
               .
            
             
               Occurse
               ,
            
             
               is
               a
               meeting
               together
               ,
               from
               
                 occurro
                 ,
                 occursum
              
               ,
            
             
               Oppobasall
               ,
            
             
               is
               said
               of
               those
               Moods
               ,
               which
               have
               a
               Catheteuretick
               Concordance
               in
               their
               Datas
               of
               the
               same
               Cathetopposite
               Angles
               ,
               and
               the
               same
               Bases
               .
            
             
               Oppocathetall
               ,
            
             
               is
               said
               of
               those
               Loxogonosphericals
               which
               have
               a
               Datisterurgetick
               Concordance
               in
               their
               Datas
               of
               the
               same
               Angles
               at
               the
               Base
               ,
               and
               the
               Perpendicular
               .
            
             
               Oppoverticall
               ,
            
             
               is
               said
               of
               those
               Moods
               which
               have
               a
               Catheteuretick
               Concordance
               in
               their
               Datas
               of
               the
               same
               Cathetopposites
               ,
               and
               verticall
               Angles
               .
            
             
               Orthogonosphericall
               ,
            
             
               is
               said
               of
               right
               angled
               Sphericals
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               rectus
               ,
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               angulus
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               gobus
               .
            
             
               Oxygonosphericall
               ,
            
             
               is
               said
               of
               acute-angled
               sphericals
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
          
           
             
               P.
               
            
             
               PArallelisme
               ,
            
             
               is
               a
               Parallel
               ,
               equality
               of
               right
               lines
               ,
               cut
               with
               a
               right
               line
               ,
               or
               of
               Sphericals
               with
               a
               Sphericall
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               equidistans
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
             
               Parallelogram
               ,
            
             
               is
               an
               oblong
               ,
               long
               square
               ,
               rectangle
               ,
               or
               figure
               made
               of
               parallel
               lines
               :
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               linea
               .
            
             
               Partiall
               ,
            
             
               is
               said
               of
               enodandas
               depending
               on
               severall
               Axioms
               .
            
             
               Particularise
               ,
            
             
               specialise
               ,
               by
               some
               especiall
               difference
               to
               contract
               the
               generality
               of
               a
               thing
               .
            
             
               Partition
               ,
            
             
               is
               said
               of
               the
               severall
               operations
               of
               every
               Loxogonosphericall
               Mood
               ,
               and
               is
               divided
               in
               praenoscendall
               ,
               catheteuretick
               and
               hysterurgetick
               .
            
             
               Permutat
            
             
               proportion
               ,
               or
               proportion
               by
               permutation
               ,
               or
               alternat
               proportion
               ,
               
               is
               when
               the
               Antecedent
               is
               compared
               to
               the
               Antecedent
               ,
               and
               the
               Consequent
               to
               the
               Consequent
               ,
               
                 vide
                 ,
                 Perturbat
              
               .
            
             
               Perpendicularity
               ,
            
             
               is
               the
               affection
               of
               the
               Perpendicular
               ,
               or
               plumb-line
               ;
               which
               comes
               from
               
                 perpendendo
                 ,
                 id
                 est
                 ,
                 explorando
                 altitudinem
                 .
              
            
             
               Perturbat
               ,
            
             
               is
               the
               same
               as
               permutat
               ,
               and
               called
               so
               because
               the
               order
               of
               the
               Analogie
               is
               perturbed
               .
            
             
               Planobliquangular
               ,
            
             
               is
               said
               of
               plaine
               Triangles
               ,
               wherein
               there
               is
               no
               right
               Angle
               at
               all
               .
            
             
               Planorectangular
               ,
            
             
               is
               said
               of
               plaine
               right-angled
               Triangles
               .
            
             
               Planotriangular
               ,
            
             
               is
               said
               of
               plaine
               Triangles
               ,
               that
               is
               ,
               such
               as
               are
               not
               Sphericall
               .
            
             
               Pleuseotechnie
               ,
            
             
               the
               Art
               of
               Navigation
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               navigatio
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               ars
               .
            
