according_a to_o their_o present_a site_n and_o positure_n without_o inversion_n of_o either_o when_o crooked_a by_o move_v the_o side_n near_a one_o another_o by_o the_o continuance_n of_o such_o motion_n they_o can_v never_o be_v bring_v to_o be_v coincident_a with_o and_o coapt_v exact_o the_o one_o unto_o the_o other_o and_o that_o by_o a_o anisoclitical_a angle_n we_o understand_v a_o plane_n angle_v contain_v under_o such_o anisoclitical_a side_n so_o all_o crooked_a line_v angle_n of_o concave_n though_o have_v side_n of_o like_a curvature_n all_o crooked_a line_v angle_n of_o convex_n though_o have_v side_n of_o like_a curvature_n all_o concavo-convexe_a angle_n who_o side_n be_v of_o different_a curvature_n all_o mix_v line_v angle_n whether_o recto-convexes_a or_o recto-concaves_a all_o mix_v crooked_a line_v angle_n whatsoever_o be_v all_o of_o they_o manifest_o anisoclitical_a angle_n and_o their_o side_n anisoclitical_a so_o in_o fig._n 4_o the_o though_o ab_fw-la and_o ac_fw-la be_v suppose_v to_o be_v of_o uniform_a answer_v and_o equal_a curvature_n and_o likewise_o in_o fig._n 2_o d._n though_o ab_fw-la and_o ac_fw-la again_o suppose_v to_o be_v of_o uniform_a answer_v and_o equal_a curvature_n and_o in_o fig._n 3_o d._n fig_n 5_o 6_o 9_o 10_o 11._o suppose_v ab_fw-la and_o ac_fw-la not_o to_o be_v of_o uniform_a answer_v and_o equal_a curvature_n by_o move_v the_o side_n ab_fw-la &_o ac_fw-la in_o that_o their_o present_a fire_n &_o positure_n upon_o the_o angular_a point_n a_o without_o the_o inversion_n of_o either_o when_o both_o be_v crooked_a to_o bring_v the_o side_n near_o than_o one_o to_o the_o other_o it_o be_v manifest_a that_o by_o the_o continuance_n of_o such_o motion_n they_o can_v never_o be_v bring_v to_o a_o coincidence_n with_o and_o to_o be_v coapt_v exact_o the_o one_o unto_o the_o other_o and_o such_o also_o be_v all_o angle_n of_o contact_n whatsoever_o as_o in_o fig._n 7_o 8_o 11._o ab_fw-la &_o ac_fw-la be_v as_o uncoincidable_a and_o uncoaptable_a as_o in_o the_o former_a except_o only_o ultradiametral_a concavo-convexe_a angle_n of_o contact_n which_o be_v of_o equal_a and_o answer_a curvature_n 18._o that_o by_o a_o angle_n of_o curvature_n or_o coincidence_n we_o understand_v a_o plane_n angle_n contain_v by_o any_o two_o part_n of_o a_o crooked_a line_n at_o the_o point_n of_o their_o concurrence_n any_o where_o to_o be_v imagine_v or_o take_v in_o crooked-line_n so_o in_o the_o 7_o the_o fig._n let_v bad_a be_v the_o circumference_n of_o a_o circle_n parabola_fw-la hyperbola_fw-la ellipsis_n etc._n etc._n at_o the_o mean_a point_n a_o the_o side_n basilius_n and_o ad_fw-la contain_v a_o angle_n of_o curvature_n or_o coincidence_n 19_o that_o by_o autoclitical_a curvature_n and_o so_o by_o a_o autoclitical_a crooked_a line_n we_o understand_v such_o a_o crooked_a line_n as_o pass_v from_o the_o angular_a point_n of_o a_o right_n line_v angle_n between_o the_o two_o side_n by_o the_o inclination_n of_o its_o curvature_n keep_v the_o convexe_a side_n of_o its_o curvature_n constant_o obvert_v to_o one_o of_o the_o side_n and_o the_o concave_a side_n of_o its_o curvature_n constant_o obvert_v to_o the_o other_o as_o in_o fig._n 17._o in_o the_o right_n line_v angle_n bac_n the_o crooked_a line_n aef_n keep_v its_o convexity_n constant_o obvert_v to_o the_o side_n ab_fw-la and_o its_o concavity_n to_o the_o side_n ac_fw-la or_o so_o much_o of_o it_o as_o be_v intercept_v between_o the_o intersection_n at_o f_o and_o the_o angular_a point_n at_o a_o the_o crooked_a line_n aef_n be_v autoclitical_a i._n e._n the_o convexity_n be_v all_o on_o one_o side_n and_o the_o concavity_n all_o on_o the_o other_o 20._o that_o by_o antanaclitical_a curvature_n and_o so_o by_o a_o antanaclitical_a curve_v line_n we_o understand_v such_o a_o crooked_a line_n as_o pass_v from_o the_o angular_a point_n of_o a_o right-lined_n angle_n between_o the_o two_o side_n by_o the_o inclination_n of_o its_o curvature_n have_v the_o convexe_a part_n of_o its_o curvature_n sometime_o towards_o the_o one_o side_n of_o the_o right_n line_v angle_n and_o sometime_o towards_o the_o other_o side_n of_o the_o right_n line_v angle_n i._n e._n the_o convexeness_n and_o the_o concaveness_n be_v not_o constant_o on_o the_o same_o several_a side_n as_o in_o fig._n 17._o in_o the_o right_n line_v angle_n bac_n the_o crooked_a line_n agh_o obvert_v the_o convexity_n at_o g_o towards_o ab_fw-la and_o at_o h_n towards_o ac_fw-la be_v antanaclitical_a these_o definition_n praemise_v to_o give_v now_o the_o true_a state_n of_o the_o controversy_n let_v there_o be_v as_o in_o fig._n 12._o two_o equal_a circle_n ahh_a and_o adk_n touch_v in_o the_o point_n a_o and_o let_v agnostus_n be_v the_o semidiameter_n of_o the_o circle_n adk_n and_o ab_fw-la be_v a_o right_a line_n tangent_fw-la touch_v both_o the_o circle_n in_o the_o point_n a_o and_o let_v ael_n be_v a_o great_a circle_n then_o either_o touch_v both_o the_o former_a and_o also_o the_o right_a line_n tangent_fw-la in_o the_o point_n a_o and_o from_o the_o point_n a_o draw_v the_o right_a line_n ac_fw-la at_o pleasure_n cut_v the_o circle_n adk_v in_o the_o point_n d_o and_o the_o circle_n ael_n in_o the_o point_n e_o now_o therefore_o whereas_o you_o say_v that_o the_o recto-convexe_a angle_n of_o contact_n bae_n and_o b_o of_o be_v not_o unequal_a and_o that_o neither_o of_o they_o be_v quantitative_a and_o that_o the_o crooked-line_n e_o a_n da_z ha_o be_v coincident_a sc._n so_o as_o to_o make_v no_o angle_n with_o the_o right_a line_n ab_fw-la or_o one_o with_o another_o and_o that_o the_o right-lined_n right_a angle_n b_o agnostus_n be_v equal_a to_o the_o mixt-lined_n i._n e._n recto-concave_a angle_n of_o the_o semicircle_n e_o agnostus_n and_o d_o agnostus_n several_o and_o that_o those_o angle_n of_o the_o semicircle_n e_o agnostus_n and_o d_o agnostus_n be_v equal_a the_o one_o unto_o the_o other_o and_o that_o the_o mix_v line_v recto-convexe_a angle_n f_o agnostus_n be_v several_o equal_a to_o all_o or_o any_o of_o the_o former_a and_o that_o there_o be_v no_o heterogeneity_n among_o plane_n angle_n but_o that_o they_o be_v all_o of_o they_o of_o the_o same_o sort_n and_o homogeneal_a and_o undevideable_a into_o part_n specifical_o different_a distinct_a and_o heterogeneal_a in_o respect_n of_o one_o another_o and_o the_o whole_a and_o that_o to_o any_o mix_v or_o crooked_a line_v angle_n whatsoever_o that_o be_v quantitative_a it_o be_v not_o impossible_a to_o give_v a_o equal_a right_n line_v angle_n i_o acknowledge_v for_o all_o these_o thing_n you_o have_v dispute_v very_o subtle_o yet_o i_o must_v with_o a_o clear_a and_o free_a judgement_n own_v and_o declare_v a_o dissent_n from_o you_o in_o they_o all_o for_o the_o reason_n to_o be_v allege_v in_o the_o ensue_a discourse_n to_o clear_v all_o which_o nothing_o can_v be_v of_o high_a consequence_n in_o this_o question_n then_o true_o to_o understand_v the_o nature_n of_o a_o angle_n what_o a_o angle_n be_v what_o be_v a_o angle_n and_o what_o be_v not_o and_o to_o exclude_v the_o consideration_n of_o angle_n herein_o unconcern_v they_o be_v plane_n angle_v i_o e._n such_o as_o be_v contain_v by_o line_n which_o lie_v whole_o and_o both_o in_o the_o same_o plane_n the_o disquisition_n of_o who_o nature_n we_o be_v now_o about_o and_o such_o be_v the_o affinity_n which_o the_o inclination_n of_o one_o line_n have_v to_o another_o in_o the_o same_o plane_n with_o the_o nature_n of_o a_o angle_n that_o without_o it_o a_o plane_n angle_n can_v be_v define_v or_o conceive_v for_o where_o there_o be_v rectitude_n and_o voidness_n of_o inclination_n as_o in_o a_o right_a line_n and_o its_o production_n there_o never_o be_v just_o suspect_v to_o be_v any_o thing_n latent_fw-la of_o a_o angular_a nature_n nor_o between_o parallel_n for_o want_v of_o mutual_a inclination_n but_o yet_o the_o inclination_n of_o line_n upon_o line_n in_o the_o same_o plane_n be_v not_o sufficient_a to_o make_v up_o the_o nature_n of_o a_o angle_n for_o in_o the_o same_o plane_n one_o line_n may_v have_v inclination_n to_o another_o and_o yet_o they_o never_o meet_v nor_o have_v in_o their_o infinite_a regular_a production_n any_o possibility_n of_o ever_o meet_v as_o the_o circumference_n of_o a_o ellipsis_n or_o circle_n to_o a_o right_a line_n lie_v whole_o without_o they_o without_o either_o section_n or_o contact_n or_o the_o asymptotes_n in_o conjugate_v section_n which_o though_o ever_o make_v a_o close_a appropinquation_n to_o the_o circumference_n of_o the_o conjugate_v figure_n yet_o infinite_o produce_v never_o attain_v a_o concurrency_n and_o though_o by_o possibility_n the_o incline_v line_n may_v meet_v yet_o if_o they_o do_v not_o a_o angle_n be_v not_o constitute_v only_o there_o be_v possibility_n of_o a_o angle_n when_o be_v produce_v they_o shall_v meet_v