             
               Plusminused
               ,
            
             
               is
               said
               of
               Moods
               which
               admit
               of
               Mensurators
               or
               whose
               illatitious
               termes
               are
               never
               the
               same
               ,
               but
               either
               more
               or
               lesse
               then
               the
               maine
               quaehtas
               .
            
             
               Poliechyrologie
               ,
            
             
               the
               Art
               of
               fortifying
               Townes
               and
               Cities
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               urbs
               ,
               civit
               as
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 munio
                 firmo
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               ratio
               .
            
             
               Possubservient
               ,
            
             
               is
               that
               which
               after
               another
               serveth
               for
               the
               resolving
               of
               a
               question
               ;
               of
               post
               ,
               and
               subserviens
               :
               of
               sub
               and
               servio
               .
            
             
               Potentia
               ,
            
             
               is
               that
               wherein
               the
               force
               and
               whole
               result
               of
               another
               thing
               lies
               .
            
             
               Power
               ,
            
             
               is
               the
               square
               ,
               quadrat
               ,
               or
               product
               of
               a
               line
               extended
               upon
               it selfe
               ,
               or
               of
               a
               number
               in
               it selfe
               multiplied
               .
            
             
               Powered
               ,
            
             
               squared
               quadrified
               .
            
             
               Precept
               ,
            
             
               document
               ,
               from
               
                 praecipio
                 ,
                 praeceptum
              
               .
            
             
               Praeroscenda
               ,
            
             
               are
               the
               termes
               ,
               which
               must
               be
               knowne
               before
               we
               can
               attaine
               to
               the
               knowledge
               of
               the
               maine
               quaesitas
               of
               prae
               and
               nosco
               .
            
             
               Praenoscendall
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               those
               Moods
               ,
               which
               agree
               in
               the
               same
               praenoscendas
               .
            
             
               Praesection
               ,
            
             
               praesectionall
               ,
               is
               concerning
               the
               digit
               towards
               the
               left
               ,
               whose
               cutting
               off
               saveth
               the
               labour
               of
               subtracting
               the
               double
               or
               single
               Radius
               .
            
             
               Praescinded
               problems
               ,
            
             
               are
               those
               speculative
               Datoquaeres
               ,
               which
               are
               not
               applied
               to
               any
               matter
               by
               way
               of
               practice
               .
            
             
               Praesubservient
               ,
            
             
               is
               said
               of
               those
               Moods
               which
               in
               the
               first
               place
               we
               must
               make
               use
               of
               for
               the
               explanation
               of
               others
               ;
               of
               prae
               ,
               and
               ●ub●ervio
               .
            
             
               Prime
               ,
            
             
               is
               said
               of
               the
               furthest
               Cathetopposite
               ,
               or
               Angle
               at
               the
               Base
               ,
               contained
               within
               the
               Triangle
               to
               be
               resolved
               .
            
             
               
               Primifie
               the
               Radius
               ,
            
             
               is
               to
               put
               the
               Radius
               in
               the
               first
               place
               ,
               
                 primumque
                 inter
                 terminos
                 collocare
                 proportionales
              
               .
            
             
               Problem
               ,
               problemet
               ,
            
             
               a
               question
               or
               datoquaere
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               unde
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               propositum
               ,
               objectaculum
               .
            
             
               Product
               ,
            
             
               is
               the
               result
               ,
               factus
               ,
               or
               operatum
               of
               a
               multiplication
               ,
               from
               
                 produco
                 ,
                 productum
              
               .
            
             
               Proportion
               ,
            
             
               proportionality
               ,
               are
               the
               same
               as
               Analogy
               ,
               and
               Analogisme
               ;
               the
               first
               being
               a
               likenesse
               of
               termes
               ,
               the
               other
               of
               proportions
               .
            
             
               Proposition
               ,
            
             
               a
               proposed
               sentence
               ,
               whether
               theorem
               or
               problem
               .
            
             
               Prosiliencie
               ,
            
             
               is
               a
               demission
               ,
               or
               falling
               of
               the
               Perpendicular
               ,
               from
               
                 prosilio
                 ,
                 ex
                 pro
                 &
                 salio
                 .
              
            
             
               Proturgetick
               ,
            
             
               is
               said
               of
               the
               first
               operation
               of
               every
               Disergetick
               Mood
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               the
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               being
               Attically
               contracted
               into
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               .
            
          
           
             
               Q.
               
            
             
               QUadrant
               ,
            
             
               the
               fourth
               part
               of
               a
               Circle
               .
            
             
               Quadranting
               ,
            
             
               the
               protracting
               of
               a
               Sphericall
               side
               unto
               a
               Quadrant
               .
            
             
               Quadrat
               ,
            
             
               a
               Square
               ,
               
                 a
                 forma
                 quadrae
              
               ,
               the
               power
               or
               possibility
               of
               a
               line
               ,
               
                 vide
                 ,
                 Power
              
               .
            
             
               Quadrobiquadraequation
               ,
            
             
               concerneth
               the
               Square
               of
               the
               subtendent
               side
               ,
               which
               is
               equall
               to
               the
               Biquadrat
               ,
               or
               two
               Squares
               of
               the
               Ambients
               .
            
             
               Quadrosubduction
               ,
            
             
               is
               concerning
               the
               subtracting
               of
               the
               Square
               of
               one
               of
               the
               Ambients
               from
               the
               Square
               of
               the
               Subtendent
               .
            
             
               Quaesitas
               ,
            
             
               the
               things
               demanded
               from
               
                 quaero
                 ▪
                 quaesitum
              
               .
            
             
               Quotient
               ,
            
             
               is
               the
               result
               of
               a
               division
               ,
               from
               quoties
               ,
               how
               many
               times
               .
            
          
           
             
               R.
               
            
             
               RAdically
               meeting
               ,
            
             
               is
               said
               of
               those
               Oblongs
               ,
               or
               Squares
               ,
               whose
               sides
               doe
               meet
               together
               .
            
             
               Radius
               ,
            
             
               ray
               ,
               or
               beame
               is
               the
               Semidiameter
               ,
               called
               so
               metaphorically
               ,
               from
               the
               spoake
               of
               a
               wheele
               which
               is
               to
               the
               limb
               thereof
               ,
               as
               the
               Semidiameter
               ,
               to
               the
               circumference
               of
               a
               circle
               .
            
             
               Reciprocall
               ,
            
             
               is
               said
               of
               proportionalities
               ,
               or
               two
               rowes
               of
               proportionals
               ,
               wherein
               the
               first
               of
               the
               first
               is
               to
               the
               first
               of
               the
               second
               ,
               as
               the
               last
               of
               
               the
               second
               is
               to
               the
               last
               of
               the
               first
               ,
               and
               contrarily
               .
            
             
               Rectangular
               ,
            
             
               is
               said
               of
               those
               figures
               ,
               which
               have
               right
               Angles
               .
            
             
               Refinedly
               ,
            
             
               is
               said
               when
               we
               go
               the
               shortest
               way
               to
               work
               by
               primifying
               the
               Radius
               .
            
             
               Renvoy
               ,
            
             
               a
               remitting
               from
               one
               place
               to
               another
               ,
               it
               comes
               from
               the
               French
               word
               Renvoyer
               .
            
             
               Representative
               ,
            
             
               is
               said
               of
               the
               letters
               ,
               which
               stand
               for
               whole
               words
               ;
               as
               E.
               for
               side
               ,
               L.
               for
               secant
               ,
               U.
               for
               subtendent
               .
            
             
               Residuat
               ,
            
             
               is
               to
               leave
               a
               remainder
               ,
               
                 nempe
                 id
                 quod
                 residet
                 &
                 superest
                 .
              
            
             
               Resolver
               ,
            
             
               is
               that
               which
               looseth
               and
               untieth
               the
               knot
               of
               a
               difficulty
               ,
               of
               re
               and
               solvo
               .
            
             
               Resolutory
               ,
            
             
               is
               said
               of
               the
               last
               partition
               of
               the
               Loxogonosphericall
               operations
               .
            
             
               Result
               ,
            
             
               is
               the
               last
               effect
               of
               a
               work
               .
            
             
               Root
               ,
            
             
               is
               the
               side
               of
               a
               Square
               ,
               Cube
               ,
               or
               any
               cossick
               figure
               .
            