so_o as_o to_o make_v up_o the_o nature_n of_o a_o angle_n two_o line_n must_v be_v incline_v one_o to_o another_o and_o also_o concur_v or_o meet_v to_o contain_v on_o their_o part_n a_o certain_a space_n or_o part_n of_o the_o plane_n between_o their_o production_n from_o their_o point_n of_o concurrence_n or_o their_o angular_a point_n and_o this_o be_v the_o general_a and_o proper_a nature_n of_o a_o plane_n angle_n as_o it_o be_v manifest_a that_o in_o a_o right_a line_n there_o be_v no_o angle_n so_o it_o be_v dubious_a what_o be_v to_o be_v judge_v of_o those_o curve_v or_o crooked_a line_n who_o curvature_n be_v either_o equal_a uniform_a or_o regular_a sc._n whether_o in_o such_o line_n there_o be_v not_o at_o every_o mean_a point_n a_o angularness_n the_o line_n still_o lie_v in_o the_o same_o positure_n it_o be_v clear_a in_o a_o right_a line_n no_o mean_a point_n can_v be_v take_v at_o which_o the_o part_n of_o the_o right_a line_n in_o their_o present_a position_n can_v be_v say_v to_o have_v any_o inclination_n one_o to_o another_o that_o though_o the_o part_n concur_v yet_o they_o want_v inclination_n but_o in_o the_o circumference_n of_o circle_n ellipse_n hyperbola_n parabola_n and_o such_o like_v curve_v or_o crooked_a line_n be_v of_o equal_a uniform_a or_o regular_a curvature_n no_o point_n can_v be_v assign_v at_o which_o the_o part_n in_o their_o present_a position_n have_v not_o a_o special_a inclination_n one_o to_o another_o so_o as_o as_o before_z inclination_n of_o line_n and_o concurrency_n make_v up_o the_o nature_n of_o a_o angle_n it_o seem_v not_o reasonable_a to_o deny_v angularity_n at_o any_o point_n of_o such_o crooked_a line_n however_o their_o curvature_n be_v either_o uniform_a equal_a or_o regular_a but_o i_o know_v it_o will_v be_v say_v that_o such_o crooked_a line_n of_o equal_a uniform_a &_o regular_a curvature_n be_v but_o one_o line_n and_o therefore_o by_o the_o definition_n of_o a_o angle_n can_v contain_v a_o angle_n which_o require_v two_o line_n to_o its_o constitution_n to_o which_o i_o answer_v that_o in_o like_a manner_n two_o concur_v right_a line_n may_v be_v take_v for_o one_o line_n continue_v though_o not_o in_o its_o rectitude_n and_o then_o the_o consideration_n of_o angularity_n between_o they_o be_v exclude_v however_o by_o reason_n of_o the_o inflexion_n and_o inclination_n at_o the_o point_n of_o incurvation_n there_o be_v a_o aptitude_n in_o that_o one_o produce_v line_n there_o to_o fall_v and_o distinguish_v itself_o into_o two_o with_o inclination_n of_o one_o unto_o the_o other_o and_o so_o to_o offer_v the_o constitutive_a nature_n of_o a_o angle_n and_o so_o it_o be_v at_o every_o point_n of_o such_o crooked_a line_n who_o curvature_n be_v equal_a or_o uniform_a and_o regular_a be_v one_o or_o more_o line_n according_a as_o by_o our_o conception_n they_o be_v continue_v or_o distinguish_v and_o as_o a_o continue_a rectitude_n such_o as_o be_v in_o right_a line_n be_v most_o inconsistent_a with_o the_o nature_n of_o a_o angle_n so_o what_o shall_v be_v judge_v more_o accept_v of_o the_o nature_n of_o angularity_n than_o curvature_n be_v be_v thereunto_o contrarious_o opposite_a and_o by_o all_o confess_v most_o what_o to_o be_v so_o save_v in_o the_o aforesaid_a case_n when_o curvature_n be_v equal_a uniform_a or_o regular_a but_o why_o shall_v equality_n uniformity_n or_o regularity_n of_o curvature_n be_v so_o urge_v in_o the_o concern_n of_o angularity_n by_o none_o of_o which_o be_v angularity_n either_o promote_v or_o hinder_v for_o there_o can_v be_v great_a equality_n uniformity_n or_o regularity_n then_o be_v to_o be_v find_v in_o rectitude_n as_o well_o as_o angle_n in_o right_a line_n as_o well_o as_o circle_n the_o thing_n constitutive_a of_o the_o nature_n of_o a_o angle_n be_v thing_n quite_o different_a from_o they_o viz._n inflexion_n or_o inclination_n and_o concurrence_n which_o be_v indifferent_o find_v in_o all_o curvatures_n equal_a and_o unequal_a uniform_a or_o not_o uniform_a regular_a or_o irregular_a and_o beside_o be_v inseparable_a from_o they_o curvature_n and_o a_o concurrent_a inflexion_n or_o angularity_n be_v but_o as_o two_o notion_n of_o the_o same_o thing_n consider_v as_o under_o several_a respect_n viz._n curvature_n be_v the_o affection_n of_o a_o line_n consider_v as_o one_o the_o same_o be_v angularity_n and_o inclination_n when_o from_o any_o point_n of_o the_o curvature_n the_o same_o line_n be_v consider_v as_o two_o so_o two_o concurrent_a right_a line_n be_v say_v to_o be_v one_o crooked_a line_n but_o when_o a_o right-lined_n angle_n be_v say_v to_o be_v contain_v by_o they_o they_o be_v then_o consider_v as_o two_o from_o the_o point_n of_o their_o incurvation_n or_o inclination_n and_o equality_n and_o uniformity_n and_o regularity_n of_o curvature_n imply_v equality_n uniformity_n and_o regularity_n of_o inflexion_n or_o inclination_n it_o be_v so_o far_o from_o conclude_v against_o angularity_n that_o it_o infer_v it_o with_o a_o additament_n viz._n of_o equal_a angularity_n uniform_a angularity_n and_o regular_a angularity_n as_o may_v be_v at_o large_a declare_v in_o the_o special_a property_n of_o several_a crooked_a line_v figure_n and_o in_o several_a uniform_a and_o regular_a crooked_a line_v figure_n there_o be_v some_o special_a point_n offer_v even_o to_o view_v and_o sense_n a_o clear_a specimen_fw-la of_o a_o more_o than_o ordinary_a angularity_n without_o any_o such_o loud_a call_n for_o the_o strong_a operation_n of_o the_o mind_n as_o the_o vertical_a point_n in_o conjugate_v figure_n in_o parabola_n and_o the_o extreme_a point_n of_o either_o axe_n in_o ellipse_n and_o the_o like_a and_o in_o the_o definition_n of_o a_o plane_n angle_n all_o the_o inclination_n which_o be_v require_v in_o the_o concur_v side_n be_v only_o that_o their_o inclination_n be_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d i._o e._n so_o as_o the_o side_n make_v not_o one_o and_o the_o same_o right_a line_n as_o it_o be_v general_o understand_v however_o so_o far_o be_v we_o from_o impose_v upon_o any_o against_o their_o judgement_n this_o special_a sort_n of_o angle_n that_o we_o ready_o acknowledge_v if_o the_o word_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d in_o the_o definition_n of_o a_o plane_n angle_n be_v force_v from_o the_o common_o receive_v sense_n of_o the_o side_n not_o lie_v in_o the_o same_o right-line_n to_o signify_v the_o non-coincidency_a of_o the_o production_n of_o the_o one_o side_n with_o the_o other_o then_o according_a to_o that_o gloss_n upon_o the_o definition_n of_o plane_n angle_n this_o whole_a sort_n of_o angle_n be_v to_o be_v reject_v wherein_o every_o one_o be_v free_o leave_v to_o his_o own_o judgement_n the_o difference_n be_v a_o question_n and_o quarrel_n about_o word_n more_o than_o matter_n and_o not_o concern_v the_o present_a controversy_n however_o from_o the_o whole_a it_o out_o of_o controversy_n appear_v that_o recto-concave_a angle_n of_o contact_n be_v true_a angle_n contain_v under_o two_o incline_v side_n concur_v also_o that_o ultradiametral_a concavo-convexe_a angle_n of_o contact_n whether_o less_o equal_a or_o great_a than_o two_o right_a right-lined_n angle_n be_v true_a angle_n for_o the_o side_n be_v incline_v and_o concur_v of_o their_o concurrence_n can_v be_v no_o doubt_n and_o that_o they_o be_v incline_v must_v of_o necessity_n be_v yield_v see_v they_o neither_o lie_n in_o rectitude_n nor_o which_o to_o some_o may_v be_v a_o causeless_a scruple_n the_o one_o in_o the_o production_n of_o the_o other_o so_o as_o their_o angularity_n be_v clear_a beyond_o all_o doubt_n the_o inclination_n understand_v in_o the_o definition_n of_o a_o angle_n be_v general_o any_o positure_n of_o one_o line_n with_o regard_n to_o another_o in_o the_o same_o plane_n so_o as_o both_o neither_o fall_n in_o one_o right_a line_n nor_o be_v situate_v parallel_n one_o to_o another_o nor_o under_o such_o a_o manner_n of_o oblique_a extension_n as_o may_v render_v their_o concurrence_n impossible_a it_o be_v not_o only_o the_o oblique_a positure_n of_o one_o line_n in_o reference_n to_o another_o as_o contradistinguish_v from_o perpendicularness_n in_o acute_a and_o obsuse_v inclination_n but_o such_o be_v the_o comprehensiveness_n of_o its_o sense_n in_o the_o present_a acceptation_n that_o every_o perpendicularness_n itself_o be_v take_v for_o a_o inclination_n so_o likewise_o it_o can_v reasonable_o be_v deny_v but_o those_o special_a angle_n of_o contact_n which_o be_v the_o chief_a subject_n of_o this_o present_a controversy_n i_o mean_v recto-convexe_a angle_n of_o contact_n be_v true_o angle_n except_o either_o the_o inclination_n of_o their_o side_n or_o their_o concurrence_n can_v be_v call_v in_o question_n nothing_o else_o be_v requisite_a unto_o the_o nature_n or_o comprise_v in_o the_o definition_n of_o a_o angle_n and_o the_o like_a be_v to_o be_v judge_v of_o all_o other_o concavo-convexe_a
of_o equal_a homogeneal_a uniform_a regular_a or_o answer_v curvature_n of_o the_o late_a sort_n be_v all_o other_o whether_o mix_v line_v angle_n or_o crooked-lined_n angle_n whether_o they_o be_v mix_v crooked-lined_n angle_n or_o unmixed_a crooked-lined_n angle_n and_o consequent_o thereupon_o beside_o the_o numerous_a distinction_n of_o angle_n in_o respect_n of_o their_o different_a inclination_n such_o as_o above_o mention_v one_o be_v more_o eminent_o material_a above_o the_o rest_n that_o the_o inclination_n of_o the_o side_n be_v sometime_o with_o a_o equability_n all_o along_o their_o production_n though_o imagine_v to_o be_v infinite_o extend_v in_o such_o line_n as_o by_o possibility_n may_v with_o reason_n be_v imagine_v so_o to_o be_v and_o sometime_o there_o be_v nothing_o of_o equability_n to_o be_v find_v in_o the_o inclination_n of_o the_o several_a part_n of_o each_o side_n to_o the_o other_o though_o it_o may_v be_v one_o of_o the_o side_n be_v a_o right-line_n or_o a_o arch_n of_o most_o equal_a uniform_a regular_a and_o homogeneal_a curvature_n and_o this_o equability_n and_o inequability_n of_o the_o inclination_n of_o the_o side_n strange_o alter_v the_o property_n of_o angle_n as_o in_o right-lined_n angle_n for_o the_o equability_n of_o the_o inclination_n of_o the_o side_n no_o part_n of_o the_o one_o side_n be_v more_o incline_v than_o the_o rest_n unto_o the_o other_o side_n and_o so_o in_o concavo-convexe_a angle_n of_o equal_a curvature_n no_o part_n of_o the_o one_o arch_n be_v more_o incline_v than_o the_o rest_n unto_o the_o other_o but_o the_o one_o arch_n be_v all_o along_o incline_v unto_o the_o other_o as_o at_o the_o angular_a point_n and_o the_o inclination_n which_o the_o one_o bear_v unto_o the_o other_o at_o the_o angular_a point_n be_v obvious_o expressable_a as_o to_o the_o quantity_n of_o the_o recess_n which_o they_o make_v one_o from_o another_o by_o the_o inclination_n of_o a_o right-line_n to_o a_o right-line_n except_o when_o the_o inclination_n of_o the_o arch_n be_v equal_a to_o or_o great_a than_o two_o right_a right-lined_n angle_n and_o in_o such_o crooked-lined_n angle_n who_o side_n have_v equability_n of_o inclination_n the_o point_n which_o from_o the_o angular_a point_n be_v at_o equal_a distance_n along_o the_o arch_n be_v also_o absolute_o at_o equal_a distance_n from_o the_o angular_a point_n along_o the_o chord_n and_o right-lined_n tangent_n at_o any_o two_o such_o homologal_a point_n where_o ever_o take_v always_o meet_v and_o contain_v a_o right-lined_n angle_n equal_a as_o to_o the_o quantity_n of_o the_o recess_n of_o the_o side_n to_o the_o crooked-lined_n angle_v contain_v by_o the_o two_o arch_n as_o be_v obvious_a to_o demonstrate_v especial_o in_o circular_a