          
           
             
               S.
               
            
             
               SCheme
               ,
            
             
               signifieth
               here
               the
               delineation
               of
               a
               Geometricall
               figure
               ,
               with
               all
               parts
               necessary
               for
               the
               illustrating
               of
               a
               demonstration
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               habeo
               .
            
             
               Sciography
               ,
            
             
               the
               Art
               of
               shadowing
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               umbra
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               scribo
               .
            
             
               Segment
               ,
            
             
               the
               portion
               of
               a
               thing
               cut
               off
               ,
               
                 quasi
                 secamentum
                 ,
                 quod
                 a
                 re
                 aliqua
                 secatur
                 .
              
            
             
               Sexagesimat
               ,
            
             
               subsexagesimat
               ,
               resubsexagesimat
               ,
               and
               biresubsexagesimat
               ,
               are
               said
               of
               the
               division
               ,
               subdivision
               ,
               resubdivision
               ,
               and
               reer-resubdivision
               of
               degrees
               into
               minuts
               ,
               seconds
               ,
               thirds
               ,
               and
               fourths
               ,
               in
               60.
               of
               each
               other
               :
               the
               devisor
               of
               the
               fore
               goer
               being
               successively
               the
               following
               dividend
               ,
               and
               the
               quotient
               alwayes
               sixty
               .
            
             
               Sharp
               ,
            
             
               is
               said
               of
               acute
               Angles
               .
            
             
               Sindiforall
               ,
            
             
               is
               said
               of
               those
               Moods
               ,
               the
               fourth
               terme
               of
               whose
               Analogie
               is
               onely
               illatitious
               to
               the
               maine
               quaesitum
               .
            
             
               Sindiforation
               ,
            
             
               is
               the
               affection
               of
               those
               foresaid
               Moods
               ,
               whereby
               the
               value
               of
               the
               mensurator
               is
               knowne
               .
            
             
               Sindiforatall
               ,
            
             
               is
               concerning
               those
               Moods
               ,
               whose
               illatitious
               terme
               is
               an
               Angle
               .
            
             
               Sindiforiutall
               ,
            
             
               is
               of
               those
               Moods
               ,
               whose
               illatitious
               terme
               is
               a
               side
               ;
               all
               these
               foure
               words
               are
               composed
               of
               representatives
               ,
               and
               (
               if
               I
               remember
               well
               )
               mentioned
               in
               my
               explanation
               .
            
             
               
               Sinocosinall
               ,
            
             
               is
               said
               of
               the
               Concordances
               of
               those
               Moods
               ,
               which
               agree
               in
               this
               ,
               that
               their
               Analogies
               run
               upon
               sines
               ,
               and
               sine-complements
               .
            
             
               Sinocotangentall
               ,
            
             
               is
               said
               of
               those
               Moods
               ,
               which
               agree
               in
               that
               the
               termes
               of
               their
               Analogie
               run
               upon
               Sines
               and
               Tangent-complements
               .
            
             
               Sinus
               ,
            
             
               is
               so
               called
               (
               I
               beleeve
               )
               because
               it
               is
               alwayes
               in
               the
               very
               bosome
               of
               the
               Circle
               .
            
             
               Sinused
               ,
            
             
               is
               said
               of
               termes
               endowed
               or
               invested
               with
               Sines
               .
            
             
               Specialized
               ,
            
             
               contracted
               to
               more
               particular
               termes
               ,
               
                 vide
                 ,
                 Individuated
              
               .
            
             
               Specifying
               ,
            
             
               determinating
               ,
               particularising
               .
            
             
               Specification
               ,
            
             
               a
               making
               more
               especiall
               ,
               by
               contracting
               the
               generality
               of
               a
               thing
               ,
               
                 vide
                 ,
                 Specialized
              
               .
            
             
               Sphericodisergeticks
               ,
            
             
               are
               the
               Sphericall
               Triangles
               of
               two
               operations
               .
            
             
               Structure
            
             
               of
               an
               operation
               ,
               is
               the
               whole
               frame
               thereof
               ,
               from
               
                 struo
                 ,
                 structum
              
               .
            
             
               Subdatoquaere
               ,
            
             
               is
               a
               particular
               datoquaere
               ,
               and
               is
               applied
               to
               the
               problems
               of
               the
               cases
               of
               every
               Sphericodisergetick
               Mood
               ,
               
                 vide
                 ,
                 Sub-problems
              
               .
            
             
               Sabajcent
               ,
            
             
               is
               the
               substerned
               side
               or
               the
               Base
               ,
               of
               
                 sub
                 &
                 jaceo
              
               ,
               vide
               ,
               
                 Sustentative
                 ,
                 Sustaining
                 side
              
               and
               Substerned
               .
            
             
               Subordinate
            
             
               problems
               ,
               is
               the
               same
               with
               subdatoquaere
               .
            
             
               Subproblems
               ,
            
             
               is
               the
               same
               with
               subordinate
               problems
               ,
               or
               problemets
               .
            
             
               Subservient
               ,
            
             
               is
               said
               of
               Moods
               which
               serve
               in
               the
               operation
               of
               other
               Moods
               .
            
             
               Substerned
               ,
            
             
               is
               the
               subjacent
               side
               or
               Base
               :
               or
               ,
               more
               generally
               ,
               any
               side
               opposite
               to
               an
               Angle
               ;
               of
               sub
               and
               
                 sterno
                 ,
                 sternere
              
               ,
               vide
               ,
               Subjacent
               .
            
             
               Subtendent
               ,
            
             
               is
               the
               side
               opposite
               to
               the
               right
               Angle
               ,
               of
               sub
               and
               tendo
               ;
               as
               if
               you
               would
               say
               ,
               Under-stretched
               .
            
             
               Subtendentine
               ,
            
             
               is
               the
               subtendent
               of
               a
               little
               rectangled
               Triangle
               ,
               comprehended
               within
               the
               Area
               of
               a
               great
               one
               ,
               and
               is
               sometimes
               called
               the
               little
               subtendent
               ,
               and
               reere
               subtendent
               .
            
             
               Subtendent
               all
               ,
            
             
               is
               the
               subtendent
               of
               a
               great
               rectangled
               Triangle
               ,
               within
               whose
               capacity
               is
               included
               a
               little
               one
               :
               it
               is
               likewise
               called
               the
               great
               subtendent
               ,
               and
               maine
               subtendent
               .
            
             
               Supernumerary
               ,
            
             
               is
               said
               of
               the
               digit
               ,
               by
               the
               which
               the
               proposed
               number
               exceeds
               in
               places
               the
               number
               of
               the
               places
               of
               the
               Radius
               .
            
             
               Supplements
               ,
            
             
               are
               the
               Oblongs
               made
               of
               the
               Segments
               of
               the
               root
               of
               a
               Square
               ;
               and
               so
               called
               ,
               because
               they
               supply
               all
               that
               the
               Diagonals
               or
               
               Squares
               of
               the
               Segments
               joyned
               together
               ,
               want
               of
               the
               whole
               lines
               square
               .
            
             
               Suppone
               severally
               ,
            
             
               is
               to
               signifie
               severall
               things
               .
            
             
               Sustaining
               side
            
             
               is
               the
               substerned
               ,
               or
               subjacent
               side
               .
            
             
               Sustentative
               ,
            
             
               is
               the
               same
               with
               sustaining
               ,
               substerned
               ,
               subjacent
               and
               Base
               .
            
             
               Sympathie
            
             
               of
               Angles
               ,
               is
               a
               similitude
               in
               their
               affection
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               passio
               ,
               vide
               homogeneall
               .
            
          
           
             
               T.
               
            
             
               TAble
               ,
            
             
               is
               an
               Index
               sometimes
               ,
               and
               sometimes
               it
               is
               taken
               for
               a
               Briefe
               and
               summary
               way
               of
               expressing
               many
               things
               .
            
             
               Tangentine
               ,
            
             
               is
               that
               which
               concerneth
               Tangents
               or
               touch-lines
               .
            
             
               Tangentosinall
               ,
            
             
               is
               said
               of
               the
               Concordance
               of
               those
               Loxogonosphericals
               ,
               the
               termes
               of
               whose
               Analogie
               runne
               upon
               Tangents
               and
               Sines
               .
            