arch_n and_o the_o right-lined_n angle_v contain_v under_o the_o two_o right-lined_n tangent_n touch_v at_o the_o two_o homologal_a and_o answer_a point_n which_o be_v equal_a to_o the_o isoclitical_a crooked-lined_n angle_n if_o the_o two_o right-lined_n tangent_n occur_v on_o that_o side_n of_o the_o right-line_n connect_v the_o homologal_a point_n on_o which_o the_o isoclitical_a angle_n fall_v than_o it_o be_v the_o angle_n contain_v by_o the_o two_o right-lined_n tangent_n into_o who_o space_n part_n of_o the_o space_n comprise_v between_o the_o two_o arch_n at_o first_o fall_v which_o be_v equal_a to_o the_o crooked-lined_n isoclitical_a angle_n but_o if_o they_o occur_v on_o the_o other_o side_n of_o the_o right-line_n connect_v the_o two_o homologal_a point_n i._n e._n averse_o from_o the_o crooked-line_n isoclitical_a angle_n than_o it_o be_v the_o compliment_n of_o such_o a_o angle_n which_o be_v equal_a to_o the_o crooked-line_n isoclitical_a angle_n but_o if_o the_o two_o right-lined_n tangent_n occur_v in_o one_o of_o the_o homologal_a point_n the_o angle_n either_o way_n contain_v under_o the_o two_o right_a line_n tangent_n be_v equal_a viz._n right_a right-lined_n angle_n either_o of_o they_o make_v forth_o what_o be_v herein_o assert_v as_o in_o fig_n 19_o under_o the_o two_o circular_a isoclitical_a arch_n bda_o and_o acn_n let_v there_o be_v constitute_v the_o isoclitical_a angle_n bac_n and_o let_v the_o right-line_n ag_fw-mi touch_v the_o arch_a acn_n in_o the_o point_n a_o and_o let_v the_o right-line_n of_o touch_n the_o arch_a adb_n in_o the_o point_n a_o so_o make_v the_o right-lined_n angle_v fag_v equal_a to_o the_o isoclitical_a concavo-convexe_a angle_n bac_n then_o take_v in_o the_o arch_n adb_v any_o point_n at_o pleasure_n viz._n the_o point_n d_o and_o draw_v the_o chord_n ad_fw-la then_o in_o the_o arch_n acn_v take_v the_o arch_n ac_fw-la subtend_v by_o the_o chord_n ac_fw-la equal_a to_o the_o chord_n ad_fw-la therefore_o because_o of_o the_o isocliticalness_n of_o the_o circular_a arch_n the_o two_o point_n d_o and_o c_o be_v two_o homologal_a i._n e._n answer_v point_v the_o one_o in_o the_o one_o arch_n the_o other_o in_o the_o other_o viz._n the_o point_n d_o in_o the_o arch_a adb_n and_o the_o point_n c_o in_o the_o arch_a acn_n then_o draw_v the_o right-line_n dc_o connect_v the_o two_o homologal_a point_n d_o and_o c._n also_o draw_v the_o right-line_n the_o touch_v the_o arch_a adb_n in_o the_o point_n d_o and_o the_o right-line_n ce_fw-fr touch_v the_o arch_a acn_n in_o the_o point_n c._n and_o let_v the_o and_o ce_fw-fr the_fw-fr two_o right-line_n tangent_n be_v produce_v till_o they_o meet_v in_o the_o point_n e_o which_o in_o this_o figure_n be_v on_o that_o side_n of_o the_o right-line_n dc_o on_o which_o the_o concavo-convexe_a angle_n cab_n lie_v i_o say_v therefore_o that_o the_o right-lined_n angle_n dec_fw-la contain_v under_o the_o two_o right-lined_n tangent_n the_o and_o ce_fw-fr touch_v the_o arch_n respective_o at_o the_o homologal_a point_n d_o and_o c_o be_v equal_a to_o the_o right-lined_n angle_n fag_n contain_v under_o the_o two_o right-line_n tangent_n fa_o and_o ga_fw-mi touch_v the_o arch_n respective_o at_o a_o the_o angular_a point_n of_o the_o isoclitical_a concavo-convexe_a angle_n for_o the_o right-lined_n tangent_fw-la fa_fw-it cut_v the_o the_o other_o right-lined_n tangent_fw-la of_o the_o same_o arch_a adb_n in_o the_o point_n h_o and_o the_o right-lined_n tangent_fw-la de_fw-la of_o the_o arch_a adb_n cut_v the_o chord_n ac_fw-la in_o the_o point_n k_o upon_o this_o construction_n the_o right-line_n da_fw-mi be_v equal_a to_o the_o right-line_n ac_fw-la and_o the_o right-line_n tangent_fw-la dh_fw-mi be_v equal_a to_o the_o right-line_n tangent_fw-la ha_o therefore_o the_o right-lined_n angle_n adh_fw-mi be_v equal_a to_o the_o right-lined_n angle_n dah_fw-mi and_o so_o to_o the_o right-lined_n angle_v ace_n and_o cag_n several_o and_o therefore_o the_o right-lined_n angle_n ahe_fw-la be_v equal_a to_o the_o two_o right-lined_n angle_n hda_fw-la and_o dah_fw-mi take_v together_o and_o the_o right-lined_n angle_n hda_fw-la be_v equal_a to_o the_o right-lined_n angle_n cag_n the_o right-lined_n angle_n ahe_fw-la be_v equal_a to_o the_o two_o right-lined_n angle_v cag_n and_o dah_fw-mi take_v together_o therefore_o that_o which_o make_v each_o equal_a to_o two_o right_a right-lined_n angle_n the_o two_o right-lined_n angle_v hka_fw-mi and_o hak_v take_v together_o be_v equal_a to_o the_o two_o right-lined_n angle_v hak_v and_o dal_n take_v together_o therefore_o the_o right-lined_n angle_n hka_fw-mi be_v equal_a to_o the_o right-lined_n angle_n dal_n therefore_o the_o right-lined_n angle_n cke_z be_v equal_a to_o the_o right-lined_n angle_n dal_n and_o therefore_o that_o which_o make_v either_o equal_a to_o two_o right_a right-lined_n angle_n the_o two_o right-lined_n angle_v kce_n and_o kec_fw-la together_o take_v be_v equal_a to_o the_o right-lined_n angle_n dag_n which_o be_v equal_a to_o the_o two_o right-lined_n angle_v dac_n and_o cag_n take_v together_o and_o the_o right-lined_n angle_n cag_n be_v equal_a to_o the_o right-lined_n angle_n kce_n therefore_o the_o right-lined_n angle_n kec_fw-la be_v equal_a to_o the_o right-lined_n angle_n dac_n therefore_o because_o the_o right-lined_n angle_v dah_fw-mi and_o cag_fw-mi be_v equal_a also_o the_o right-lined_n angle_n kec_fw-fr s_o c._n dec_fw-la be_v equal_a to_o the_o right-lined_n angle_n hag_n s_o c._n fag_n which_o be_v to_o be_v demonstrate_v but_o if_o the_o two_o right-lined_n tangent_n the_o and_o ce_fw-fr as_o in_o fig._n 20._o do_v not_o occur_v towards_o the_o concavo-convexe_a angle_n bac_n but_o on_o the_o other_o side_n of_o the_o right-line_n dc_o in_o the_o point_n e_o then_o be_v the_o right_n line_v angle_n dec_fw-la contain_v under_o the_o two_o right_a line_n tangent_n the_o and_o ce_fw-fr touch_v at_o the_o homologal_a point_n d_o and_o c_o not_o equal_a to_o the_o concavo-convexe_a isoclitical_a angle_n bac_n or_o the_o right_n line_v angle_n equal_a unto_o it_o fag_n but_o to_o its_o compliment_n unto_o two_o right_a right-lined_n angle_n viz._n unto_o the_o right_n line_v angle_n fall_v the_o right_a line_n la_fw-fr
the_o homogeneity_n of_o their_o figure_n and_o as_o right-line_n and_o crooked-line_n be_v heterogeneal_a as_o above_o not_o possible_o to_o be_v coapt_v with_o the_o precedent_a limitation_n so_o also_o be_v all_o curve_v line_n who_o curvature_n be_v unequal_a and_o unlike_a nay_o though_o it_o may_v be_v they_o be_v but_o several_a part_n of_o the_o same_o line_n or_o though_o the_o curvature_n of_o both_o be_v every_o way_n and_o every_o where_o equal_a and_o like_a yet_o if_o the_o convexe_a and_o concave_a part_n of_o the_o one_o be_v not_o alike_o posit_v as_o in_o the_o other_o there_o will_v be_v a_o manifest_a heterogeneity_n in_o they_o and_o a_o impossibility_n of_o coapt_a they_o observe_v the_o limitation_n as_o above_o and_o why_o do_v the_o heterogeneity_n in_o the_o side_n make_v heterogeneity_n in_o the_o angle_n but_o because_o thereby_o be_v found_v a_o heterogeneity_n in_o the_o inclination_n or_o rather_o in_o the_o inclinableness_n of_o the_o one_o side_n to_o the_o other_o for_o here_o it_o be_v not_o the_o several_a degree_n of_o the_o same_o kind_n of_o inclination_n that_o be_v intend_v for_o then_o all_o unequal_a right-lined_n angle_n shall_v be_v altogether_o heterogeneal_a one_o to_o another_o but_o it_o be_v a_o more_o than_o gradual_a a_o specific_a distinctness_n in_o their_o inclinableness_n which_o we_o be_v now_o discover_v to_o make_v the_o figuration_n of_o the_o angle_n more_o fair_o and_o full_o heterogeneal_a and_o as_o inclination_n be_v the_o habitude_n of_o line_n to_o line_n not_o be_v posit_v in_o the_o same_o right_a line_n nor_o parallel_n for_o even_o perpendicular_o be_v in_o this_o sense_n here_o say_v to_o be_v incline_v so_o as_o above_z from_o the_o heterogeneity_n of_o the_o line_n will_v arise_v a_o heterogeneity_n of_o inclination_n and_o indeed_o for_o no_o other_o reason_n do_v heterogeneal_a line_n make_v heterogeneal_a angle_n but_o because_o their_o inclination_n be_v necessary_o heterogeneal_a and_o as_o above_z heterogeneal_a inclination_n be_v respective_o equal_a as_o in_o some_o right_n line_v and_o crooked_a line_v angle_n this_o do_v not_o in_o the_o least_o annul_v the_o heterogeneity_n of_o their_o inclination_n as_o a_o right_a line_n and_o a_o crooked_a line_n may_v be_v equal_a yet_o as_o to_o the_o positure_n of_o their_o extension_n they_o be_v heterogeneal_a and_o as_o heterogeneity_n of_o side_n or_o inclination_n make_v heterogeneity_n of_o angle_n so_o likewise_o do_v any_o heterogenealnes_n in_o the_o other_o point_n requisite_a to_o the_o nature_n of_o a_o angle_n which_o be_v the_o manner_n of_o the_o side_n concurrence_n and_o there_o be_v only_o three_o way_n in_o the_o concurrence_n of_o the_o side_n of_o angle_n according_a to_o which_o they_o can_v be_v heterogeneal_a one_o to_o another_o for_o either_o the_o production_n of_o the_o one_o concur_v side_n become_v coincident_a with_o the_o other_o concur_v side_n or_o else_o it_o depart_v from_o it_o on_o the_o same_o side_n on_o which_o it_o do_v occur_v or_o else_o it_o depart_v from_o it_o on_o the_o contrary_a side_n to_o that_o on_o which_o it_o do_v occur_v all_o which_o be_v clear_o not_o several_a degree_n of_o the_o same_o manner_n of_o concurrence_n but_o several_a kind_n of_o concurrence_n viz._n the_o one_o by_o way_n of_o contact_n the_o other_o by_o way_n of_o section_n and_o a_o three_o by_o way_n of_o curvature_n or_o coincidence_n that_o as_o these_o be_v diversify_v in_o angle_n i_o mean_v from_o kind_n to_o kind_n not_o from_o degree_n to_o degree_n so_o there_o be_v thereby_o lodge_v in_o their_o figuration_n a_o heterogeneity_n though_o in_o some_o mathematical_a respect_n neither_o side_n nor_o inclination_n can_v sometime_o be_v deny_v to_o be_v however_o homogeneal_a so_o particular_o angle_v of_o contact_n in_o respect_n of_o their_o figuration_n must_v necessary_o be_v acknowledge_v clear_a of_o another_o kind_n than_o all_o other_o angle_n because_o the_o inclination_n of_o their_o side_n be_v tangent_fw-la concur_v only_o in_o a_o punctual_a touch_n whereas_o