             
               Tenet
               ,
            
             
               is
               a
               secondary
               maxim
               ,
               and
               is
               onely
               said
               here
               of
               Cathetothetick
               principles
               .
            
             
               Theorematick
               ,
            
             
               speculative
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 a
                 speculation
              
               ,
               which
               cometh
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               or
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               speculare
               ,
               or
               contemplare
               .
            
             
               Topanglet
               ,
            
             
               and
               verticalin
               are
               the
               same
               .
            
             
               Trigonometry
               ,
            
             
               is
               the
               Art
               of
               calculating
               ,
               and
               measuring
               of
               Triangles
               ,
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               triangulus
               ,
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               metior
               .
            
             
               Trissotetras
               ,
            
             
               is
               that
               which
               runneth
               all
               along
               upon
               threes
               and
               foures
               of
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               and
               
                 in
                 plurali
              
               ,
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 tertius
                 ,
                 trinus
                 ,
                 triplex
                 ,
                 tres
                 ,
              
               and
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               
                 numerus
                 quaternarius
              
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               quatuor
               .
            
          
           
             
               U.
               
            
             
               VAriator
               ,
            
             
               is
               from
               
                 vario
                 ,
                 variare
              
               ,
               to
               diver
               sifie
               ,
               and
               is
               said
               of
               cases
               ,
               which
               upon
               the
               same
               Datas
               are
               onely
               diverse
               in
               the
               manner
               of
               resolving
               the
               Quaesitum
               .
            
             
             
             
             
             
               
               Verticaline
               ,
            
             
               verticall
               ,
               verticalet
               ,
               are
               the
               top-angles
               ,
               and
               top-anglets
               ,
               from
               
                 vertex
                 ,
                 verticis
              
               .
            
             
               Underproblem
               ,
            
             
               problemet
               ,
               subordinate
               problem
               ,
               sub-problem
               ,
               under-datoquaere
               ,
               and
               sub
               ▪
               datoquaere
               are
               ,
               all
               the
               same
               thing
               .
            
             
               Unradiated
               ,
            
             
               or
               unradiused
               ,
               is
               said
               of
               a
               summe
               of
               Logarithms
               from
               which
               the
               Radius
               is
               abstracted
               .
            
          
           
             
               Z.
               
            
             
               ZEterick
               ,
            
             
               is
               said
               of
               Loxogonosphericall
               Moods
               which
               agree
               in
               the
               same
               quaesitas
               ,
               from
               
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
                 〈◊〉
              
               ,
               quaero
               ,
               inquiro
               .
            
          
        
      
       
         
         
           The
           finall
           Conclusion
           .
        
         
           IF
           the
           novelty
           of
           this
           my
           Invention
           be
           acceptable
           (
           as
           I
           doubt
           not
           but
           it
           will
           )
           to
           the
           most
           Learned
           ,
           and
           judicious
           Mathematicians
           ,
           I
           have
           already
           reaped
           all
           the
           benefit
           I
           expected
           by
           it
           ,
           and
           shall
           hereafter
           ,
           (
           God
           willing
           )
           without
           hope
           of
           any
           further
           recompence
           ,
           cheerfully
           under-goe
           more
           laborious
           employments
           ,
           of
           the
           like
           nature
           ,
           to
           doe
           them
           service
           :
           But
           as
           for
           such
           ,
           who
           ,
           either
           understanding
           it
           not
           ,
           or
           vain-gloriously
           being
           accustomed
           to
           Criticise
           on
           the
           Works
           of
           others
           ,
           will
           presume
           to
           carp
           therein
           at
           what
           they
           cannot
           amend
           ,
           I
           pray
           God
           to
           illuminate
           their
           Judgements
           ,
           and
           rectifie
           their
           Wils
           ,
           that
           they
           may
           know
           more
           ,
           and
           censure
           lesse
           ;
           for
           so
           by
           forbearing
           detraction
           ,
           the
           venom
           whereof
           must
           needs
           reflect
           upon
           themselves
           ,
           they
           will
           come
           to
           approve
           better
           of
           the
           endeavours
           of
           those
           ,
           that
           wish
           them
           no
           harme
           .
        