the_o inclination_n of_o the_o side_n of_o all_o other_o angle_n be_v secant_fw-la and_o at_o the_o point_n of_o their_o concurrence_n by_o reason_n of_o their_o inclination_n they_o cut_v one_o another_o or_o else_o they_o be_v coincident_a then_o which_o what_o can_v make_v a_o more_o material_a difference_n in_o the_o inclination_n of_o the_o side_n and_o as_o more_o especial_o relate_v to_o that_o so_o much_o urge_v analogy_n between_o right-lined_n angle_n and_o angle_n of_o contact_n the_o inclination_n of_o the_o contain_v side_n in_o every_o angle_n of_o contact_n be_v such_o as_o be_v impossible_a to_o be_v between_o the_o side_n of_o any_o right-lined_n angle_n for_o the_o side_n of_o no_o right-lined_n angle_n can_v touch_v without_o cut_v and_o what_o more_o manifest_a and_o material_a difference_n can_v be_v in_o the_o inclination_n make_v upon_o or_o unto_o a_o right-line_n then_o if_o in_o the_o one_o case_n a_o right-line_n be_v incline_v unto_o it_o and_o in_o the_o other_o a_o circumference_n or_o other_o crooked-line_n yet_o further_o to_o clear_v that_o difference_n of_o kind_n which_o be_v between_o angle_n of_o contact_n and_o other_o angle_n i_o think_v on_o all_o side_n it_o will_v be_v judge_v unreasonable_a to_o make_v those_o angle_n of_o the_o same_o kind_n which_o have_v neither_o one_o common_a way_n of_o measure_v nor_o be_v coaptable_a nor_o any_o way_n proportionable_a one_o to_o another_o nor_o can_v any_o way_n by_o the_o contraction_n or_o dilatation_n of_o their_o side_n be_v make_v equal_a one_o to_o another_o and_o this_o we_o shall_v find_v to_o be_v the_o condition_n of_o many_o angle_v one_o in_o relation_n to_o another_o however_o mis-understand_a i_o not_o as_o if_o i_o make_v any_o commensurability_n a_o full_a mark_n of_o a_o full_a homogeneity_n for_o as_o before_o crooked-lined_n and_o right-lined_n angle_n may_v be_v equal_a and_o of_o different_a kind_n have_v their_o inclination_n different_a in_o the_o kind_n of_o their_o figuration_n though_o equal_a in_o the_o recess_n of_o the_o side_n and_o thus_o have_v at_o large_a deduce_v the_o grand_a difference_n which_o be_v between_o the_o mathematical_a heterogeneity_n of_o angle_n and_o their_o heterogeneity_n in_o respect_n of_o their_o figuration_n it_o will_v now_o be_v easy_a for_o we_o to_o extricate_v ourselves_o out_o of_o all_o the_o difficulty_n with_o which_o former_a disquisition_n upon_o this_o subject_n have_v be_v involve_v as_o first_o what_o be_v to_o be_v understand_v by_o the_o equality_n which_o be_v assert_v to_o be_v between_o right-lined_n and_o isoclitical_a concavo-convexe_a angle_n for_o it_o be_v out_o of_o controversy_n and_o on_o all_o hand_n yield_v that_o to_o any_o right-lined_n angle_v give_v may_v be_v give_v also_o a_o concavo-convexe_a isoclitical_a angle_n equal_a and_o that_o also_o in_o a_o thousand_o variety_n as_o be_v most_o manifest_a in_o the_o circumference_n of_o any_o two_o and_o two_o equal_a circle_n or_o any_o two_o and_o two_o equal_a arch_n and_o so_o in_o a_o converse_n manner_n to_o any_o isoclitical_a concavo-convexe_a angle_n give_v who_o side_n make_v their_o recess_n one_o from_o the_o other_o by_o a_o arch_n less_o than_o a_o semi_a circle_n may_v be_v give_v a_o equal_a right-lined_n angle_n although_o in_o the_o infinite_a number_n of_o right_n line_v angle_n it_o be_v impossible_a to_o find_v any_o more_o than_o one_o right-lined_n angle_v equal_a to_o the_o give_v concavo-convexe_a angle_n because_o in_o rectitude_n there_o can_v be_v no_o diversity_n as_o there_o may_v and_o be_v in_o curvature_n now_o in_o the_o above_o recite_v case_n why_o be_v equality_n between_o such_o different_a angle_n assert_v possible_a and_o what_o be_v mean_v by_o their_o equality_n and_o whence_o and_o how_o be_v the_o equality_n of_o they_o to_o be_v demonstrate_v of_o necessity_n it_o must_v be_v found_v upon_o some_o special_a method_n of_o measure_v angle_n or_o of_o somewhat_o which_o be_v in_o some_o if_o not_o in_o all_o angle_n of_o which_o in_o common_a both_o these_o different_a sort_n of_o angle_n be_v natural_o and_o indifferent_o capable_a and_o to_o be_v short_a particular_a and_o plain_a all_o the_o mysteriousness_n of_o this_o their_o equality_n be_v found_v upon_o this_o that_o these_o two_o sort_n of_o angle_n right_o line_v angle_n and_o concavo-convexe_a angle_n of_o equal_a arch_n they_o both_o have_v in_o common_a one_o special_a property_n of_o which_o all_o other_o sort_n of_o plane_n angle_n whatsoever_o be_v destitute_a viz._n that_o each_o in_o their_o kind_n be_v isoclitical_a angle_n and_o the_o side_n in_o each_o be_v isoclitical_a and_o in_o each_o angle_n the_o one_o side_n by_o the_o adduction_n contraction_n and_o draw_v together_o of_o the_o side_n will_v be_v coincident_a and_o coapt_v unto_o the_o other_o and_o as_o the_o coincidence_n of_o right_a
line_n the_o one_o upon_o the_o other_o make_v a_o right_n line_v angle_n of_o contact_n impossible_a so_o the_o coincidence_n of_o isoclitical_a crooked-line_n the_o one_o upon_o the_o other_o make_v a_o isoclitical_a angle_n of_o contact_n impossible_a except_o only_o in_o a_o ultradiametral_a positure_n and_o as_o the_o mensuration_n of_o right_n line_v angle_n be_v by_o the_o arch_n of_o circle_n draw_v upon_o the_o angular_a point_n intercept_v between_o the_o two_o isoclitical_a side_n to_o show_v how_o far_o they_o be_v depart_v from_o their_o coincidence_n so_o in_o isoclitical_a crooked_a line_v angle_n by_o the_o same_o way_n of_o mensuration_n a_o account_n may_v be_v take_v of_o the_o departure_n which_o each_o isoclitical_a side_n have_v make_v from_o the_o other_o since_o their_o coincidence_n and_o this_o be_v the_o point_n in_o which_o their_o equality_n consist_v and_o be_v account_v and_o which_o find_v the_o mathematical_a homogeneity_n which_o be_v between_o they_o 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intercept_v between_o the_o two_o isoclitical_a crooked-line_n abc_n and_o afh_n be_v equal_a to_o the_o respective_a arch_n ei_o intercept_v between_o the_o two_o isoclitical_a right-line_n and_o this_o wheresoever_o the_o point_n i_o be_v take_v in_o the_o right-line_n ah_o so_o as_o by_o this_o common_a way_n of_o mensuration_n common_a to_o both_o these_o sort_n of_o angle_n by_o reason_n of_o the_o isocliticalness_n and_o the_o coaptability_n and_o coincidibleness_n of_o the_o side_n in_o each_o the_o one_o be_v a_o isoclitical_a concavo_fw-la convexe_a angle_n be_v copious_o demonstrate_v to_o be_v equal_a to_o the_o other_o be_v a_o right-lined_n angle_n but_o now_o after_o what_o manner_n be_v we_o to_o understand_v this_o equality_n assert_v between_o such_o right-lined_n and_o isoclitical_a concavo-convexe_a angle_n it_o be_v not_o a_o every_o way_n absolute_a equality_n which_o be_v between_o the_o angle_n such_o as_o be_v between_o two_o equal_a square_n or_o two_o like_a 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magnitude_n which_o be_v otherwise_o in_o right-lined_n angle_n and_o
the_o other_o in_o the_o other_o be_v very_o near_o show_v and_o as_o near_o as_o be_v possible_a in_o right-line_n by_o the_o right-line_n tangent_n of_o those_o respective_a point_n but_o in_o mix_v line_v and_o mix_v crooked-lined_n angle_n by_o several_a way_n of_o account_v several_a point_n be_v make_v to_o answer_v one_o another_o as_o by_o account_v by_o distance_n from_o the_o angular_a point_n or_o by_o account_v by_o equalness_n of_o line_n along_o the_o side_n etc._n etc._n mix_v line_v angle_n of_o contact_n when_o they_o can_v be_v and_o be_v divide_v by_o a_o right-line_n the_o part_n be_v heterogeneal_a and_o unequal_a and_o one_o of_o the_o unequal_a part_n be_v a_o right-lined_n angle_n every_o recto-convexe_a and_o convexo-convexe_a or_o citradiametral_a concavo-convexe_a angle_n of_o contact_n be_v the_o least_o possible_a under_o those_o side_n rectilineary_a mensurableness_n in_o mix_v line_v crooked-lined_n angle_v concur_v by_o way_n of_o section_n begin_v 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extend_v the_o habitude_n of_o it_o far_o into_o
and_o convexo-convexe_a angle_n of_o contact_n but_o you_o say_v that_o in_o the_o recto-convexe_a angle_n of_o contact_n the_o right-line_n tangent_fw-la and_o circumference_n make_v no_o angle_n because_o the_o tangent_fw-la be_v not_o incline_v to_o the_o circumference_n but_o coincident_a with_o it_o what_o mystery_n of_o reason_n or_o force_v of_o argument_n shall_v be_v in_o this_o deduction_n if_o you_o say_v it_o be_v coincidence_n you_o mean_v by_o non-inclination_n i_o ready_o yield_v where_o two_o line_n become_v coincident_a their_o former_a angle_n be_v thereby_o extinct_a as_o thereby_o they_o come_v under_o the_o consideration_n but_o of_o one_o line_n as_o when_o two_o right-line_n or_o two_o isoclitical_a crooked_a line_n be_v one_o of_o each_o of_o they_o so_o move_v about_o the_o angular_a point_n till_o the_o two_o line_n become_v one_o but_o where_o be_v any_o such_o coincidency_n between_o right-line_n tangent_n and_o circumference_n or_o what_o possibility_n be_v there_o of_o any_o such_o coincidency_n a_o crooked_a line_n and_o a_o right-line_n may_v no_o doubt_n be_v commensurate_a or_o proportionable_a in_o length_n but_o in_o position_n it_o be_v impossible_a and_o if_o we_o imagine_v the_o tangent_fw-la bow_v to_o such_o a_o coincidence_n than_o it_o be_v not_o any_o long_o a_o tangent_fw-la or_o right-line_n but_o a_o circumference_n and_o though_o as_o you_o urge_v a_o right-line_n circumduce_v about_o any_o middle_a point_n in_o the_o side_n of_o a_o regular_a polygone_a at_o last_o become_v coincident_a with_o the_o side_n and_o lose_v all_o inclination_n and_o angularity_n with_o the_o side_n what_o do_v that_o concern_n or_o how_o do_v that_o prove_v the_o non-inclination_n of_o the_o tangent_fw-la to_o the_o circumference_n yourselves_o sometime_o in_o every_o point_n save_v only_o in_o the_o point_n of_o contact_n acknowledge_v a_o inclination_n and_o as_o be_v else_o where_o hint_v in_o the_o point_n of_o contact_n alone_o and_o abstract_o the_o inclination_n itself_o which_o be_v the_o habitude_n of_o the_o concur_v incline_v side_n be_v not_o to_o be_v seek_v but_o only_o the_o particular_a termination_n of_o their_o inclination_n there_o but_o you_o will_v say_v it_o be_v non-secancy_a which_o be_v mean_v by_o this_o coincidency_n and_o what_o i_o pray_v be_v that_o more_o than_o the_o lateran_n bell_n to_o the_o concern_v or_o constitution_n of_o angle_n be_v there_o not_o many_o regular_a curve_v line_n produce_v some_o infinite_o without_o section_n in_o which_o especial_o the_o circumference_n of_o circle_n you_o be_v please_v sometime_o every_o where_o to_o think_v you_o have_v cause_n to_o imagine_v a_o angle_n it_o be_v meet_v to_o know_v the_o meaning_n of_o such_o odd_o connext_v term_n before_o reason_v upon_o they_o be_v regard_v in_o question_n of_o weight_n that_o the_o contact_n of_o the_o angular_a side_n be_v as_o different_a from_o coincidency_n as_o from_o secancy_n be_v most_o unquestionable_o apparent_a in_o angle_n of_o such_o side_n as_o be_v capable_a of_o all_o three_o viz._n contact_n secancy_n and_o coincidence_n for_o example_n in_o fig._n 16._o let_v there_o be_v two_o arch_n of_o equal_a circle_n daf_a and_o bac_n touch_v one_o another_o in_o the_o point_n a._n if_o the_o arch_a daf_n be_v circumduce_v about_o the_o point_n a_o as_o a_o unmoveable_a centre_n at_o after_o a_o infinite_a succession_n of_o secancy_n at_o last_o all_o will_v terminate_v in_o a_o manifest_a coincidency_n and_o the_o arch_n of_o be_v coincident_a with_o the_o arch_n ab_fw-la and_o the_o arch_n ad_fw-la be_v coincident_a with_o the_o arch_n ac_fw-la so_o as_o secancy_n contact_n and_o coincidency_n be_v distinguishable_a one_o from_o another_o with_o as_o much_o ease_n and_o cleareness_n as_o a_o odd_a number_n from_o a_o even_o but_o if_o it_o be_v urge_v that_o you_o assert_v not_o a_o coincidency_n between_o the_o arch_n d_o of_o and_o the_o arch_n bac_n but_o only_o that_o gh_a be_v a_o right_a line_n and_o touch_v the_o arch_n b_o ac_fw-la in_o the_o point_n a_o that_o i_o say_v the_o right_a line_n gh_o and_o the_o arch_n bac_n be_v coincident_a the_o vanity_n of_o this_o may_v be_v evince_v in_o that_o by_o the_o same_o reason_n it_o follow_v that_o the_o right_a line_n gh_o must_v be_v coincident_a with_o the_o arch_a daf_n and_o so_o the_o arch_a daf_n coincident_a with_o the_o arch_n b_o ac_fw-la the_o contrary_a of_o which_o be_v above-shewn_a and_o confess_v and_o beside_o hereupon_o shall_v two_o arch_n b_o ac_fw-la and_o daf_n be_v convexo-convexe_o posit_v and_o the_o right_a line_n tangent_fw-la gh_n be_v all_o coincident_a which_o i_o leave_v for_o other_o to_o say_v rather_o than_o myself_o when_o a_o right_a line_n tangent_fw-la and_o many_o crooked_a line_n of_o different_a curvature_n all_o touch_n together_o in_o one_o and_o the_o same_o point_n as_o in_o fig._n 13_o you_o say_v though_o without_o the_o angular_a point_n of_o contact_n the_o side_n be_v various_o divaricate_v one_o from_o another_o yet_o in_o the_o point_n of_o contact_n they_o have_v not_o several_a inclination_n for_o you_o say_v they_o have_v no_o inclination_n at_o all_o the_o truth_n be_v the_o angular_a point_n of_o its_o self_n be_v not_o capable_a of_o inclination_n nor_o for_o the_o indivisibility_n of_o its_o nature_n can_v by_o any_o possibility_n comprehend_v they_o yet_o that_o line_n concur_v in_o one_o only_a point_n and_o present_o after_o recede_v each_o from_o the_o other_o shall_v not_o be_v incline_v each_o to_o other_o in_o or_o at_o the_o point_n of_o their_o concurrency_n whether_o it_o be_v by_o contact_n or_o section_n though_o in_o the_o case_n of_o contact_n the_o inclination_n be_v less_o then_o can_v be_v express_v by_o the_o inclination_n of_o any_o right-line_n upon_o a_o right-line_n be_v absolute_o unconceivable_a there_o be_v no_o lineary_a coincidence_n but_o only_o of_o one_o point_n between_o they_o for_o as_o else_o where_o in_o the_o same_o plane_n neither_o of_o point_n to_o point_n nor_o of_o point_n to_o line_n nor_o of_o point_n to_o plane_n can_v be_v any_o inclination_n but_o in_o the_o present_a case_n of_o plane_n angle_n inclination_n must_v be_v of_o line_n and_o may_v be_v of_o they_o in_o the_o very_a point_n of_o concurrence_n or_o else_o from_o the_o point_n of_o concurrence_n they_o will_v not_o part_v several_a way_n for_o it_o be_v their_o diverse_a inclination_n at_o that_o point_n which_o make_v their_o departure_n one_o from_o another_o when_o they_o depart_v from_o thence_o and_o as_o even_o in_o the_o angular_a point_n of_o right-lined_n angle_v the_o line_n have_v the_o same_o inclination_n as_o else_o where_o so_o in_o all_o other_o angle_n save_v only_o such_o angle_n of_o contact_n as_o be_v less_o than_o the_o least_o right-lined_n angle_n by_o draw_v right-lined_n tangent_n to_o the_o arch_n or_o arch_n at_o the_o angular_a point_n be_v show_v in_o right-line_n either_o the_o very_a inclination_n of_o the_o side_n in_o or_o rather_o at_o the_o angular_a point_n or_o else_o the_o least_o right-lined_n inclination_n which_o be_v great_a or_o the_o great_a right-lined_n inclination_n which_o be_v less_o for_o though_o they_o may_v differ_v much_o in_o their_o distance_n and_o divarication_n one_o from_o another_o without_o the_o point_n of_o their_o concurrence_n in_o the_o point_n of_o their_o concurrence_n without_o much_o absurdity_n they_o may_v be_v say_v to_o be_v equal_o distant_a i._n e._n not_o at_o all_o distant_a there_o one_o from_o another_o for_o there_o they_o be_v not_o indeed_o at_o all_o distant_a any_o of_o they_o from_o the_o rest_n yet_o it_o do_v not_o hence_o any_o way_n follow_v that_o in_o like_a manner_n it_o may_v be_v say_v of_o they_o without_o absurdity_n that_o in_o the_o point_n of_o concurrence_n they_o have_v equal_a inclination_n i._n e._n no_o inclination_n one_o to_o another_o for_o though_o in_o the_o point_n of_o concurrence_n it_o be_v truth_n they_o have_v no_o distance_n yet_o it_o no_o way_n as_o may_v appear_v to_o those_o that_o will_v consider_v equal_o follow_v that_o they_o shall_v there_o have_v no_o inclination_n beside_o that_o the_o urge_n of_o the_o coincidency_n of_o the_o side_n in_o recto-convexe_a angle_n of_o contact_n be_v most_o direct_o opposite_a to_o the_o nature_n and_o property_n of_o the_o special_a line_n under_o which_o such_o angle_n be_v contain_v for_o it_o be_v the_o special_a property_n of_o some_o line_n that_o they_o can_v touch_v but_o they_o can_v no_o way_n becoincident_a as_o arch_n of_o unequal_a and_o unlike_a curvature_n and_o a_o right-line_n and_o a_o crooked_a line_n some_o can_v be_v coincident_a and_o can_v no_o way_n touch_v as_o right_a line_n some_o can_v both_o as_o arch_n of_o equal_a and_o answer_a curvature_n which_o set_v concavo-convexe_o and_o citradiametral_o can_v touch_v but_o will_v fall_v into_o a_o coincidence_n
but_o posit_v convexo-convexe_o can_v be_v coincident_a but_o may_v construct_v a_o angle_n of_o contact_n so_o as_o the_o chimaera_n of_o the_o coincidency_n of_o the_o side_n in_o recto-convexe_a angle_n of_o contact_n if_o persist_v in_o be_v worth_a laugh_v at_o and_o like_o his_o philosophy_n who_o when_o every_o one_o be_v at_o his_o high_a lavolta_n deny_v the_o possibility_n of_o motion_n in_o the_o world_n but_o to_o justify_v the_o non-inclination_n of_o the_o side_n against_o the_o eye_n and_o reason_n this_o horrid_o distort_a monster_n of_o their_o coincidency_n be_v introduce_v indeed_o if_o they_o be_v coincident_a they_o make_v no_o angle_n but_o it_o will_v cramp_v the_o understanding_n of_o a_o oedipus_n to_o declare_v how_o either_o a_o right-line_n or_o a_o crooked-line_n touch_v another_o crooked-line_n in_o one_o only_a point_n and_o no_o more_o shall_v ever_o be_v conceive_v notwithstanding_o to_o be_v coincident_a with_o the_o production_n of_o the_o other_o crooked-line_n whether_o the_o tangencie_n of_o the_o crooked-line_n be_v concavo-convexe_a or_o convexo-convexe_a i._n e._n the_o one_o within_o the_o other_o or_o else_o the_o one_o turn_v away_o from_o the_o other_o but_o you_o will_v say_v you_o assert_v coincidency_n only_o in_o the_o point_n of_o contact_n i_o answer_v that_o be_v frivolous_a not_o to_o say_v ridiculous_a and_o impertinent_a for_o coincidency_n in_o the_o present_a question_n of_o angle_n be_v take_v as_o oppose_v to_o inclination_n which_o be_v a_o affection_n and_o propriety_n of_o the_o concur_v side_n of_o the_o angle_n not_o only_o of_o the_o angular_a point_n take_v by_o itself_o abstract_o as_o inclination_n can_v be_v in_o a_o point_n though_o it_o may_v be_v at_o a_o point_n so_o a_o point_n can_v be_v say_v to_o be_v incline_v unto_o a_o line_n especial_o itself_o be_v in_o the_o same_o line_n it_o may_v be_v say_v to_o hold_v such_o and_o such_o a_o distance_n from_o the_o line_n when_o it_o be_v without_o it_o but_o not_o to_o be_v incline_v unto_o it_o and_o if_o the_o be_v of_o the_o point_n of_o contact_n general_o as_o a_o point_n of_o the_o one_o side_n in_o the_o tangent_fw-la line_n as_o in_o the_o other_o side_n whether_o right_a or_o crooked_a make_v a_o coincidency_n destroy_v inclination_n then_o all_o inclination_n and_o angle_n whatsoever_o be_v destroy_v and_o every_o where_o will_v be_v a_o coincidency_n for_o that_o be_v common_a to_o every_o angle_n to_o have_v the_o angular_a point_n still_o common_a to_o both_o side_n and_o the_o secant_fw-la angle_v might_n as_o well_o be_v say_v coincident_a as_o the_o tangent_fw-la angle_n for_o what_o you_o say_v that_o it_o be_v tangency_n as_o oppose_v to_o secancy_n that_o you_o mean_v by_o coincidency_n i_o answer_v the_o gloss_n be_v improper_a and_o beside_o the_o anvil_n and_o tangency_n undeny_v but_o in_o this_o case_n impertinent_o by_o you_o allege_v till_o it_o be_v prove_v that_o tangency_n in_o one_o only_a point_n and_o no_o more_o do_v quite_o annul_v and_o destroy_v the_o inclination_n of_o the_o line_n though_o on_o both_o side_n of_o the_o contact_n clear_o recede_v the_o one_o from_o the_o other_o till_o which_o be_v do_v happiness_n to_o my_o friend_n and_o no_o long_o i_o may_v add_v that_o it_o be_v touch_v which_o be_v only_o mention_v in_o the_o definition_n of_o plane_n angle_n but_o i_o shall_v dispatch_v all_o by_o set_v the_o case_n before_o you_o in_o this_o diagramme_n in_o fig._n 12._o if_o the_o right_a line_n basilius_n touch_v the_o arch_n la_fw-fr in_o the_o point_n