         
           Sit
           Deo
           Gloria
           .
        
      
       
         
         
           The
           Diorthosis
           .
        
         
           THe
           mistakes
           of
           the
           Presse
           ,
           can
           breed
           but
           little
           obstruction
           to
           the
           progresse
           of
           the
           ingenious
           Reader
           ,
           if
           with
           his
           Pen
           ,
           before
           he
           enter
           upon
           the
           perusall
           of
           this
           Treatise
           ,
           he
           be
           pleased
           thus
           to
           correct
           (
           as
           I
           hope
           he
           will
           )
           these
           ensuing
        
         
           
             
               Erratas
               .
            
             
               
                 Pag.
                 
              
               
                 lin
                 .
              
               
                 Errata
                 .
              
               
                 Emendata
                 .
              
               
                 Pag.
                 
              
               
                 lin
                 .
              
               
                 Errata
                 .
              
               
                 Emendata
                 .
              
            
             
               
                 8
              
               
                 25
              
               
                 Talfagro
                 Talzo
                 .
              
               
                 Talfagros
                 Talzos
                 .
              
               
                 16
              
               
                 25
              
               
                 This
                 Cheme
                 .
              
               
                 the
                 Schemes
                 &
                 dining
                 room
                 totall
                 summe
                 .
              
            
             
               
                 10
              
               
                 17
              
               
                 Niubprodesver
                 .
              
               
                 Niubprodnesver
                 .
              
               
                 16
              
               
                 31
              
               
                 or
                 dining
                 roome
                 totall
                 Sine
                 .
              
               
                  
              
            
             
               
                 11
              
               
                 29
              
               
                 Natfr
                 .
                 Autir
                 .
              
               
                 Natfr
                 .
                 Eutir
                 .
              
               
                 23
              
               
                 23
              
               
                  
              
               
                  
              
            
             
               
                 11
              
               
                 35
              
               
                 Nat.
                 Nad
                 Nath.
                 
              
               
                 Naet
                 .
                 Naed
                 .
                 Naeth
                 .
              
               
                 26
              
               
                 6
              
               
                 as
                 the
                 Sine
                 of
                 the
                 cosubtendent
                 .
              
               
                 as
                 the
                 cosubtendent
                 .
              
            
             
               
                 11
              
               
                 36
              
               
                 Eheromabme
                 .
              
               
                 Eherolabme
                 .
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
            
             
               
                 11
              
               
                 37
              
               
                 Being
                 Allotted
                 .
              
               
                 Being
                 abinarie
                 allotted
                 .
              
               
                  
              
               
                  
              
               
                  
              
               
                 second
                 basidion
                 .
              
            
             
               
                 16
              
               
                 8
              
               
                  
              
               
                  
              
               
                 80
              
               
                 9
              
               
                 second
                 Base
                 .
              
               
                  
              
            
          
        
         
           What
           errors
           else
           (
           if
           any
           )
           have
           slipt
           animadversion
           (
           besides
           their
           not
           being
           very
           materiall
           )
           are
           so
           intelligible
           ,
           that
           being
           by
           the
           easiest
           judgement
           with
           as
           much
           facility
           eschewable
           ,
           as
           I
           can
           observe
           them
           ,
           not
           to
           mention
           the
           commission
           of
           such
           faults
           is
           no
           great
           omission
           ;
           and
           therefore
           will
           I
           heartily
           (
           without
           further
           ceremony
           )
           conduct
           the
           Student
           (
           who
           making
           this
           the
           beginning
           of
           the
           Book
           ,
           as
           it
           is
           most
           fit
           he
           doe
           ,
           seeing
           a
           Ruler
           should
           be
           made
           streight
           before
           any
           thing
           be
           ruled
           by
           it
           )
           is
           willing
           to
           go
           along
           with
           me
           from
           hence
           circularly
           through
           the
           title
           ,
           to
           the
           end
           of
           the
           Treatise
           in
           the
           proposed
           way
           ,
           as
           followes
           .
        
         
           And
           so
           God
           blesse
           us
           both
           .