a_o and_o the_o right-line_n lb_n be_v so_o draw_v as_o to_o touch_v the_o arch_n la_fw-fr in_o the_o point_n l_o hear_v now_o be_v a_o plane_n on_o every_o side_n bound_v by_o two_o right-line_n viz._n bl_n and_o basilius_n and_o one_o crooked-line_n viz._n the_o arch_n la._n that_o lba_n be_v a_o angle_n will_v not_o be_v doubt_v and_o because_o the_o three_o line_n perfect_o bind_v in_o and_o limit_v a_o plane_n on_o all_o side_n the_o arch_n la_fw-fr can_v neither_o be_v coincident_a with_o the_o right_a line_n lb_n nor_o with_o the_o right-line_n ab_fw-la and_o in_o the_o angle_n labernele_n the_o line_n la_fw-fr and_o ba_z and_o in_o the_o angle_n alb_n the_o line_n la_fw-fr and_o bl_n concur_v without_o coincidency_n and_o without_o lie_v in_o the_o same_o right-line_n or_o the_o one_o so_o much_o as_o in_o the_o production_n of_o the_o other_o inclination_n and_o so_o the_o true_a nature_n of_o a_o angle_n can_v be_v deny_v they_o according_a to_o the_o most_o severe_a limitation_n and_o hard_a gloss_n that_o can_v with_o any_o reason_n be_v deduce_v from_o the_o definition_n of_o a_o plane_n angle_n and_o to_o make_v all_o clear_a let_v a_o paper_n or_o other_o plane_n be_v cut_v in_o the_o form_n of_o the_o mixed-lined_n triangle_n alb_n and_o the_o wildness_n of_o question_v the_o angularity_n of_o the_o two_o recto-convexe_a angle_n of_o contact_n labernele_n and_o blanvel_n will_v be_v clear_a to_o all_o person_n both_o rude_a and_o learned_a i_o take_v it_o therefore_o for_o grant_v that_o all_o suspicion_n of_o coincidency_n and_o non-inclination_n in_o whatsoever_o pertinent_a sense_n of_o contact-angle_n side_n be_v evict_v and_o send_v of_o the_o scene_n all_o recto-convexe_a angle_n of_o contact_n be_v true_o angle_v to_o pass_v now_o unto_o another_o of_o your_o thesis_n in_o which_o you_o peremptory_o conclude_v recto-convexe_a angle_n of_o contact_n to_o be_v devoid_a of_o all_o quantitativeness_n when_o i_o urge_v their_o quantitativeness_n i_o mean_v not_o that_o they_o can_v at_o pleasure_n be_v divide_v into_o part_n in_o any_o give_v and_o limit_a proportion_n or_o by_o a_o mathematical_a homogeneity_n hold_v any_o proportion_n with_o the_o angle_n of_o contact_n divide_v only_o that_o from_o the_o angular_a point_n between_o the_o side_n of_o the_o least_o recto-convexe_a angle_n of_o contact_n infinite_a other_o line_n may_v be_v draw_v divide_v the_o angle_n though_o heterogeneal_o and_o certain_o its_o be_v a_o angle_n of_o contact_n can_v in_o the_o least_o prejudice_n its_o quantitativeness_n for_o it_o be_v most_o apparent_a that_o a_o convexo-convexe_a angle_n of_o contact_n contain_v under_o arch_n of_o equal_a curvature_n may_v be_v divide_v into_o two_o mixed-lined_n i._n e._n recto-convexe_a angle_n of_o contact_n which_o hold_v proportion_n of_o equality_n one_o with_o another_o and_o each_o of_o they_o be_v in_o subduple_n proportion_n to_o the_o convexo-convexe_a angle_n of_o contact_n which_o be_v divide_v and_o infinite_a number_n of_o contact-angle_n of_o several_a sort_n may_v be_v adjoin_v one_o unto_o another_o distinct_a in_o their_o situation_n without_o drown_v and_o extinguish_v one_o another_o and_o each_o lie_v without_o the_o other_o which_o be_v not_o true_a of_o indivisibles_n when_o they_o be_v adjoin_v to_o and_o touch_v one_o another_o beside_o the_o recto-concave_a angle_n of_o contact_n be_v great_a than_o the_o great_a right-lined_n angle_n at_o the_o angular_a point_n of_o contact_n have_v a_o manifest_a inclination_n of_o the_o side_n for_o concur_v they_o neither_o be_v one_o right-line_n nor_o one_o crooked-line_n moreover_o ultradiametral_a concavo-convexe_a angle_n of_o contact_n may_v be_v equal_a to_o two_o right_a right-lined_n angle_n or_o great_a sometime_o as_o in_o fig._n 12._o if_o the_o two_o circle_n fah_o and_o adk_v be_v equal_a and_o touch_v in_o the_o point_n a_o and_o ak_v be_v diameter_n and_o be_v be_v another_o arch_n fall_v beyond_o the_o diameter_n ak_v and_o beyond_o the_o right_a line_n tangent_fw-la ab_fw-la it_o be_v manifest_a that_o the_o concavo-convexe_a ultradiametral_a angle_n of_o contact_n dah_n be_v as_o to_o the_o recess_n of_o the_o side_n equal_a to_o two_o right-lined_n angle_n for_o the_o recto-concave_a angle_n of_o contact_n bah_o be_v equal_a to_o two_o right_a right-lined_n angle_n deduct_v the_o recto-convexe_a angle_n of_o contact_n have_v and_o the_o recto-convexe_a angle_n of_o contact_n bad_a be_v equal_a to_o the_o recto-convexe_a angle_n of_o contact_n h_n as_o therefore_o the_o ultradiametral_a concavo-convexe_a angle_n of_o contact_n d_o ah_o be_o equal_a to_o two_o right_a right-lined_n angle_n and_o therefore_o the_o ultradiametral_a concavo-convexe_a angle_n of_o contact_n d_o ar_n be_v great_a than_o two_o right_a right-lined_n angle_n and_o then_o what_o great_a monster_n be_v discoverable_a in_o the_o doctrine_n of_o the_o quantitativeness_n of_o the_o recto-convexe_a angle_n of_o contact_n it_o be_v demonstrate_v that_o between_o the_o right-line_n tangent_fw-la and_o the_o arch_n which_o it_o touch_v no_o right-line_n can_v pass_v i._n e._n the_o recto-convexe_a angle_n of_o contact_n be_v less_o than_o the_o least_o right-lined_n angle_n but_o why_o shall_v hence_o be_v infer_v that_o the_o recto-convexe_a angle_n of_o contact_n have_v no_o true_a quantity_n you_o will_v say_v because_o a_o right-lined_n
if_o you_o say_v how_o can_v recto-convexe_a angle_n of_o contact_n be_v say_v to_o be_v part_n of_o such_o concrete_a and_o composite_a angle_n if_o unable_a by_o any_o multiplicity_n to_o exhaust_v the_o composite_a angle_n omit_v the_o answer_n that_o part_n be_v sometime_o essential_a and_o of_o the_o definition_n and_o yet_o by_o no_o multiplicity_n can_v equalise_v the_o whole_a as_o four_o angle_n in_o the_o definition_n of_o a_o tetragone_fw-mi and_o a_o foot_n line_n in_o the_o definition_n of_o a_o foot_n cube_n i_o answer_v chief_o that_o where_o the_o integral_n be_v heterogeneal_a as_o here_o and_o make_v up_o and_o proper_o and_o natural_o resoluble_a into_o several_a heterogeneal_a part_n and_o can_v be_v divide_v into_o any_o number_n at_o pleasure_n of_o part_n all_o homogeneal_a there_o some_o part_n may_v never_o be_v able_a by_o any_o multiplicity_n to_o equalise_v the_o whole_a or_o some_o other_o heterogeneal_a part_n and_o elss-where_a that_o under_o the_o same_o 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of_o their_o superficial_a positure_n in_o body_n in_o respect_n of_o their_o solid_a positure_n and_o in_o the_o cast_n of_o each_o of_o their_o schematisme_n quantity_n be_v involve_v as_o length_n breadth_n depth_n viz._n their_o quantity_n and_o the_o quantity_n which_o be_v compound_v of_o several_a or_o all_o of_o they_o together_o and_o here_o by_o the_o figuration_n which_o we_o assert_v in_o angle_n we_o can_v be_v think_v to_o mean_v that_o any_o right-lined_n figure_n can_v be_v complete_v perfect_o to_o bind_v up_o a_o plane_n on_o all_o side_n by_o one_o angle_n it_o be_v beyond_o the_o power_n of_o two_o and_o three_o be_v the_o least_o number_n of_o angle_n requisite_a so_o to_o constitute_v and_o perfect_o limit_v out_o a_o right-lined_n plane_n figure_n and_o though_o some_o plane_n figure_n be_v perfect_v and_o perfect_o bound_v without_o any_o such_o angle_n as_o be_v contain_v by_o side_n concur_v by_o way_n either_o of_o section_n or_o contact_n as_o nam_o all_o circle_n and_o ellipse_n yet_o the_o angularity_n of_o curve_v coincidence_n be_v every_o where_o find_v or_o at_o pleasure_n assignable_a in_o the_o bounder_n of_o such_o figure_n but_o our_o meaning_n be_v a_o plane_n angle_n though_o most_o what_o it_o do_v not_o by_o the_o continuation_n and_o production_n of_o its_o side_n perfect_o bind_v in_o and_o limit_v out_o a_o certain_a plane_n and_o space_n on_o every_o side_n however_o be_v the_o mutual_a habitude_n of_o concur_v line_n it_o give_v a_o imperfect_a figuration_n to_o the_o plane_n and_o space_n on_o its_o part_n and_o as_o a_o bound_a plane_n can_v be_v without_o some_o kind_n of_o plane_n figure_n so_o a_o limit_a angle_n ever_o imply_v in_o it_o a_o imperfect_a figuration_n of_o some_o sort_n or_o other_o for_o figuration_n be_v the_o consectary_n of_o material_a finiteness_n and_o limitation_n in_o the_o position_n of_o length_n breadth_n depth_n surface_n and_o solidity_n that_o every_o angle_n have_v its_o limit_n and_o bound_n can_v be_v thereof_o destitute_a and_o if_o the_o name_n of_o figure_n be_v so_o frequent_o give_v to_o hyperbola_n parabola_n and_o the_o like_a which_o neither_o do_v nor_o ever_o can_v by_o any_o possible_a production_n perfect_o bind_v in_o their_o plane_n what_o reason_n be_v there_o then_o why_o angle_n shall_v be_v deny_v a_o imperfect_a interest_n in_o the_o name_n beside_o as_o a_o plane_n in_o its_o own_o general_a nature_n at_o large_a do_v not_o denote_v any_o special_a plane_n figure_n but_o the_o rise_n of_o figure_n i_o mean_v plane_n figure_n be_v from_o the_o bound_a of_o the_o plane_n so_o it_o be_v in_o angle_n as_o they_o by_o the_o mutual_a habitude_n of_o their_o concur_v side_n give_v imperfect_a limit_n and_o bound_n unto_o the_o space_n and_o plane_n so_o they_o therein_o make_v a_o imperfect_a figuration_n that_o in_o angle_n something_o of_o form_n and_o figure_n be_v to_o be_v note_v as_o well_o as_o magnitude_n and_o one_o line_n can_v concur_v with_o and_o be_v incline_v upon_o another_o but_o a_o imperfect_a figuration_n will_v arise_v from_o that_o their_o mutual_a inclination_n and_o the_o same_o two_o angle_n may_v have_v the_o inclination_n i._n e._n the_o recess_n of_o their_o respective_a side_n one_o from_o another_o equal_a though_o there_o be_v no_o analogy_n between_o the_o figuration_n of_o the_o angle_n or_o the_o shape_n in_o the_o which_o the_o side_n be_v incline_v in_o the_o one_o and_o in_o the_o other_o for_o by_o reason_n that_o in_o angle_n form_n and_o figure_n be_v to_o be_v observe_v as_o well_o as_o quantity_n crooked-lined_n and_o right-lined_n angle_n may_v be_v equal_a in_o some_o particular_a quantity_n yet_o otherwise_o not_o of_o the_o same_o kind_n they_o have_v equality_n in_o some_o magnitude_n but_o be_v distinct_a in_o the_o manner_n of_o their_o form_n figuration_n and_o constitution_n as_o equality_n may_v be_v between_o a_o square_a and_o a_o triangle_n though_o figure_n altogether_o different_a in_o kind_n and_o in_o respect_n of_o such_o their_o figuration_n plane_n angle_v receive_v distinction_n either_o from_o the_o diverse_a manner_n in_o which_o their_o contain_v side_n do_v concur_v or_o else_o from_o the_o diverse_a nature_n and_o figuration_n of_o the_o line_n under_o which_o they_o be_v contain_v or_o which_o be_v tantamount_n from_o the_o diversity_n of_o the_o inclination_n and_o inflexion_n or_o rather_o inclinableness_n and_o imflexibleness_n by_o which_o they_o be_v incline_v each_o to_o other_o or_o from_o several_a of_o these_o ground_n of_o distinction_n take_v together_o plane_n angle_v from_o the_o different_a manner_n of_o their_o side_n concur_v may_v apt_o be_v thus_o distinguish_v viz._n into_o angle_n who_o side_n concur_v by_o way_n of_o section_n or_o else_o that_o have_v their_o side_n concur_v only_o by_o way_n of_o touch_n in_o some_o single_a singular_a point_n without_o mutual_a section_n or_o else_o their_o concurrence_n be_v in_o curvature_n where_o after_o the_o meeting_n of_o their_o side_n in_o the_o angular_a point_n the_o side_n do_v not_o in_o their_o production_n depart_v one_o from_o another_o neither_o by_o way_n of_o touch_n nor_o section_n but_o become_v the_o production_n of_o the_o one_o side_n coincident_a with_o the_o other_o side_n so_o as_o this_o kind_n of_o angle_n may_v apt_o be_v call_v angle_n of_o coincidency_n or_o angle_n of_o curvature_n and_o in_o these_o lie_v the_o genuine_a ratio_fw-la and_o true_a account_n of_o the_o curvature_n from_o the_o diverse_a figuration_n of_o the_o line_n under_o which_o a_o plane_n angle_n be_v contain_v very_o many_o difference_n of_o angle_n may_v arise_v according_a to_o the_o various_a distinction_n of_o which_o line_n themselves_o be_v capable_a i_o mean_v such_o line_n as_o fall_v not_o without_o the_o capacity_n and_o comprehensivenes_n of_o the_o same_o plane_n as_o that_o some_o be_v helicoidal_a some_o parabolar_n some_o elliptitical_a etc._n etc._n but_o as_o of_o line_n so_o hence_o of_o angle_n the_o chief_a and_o primary_n distinction_n be_v especial_o these_o viz._n that_o plane_n angle_n be_v either_o right-lined_n angle_n contain_v under_o two_o incline_v right-line_n or_o not_o right-lined_n angle_n not_o right-lined_n angle_n be_v either_o mix_v line_v angle_n contain_v under_o one_o right-line_n and_o one_o crooked-line_n or_o crooked-lined_n angle_v contain_v under_o two_o crooked-lined_n side_n and_o from_o the_o several_a kind_n of_o special_a or_o ordinary_a curvature_n as_o circular_a elliptical_a hyperbolar_n etc._n etc._n the_o mixt-lined_n and_o crooked-lined_n angle_n be_v capable_a of_o many_o far_o and_o more_o particular_a distinction_n but_o especial_o from_o the_o site_n of_o the_o convexeness_n or_o concaveness_n of_o the_o line_n to_o or_o from_o the_o angle_n side_n though_o all_o such_o secondary_a distinction_n rise_v from_o these_o two_o last_o mention_v head_n be_v as_o proper_o and_o pertinent_o referable_a to_o the_o other_o ground_n of_o distinguish_a plane_n angle_n take_v from_o the_o difference_n which_o may_v be_v in_o the_o inclination_n of_o the_o one_o contain_v side_n to_o the_o other_o for_o a_o vast_a difference_n be_v in_o the_o inclination_n of_o a_o crooked-line_n by_o obvert_v the_o concave_a or_o convexe_a side_n to_o any_o other_o line_n so_o the_o constitute_v a_o circular_a or_o elliptical_a arch_n etc._n etc._n for_o one_o side_n make_v a_o vast_a difference_n in_o the_o inclination_n because_o of_o the_o difference_n in_o their_o curvature_n also_o another_o principal_a distinction_n of_o angle_n from_o their_o side_n may_v be_v into_o angle_n who_o side_n be_v coaptable_a and_o by_o possibility_n may_v be_v coincident_a one_o with_o another_o or_o else_o such_o as_o have_v between_o they_o no_o possibility_n of_o coaptation_n and_o coincidence_n of_o the_o former_a sort_n be_v all_o right-lined_n angle_n and_o all_o concavo-convexe_a angle_n contain_v by_o arch_n
neither_o the_o curvature_n of_o the_o mix_v line_v inclination_n have_v any_o thing_n in_o it_o conform_v or_o proportionable_a to_o the_o rectitude_n of_o the_o right-lined_n inclination_n nor_o the_o rectitude_n of_o the_o right-lined_n inclination_n to_o the_o curvature_n of_o the_o mix_v line_v inclination_n in_o a_o word_n so_o different_a be_v the_o inclinableness_n of_o a_o crooked_a line_n upon_o a_o right_a line_n from_o the_o inclinableness_n of_o a_o right_a line_n upon_o a_o right_a line_n that_o it_o be_v impossible_a for_o the_o one_o ever_o to_o be_v either_o equal_a or_o any_o way_n determinate_o proportionable_a unto_o the_o other_o because_o the_o coaptation_n of_o a_o right_a line_n as_o a_o right_a line_n to_o a_o crooked-line_n as_o a_o crooked_a jine_n be_v against_o the_o property_n of_o their_o figuration_n kind_n and_o nature_n and_o for_o what_o reason_n shall_v there_o be_v lesser_a difference_n between_o a_o crooked-lined_n inclination_n and_o a_o right_n line_v inclination_n than_o there_o be_v between_o a_o crooked-line_n and_o a_o right_a line_n yet_o all_o this_o their_o distinctness_n concern_v only_o a_o heterogeneity_n in_o their_o figuration_n and_o not_o at_o all_o or_o not_o primary_o their_o quantity_n the_o argument_n if_o they_o be_v angle_n or_o plane_n angle_n they_o be_v homogeneal_a and_o of_o the_o same_o kind_n be_v of_o no_o more_o force_n than_o this_o consequence_n if_o they_o be_v quantity_n or_o continuous_a magnitude_n they_o be_v homogeneal_a and_o of_o the_o same_o kind_n and_o they_o that_o deny_v all_o heterogeneity_n in_o angle_n because_o they_o be_v all_o angle_n will_v find_v it_o a_o hard_a task_n upon_o the_o same_o ground_n to_o maintain_v a_o analogous_n homogeneity_n or_o any_o other_o considerable_a homogeneity_n between_o right_n line_v angle_n and_o spherical_a angle_n or_o any_o other_o angle_n make_v by_o plane_n cut_v the_o heterepipedal_n surface_v of_o solid_n and_o especial_o solid_a angle_n of_o what_o sort_n soever_o and_o to_o yield_v that_o all_o plane_n angle_n be_v homogeneal_a for_o it_o be_v true_a and_o the_o most_o absolute_a proper_a and_o genuine_a homogeneity_n be_v among_o plane_n angle_n i_o e._n no_o part_n of_o a_o plane_n angle_n can_v be_v any_o other_o than_o a_o plane_n angle_n how_o great_a or_o little_a soever_o and_o whether_o proportionable_a or_o improportionable_a one_o to_o another_o i._n e._n whither_o mathematical_o homogeneal_a or_o heterogeneal_a yet_o if_o we_o serious_o consider_v what_o be_v this_o homogeneity_n which_o be_v among_o plane_n angle_n that_o all_o their_o part_n be_v plane_n angle_n it_o be_v not_o as_o be_v say_v any_o quantitative_a or_o mathematical_a homogeneity_n the_o contrary_a of_o which_o be_v plentiful_o demonstrate_v in_o geometry_n to_o be_v possible_a nor_o any_o such_o homogeneity_n in_o respect_n of_o the_o manner_n of_o their_o positure_n and_o figuration_n as_o to_o exclude_v all_o far_a distinguishableness_n in_o respect_n of_o figuration_n but_o only_o denote_v that_o in_o every_o plane_n angle_n and_o in_o every_o part_n of_o every_o plane_n angle_n the_o side_n lie_v still_o in_o the_o same_o plane_n which_o homogeneity_n as_o be_v plain_a exclude_v neither_o heterogeneity_n in_o respect_n of_o figuration_n nor_o in_o respect_n of_o proportion_n and_o identity_n in_o the_o way_n of_o measure_v their_o quantity_n to_o the_o objection_n that_o in_o fig._n 12._o the_o recto-convexe_a angle_n of_o contact_n baf_n can_v be_v add_v to_o the_o right_a right-lined_n angle_v bag_n so_o make_v the_o outer_a angle_n of_o a_o semi_a circle_n f_o agnostus_n or_o take_v out_o of_o it_o sc._n the_o recto-convexe_a angle_n of_o contact_n bid_v out_o of_o the_o same_o right_a right-lined_n angle_v bag_n so_o make_v the_o inner_a angle_n of_o a_o semi_a circle_n gad_n and_o that_o therefore_o the_o recto-convexe_a angle_n of_o contact_n f_o ab_fw-la and_o b_o ad_fw-la and_o the_o 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incommensurable_n and_o specious_a quantity_n of_o which_o it_o be_v not_o know_v whether_o they_o be_v homogeneal_a or_o heterogeneal_a and_o out_o of_o a_o heterogeneal_a sum_n as_o a_o store-house_n why_o can_v some_o of_o the_o heterogeneal_n be_v subduct_v the_o rest_n remain_v and_o what_o be_v more_o usual_a than_o the_o add_v of_o heterogeneal_a figure_n one_o unto_o another_o and_o subduct_v out_o of_o a_o give_v figure_n some_o other_o figure_n which_o be_v quite_o heterogeneal_a to_o the_o first_o give_v figure_n so_o to_o add_v together_o number_n and_o measure_n and_o weight_n the_o sum_n of_o which_o may_v be_v divide_v multiply_a increase_v or_o lessen_v notwithstanding_o its_o heterogeneity_n as_o suppose_v a_o b_o c_o d_o all_o heterogeneal_a as_o be_v usual_a in_o analytic_n the_o half_a or_o three_o part_n or_o any_o proportionable_a part_n of_o this_o heterogeneal_a sum_n may_v be_v give_v and_o any_o one_o of_o the_o heterogeneal_a magnitude_n subduct_v the_o rest_n remain_v or_o a_o five_o heterogeneal_a magnitude_n add_v to_o the_o former_a sum_n or_o any_o algorythme_n ever_o specious_o sometime_o complete_o and_o absolute_o thereupon_o perform_v beside_o upon_o geometrical_a demonstration_n and_o your_o own_o confession_n all_o recto-convexe_a convexo-convexe_a and_o citradiametral_a concavo-convexe_a angle_n of_o contact_n must_v necessary_o by_o your_o own_o principle_n be_v allow_v to_o be_v absolute_o heterogeneal_a to_o all_o right-lined_n angle_n whatsoever_o yourself_o acknowledge_v that_o neither_o in_o equality_n nor_o in_o any_o kind_n of_o multiplicity_n or_o submultiplicity_n be_v any_o proportionableness_n possible_a among_o they_o and_o where_o between_o angle_n a_o mathematical_a homogeneity_n be_v confess_v and_o allow_v yet_o heterogeneity_n in_o respect_n of_o their_o schematisme_n and_o figuration_n be_v undeniable_a the_o thing_n therefore_o constitute_v and_o distinguish_v plane_n angle_v in_o respect_n of_o their_o figuration_n be_v as_o above_z their_o side_n their_o inclination_n or_o rather_o inclineableness_n and_o their_o concurrence_n that_o when_o two_o angle_n have_v all_o these_o in_o the_o same_o respective_a kind_n the_o angle_n be_v upon_o good_a reason_n in_o this_o sense_n conclude_v to_o be_v homogeneal_a but_o when_o between_o two_o angle_n be_v a_o heterogeneity_n in_o any_o of_o these_o thing_n which_o be_v of_o the_o essence_n and_o constitution_n of_o a_o angle_n those_o angle_n may_v just_o be_v judge_v in_o this_o sense_n to_o be_v heterogeneal_a and_o that_o such_o a_o specific_a heterogeneity_n may_v be_v in_o each_o of_o these_o may_v easy_o be_v declare_v as_o above_o as_o first_o in_o line_n which_o be_v the_o contain_v side_n how_o easy_a be_v it_o to_o discover_v such_o a_o heterogeneity_n for_o though_o a_o right_a line_n and_o a_o crooked-line_n agree_v undeniable_o in_o the_o general_a nature_n of_o a_o line_n and_o of_o length_n and_o of_o extension_n yet_o the_o rectitude_n of_o the_o one_o and_o the_o curvature_n of_o the_o other_o be_v several_a kind_n of_o positure_n into_o which_o the_o length_n of_o the_o one_o and_o of_o the_o other_o be_v dispose_v that_o except_o in_o contrariety_n we_o can_v see_v nothing_o but_o homogeneity_n such_o a_o heterogeneity_n must_v needs_o be_v acknowledge_v between_o they_o and_o whereas_o homogeneity_n as_o to_o side_n inclination_n and_o concurrence_n be_v require_v to_o the_o homogeneal_a figuration_n of_o angle_n the_o heterogeneity_n of_o the_o side_n hinder_v the_o possibility_n of_o ever_o make_v they_o out_o to_o be_v such_o or_o that_o by_o any_o alter_n their_o divarication_n keep_v their_o present_a property_n they_o can_v be_v coaptable_a and_o that_o angle_v contain_v by_o heterogeneal_a side_n may_v be_v equal_a prove_v only_o the_o equality_n of_o the_o inclination_n in_o either_o but_o not_o the_o homogeneity_n of_o the_o figuration_n of_o the_o angle_n or_o inclination_n as_o the_o equality_n between_o a_o square_a and_o triangle_n in_o respect_n of_o their_o equal_a perimeter_n area_n height_n base_n prove_v not_o in_o the_o least_o
seldom_o be_v we_o be_v not_o to_o maintain_v controversy_n as_o if_o they_o be_v the_o way_n of_o nature_n this_o sir_n i_o have_v write_v not_o to_o inform_v but_o confirm_v your_o judgement_n which_o i_o know_v so_o well_o verse_v in_o the_o syntaxe_n of_o this_o our_o humane_a body_n that_o it_o can_v dissent_v from_o what_o be_v the_o five_o answer_n that_o immense_a volatility_n may_v consist_v with_o immense_a ponderousness_n that_o tinctur_n may_v be_v alter_v by_o maturation_n without_o any_o addition_n whether_o the_o appear_a and_o motion_n of_o comet_n may_v be_v before_o their_o appearance_n predict_v that_o no_o such_o particular_a prediction_n can_v be_v make_v concern_v the_o meteor_n which_o be_v in_o the_o atmosphere_n of_o the_o earth_n nor_o of_o the_o first_o appearance_n of_o such_o comet_n as_o be_v suppose_v to_o have_v their_o original_a from_o new_a amassement_n of_o cometical_a matter_n in_o other_o atmospheare_n though_o after_o their_o first_o appearance_n upon_o some_o observation_n accurate_o make_v somewhat_o though_o nothing_o so_o peremptory_o as_o in_o other_o case_n may_v be_v predict_v relate_v to_o their_o future_a motion_n that_o it_o be_v not_o impossible_a but_o concern_v comet_n which_o be_v permanent_a body_n and_o not_o new_a amassement_n prediction_n may_v be_v make_v long_o before_o of_o their_o future_a appearance_n and_o motion_n sir_n i_o take_v your_o satisfaction_n upon_o my_o last_o propose_v clear_a and_o doubtless_o experiment_n now_o since_o by_o you_o prove_v and_o approve_v as_o a_o fair_a acknowledgement_n of_o that_o truth_n which_o however_o to_o i_o upon_o its_o former_a evidence_n need_v no_o far_a confirmation_n viz._n that_o so_o different_a be_v the_o genuine_a notion_n and_o quality_n of_o fixednes_n and_o gravity_n that_o immense_a volatility_n may_v and_o be_v ordinary_o consistent_a with_o a_o immense_a specific_a ponderousness_n arise_v not_o from_o the_o mole_n and_o quantity_n of_o the_o body_n under_o consideration_n but_o rather_o from_o their_o nature_n and_o kind_n that_o what_o in_o your_o first_o velitation_n you_o assume_v as_o absurd_o gross_a inconsistent_a and_o impossible_a be_v now_o upon_o your_o own_o acknowledgement_n most_o easy_o and_o obvious_o demonstrable_a by_o experience_n no_o less_o certain_o be_v to_o be_v acknowledge_v in_o what_o you_o propose_v for_o the_o tinge_v of_o metalline_a body_n only_o by_o order_v and_o attend_v they_o in_o the_o management_n of_o their_o maturation_n without_o the_o addition_n either_o of_o body_n or_o spirit_n as_o be_v all_o the_o time_n of_o this_o operation_n under_o the_o sure_a seal_n can_v i_o persuade_v myself_o it_o shall_v not_o be_v resent_v as_o a_o grand_a unkindness_n to_o be_v silent_a in_o what_o you_o call_v for_o my_o thought_n in_o in_o the_o close_a of_o you_o viz._n whether_o prognostic_n at_o certainty_n such_o as_o be_v of_o eclipse_n of_o coition_n opposition_n motion_n of_o other_o star_n whether_o i_o say_v such_o prognostic_n and_o of_o like_a certainty_n may_v not_o also_o be_v make_v of_o comet_n of_o their_o appearance_n common_a impediment_n remove_v and_o of_o their_o motion_n i_o shall_v if_o it_o may_v be_v herein_o lay_v harpocrates_n finger_n upon_o my_o lip_n and_o seal_v they_o up_o into_o a_o pertination_n silence_n not_o but_o i_o be_o desirous_a to_o know_v and_o willing_a to_o search_v after_o truth_n only_o i_o fear_v i_o these_o be_v secret_n of_o nature_n by_o their_o peculiar_a mysteriousnes_n sink_v themselves_o so_o low_o into_o the_o pit_n of_o obscurity_n that_o the_o stock_n of_o observation_n and_o disquisition_n about_o they_o which_o the_o world_n as_o yet_o have_v be_v not_o able_a to_o raise_v they_o so_o high_a and_o place_v they_o so_o near_o day_n as_o to_o be_v within_o humane_a reach_n and_o discovery_n what_o i_o now_o offer_n be_v a_o caesarean_a birth_n of_o the_o mind_n not_o bring_v forth_o by_o i_o but_o cut_v out_o of_o the_o womb_n by_o your_o importunity_n if_o it_o want_v shape_n lick_v and_o lineament_n accept_v it_o as_o a_o unripe_a abort_n and_o either_o hatch_v it_o to_o perfection_n in_o your_o thigh_n or_o give_v it_o a_o little_a dust_n to_o cover_v it_o i_o take_v by_o way_n of_o presumption_n that_o in_o this_o quaery_n and_o question_v you_o mean_v not_o by_o comet_n any_o of_o those_o more_o usual_a less_o permanent_a be_v regular_o move_v meteor_n breed_v gather_v fire_v and_o burn_v in_o the_o atmosphaere_n of_o our_o earth_n after_o their_o appearance_n there_o can_v be_v much_o certainty_n of_o their_o motion_n though_o sometime_o there_o may_v be_v conjecture_n probable_a enough_o and_o in_o the_o event_n by_o observation_n justify_v when_o the_o fuel_n or_o fovent_fw-la matter_n of_o such_o meteor_n be_v manifest_o upon_o what_o account_n soever_o know_v to_o be_v only_o or_o most_o copious_o situate_a and_o dispose_v some_o one_o particular_a way_n but_o of_o the_o generation_n and_o first_o appearance_n of_o such_o meteor_n particular_o the_o moment_n when_o and_o the_o point_n or_o exact_a place_n where_o they_o shall_v appear_v much_o less_o of_o certainty_n in_o such_o pronostick_n be_v to_o be_v expect_v there_o be_v so_o much_o variety_n contingency_n and_o uncertainty_n in_o the_o cause_n and_o meet_v together_o of_o those_o cause_n which_o contribute_v to_o their_o production_n and_o though_o there_o be_v and_o have_v be_v many_o prediction_n in_o general_a and_o rule_n of_o praedict_v astrological_o or_o physiological_o from_o the_o season_n of_o the_o year_n the_o temperature_n and_o distemper_n of_o season_n etc._n etc._n concern_v such_o meteor_n in_o general_a yet_o in_o a_o matter_n so_o unally_v unto_o certainty_n replenish_v with_o all_o manner_n of_o casualty_n to_o promote_v or_o retard_v such_o production_n i_o have_v not_o know_v any_o offer_n peremptory_a pronostick_n of_o the_o kind_n of_o the_o meteor_n its_o shape_n magnitude_n duration_n motion_n with_o absolute_a determination_n of_o its_o time_n and_o place_n at_o any_o time_n beforehand_o the_o quality_n and_o condition_n of_o the_o subject_a matter_n make_v it_o as_o impossible_a to_o bolt_n out_o scientifical_a and_o oracular_o certain_a prediction_n of_o such_o meteor_n as_o it_o be_v a_o year_n before_o to_o prove_v or_o show_v that_o in_o such_o a_o hour_n and_o in_o such_o a_o quarter_n shall_v be_v a_o rainbow_n so_o colour_v so_o continue_v or_o discontinue_v and_o of_o such_o limit_a dimension_n or_o that_o such_o a_o moment_n from_o such_o a_o point_n of_o such_o a_o azymuth_n shall_v a_o devolant_fw-fr star_n spring_n forth_o which_o in_o its_o fall_n shall_v run_v oblique_o through_o such_o and_o such_o azymuth_n and_o expire_v at_o such_o a_o height_n above_o the_o horizon_n or_o that_o such_o a_o hour_n in_o such_o a_o longitude_n and_o latitude_n shall_v in_o such_o altitudet_fw-la and_o position_n and_o of_o such_o dimension_n be_v see_v by_o day_n four_o sun_n or_o by_o night_n three_o moon_n to_o lay_v aside_o therefore_o the_o consideration_n of_o these_o as_o suppose_v impertinent_a to_o our_o present_a purpose_n what_o may_v judicious_o be_v conclude_v upon_o your_o question_n concern_v those_o other_o comet_n which_o lie_v without_o the_o compass_n of_o this_o earth_n atmospaere_n if_o there_o be_v not_o two_o sort_n of_o such_o celestial_a comet_n there_o be_v at_o least_o two_o several_a and_o very_o different_a hypothesis_n and_o notion_n under_o which_o they_o be_v consider_v by_o artist_n and_o artist_n of_o equal_a worth_n and_o fame_n order_n their_o reason_n some_o upon_o the_o one_o supposition_n some_o upon_o the_o other_o viz._n some_o as_o if_o comet_n be_v new_o make_v body_n amass_v and_o gather_v in_o some_o of_o the_o superior_a atmosphaere_n many_o of_o which_o be_v not_o without_o fair_a reason_n suppose_v to_o be_v in_o those_o vast_o remote_a aethereal_a region_n other_o as_o if_o comet_n be_v coaeval_fw-fr to_z and_o neither_o less_o permanent_a nor_o more_o new_a than_o the_o rest_n of_o the_o star_n only_o seldom_o see_v and_o when_o see_v soon_o pass_v again_o out_o of_o sight_n by_o reason_n of_o the_o line_n upon_o which_o their_o centre_n be_v move_v and_o nothing_o as_o yet_o appear_v hinder_v the_o truth_n of_o the_o possibility_n and_o consistency_n of_o both_o these_o opinion_n leave_v it_o especial_o indifferent_a in_o the_o late_a hypothesis_n to_o call_v such_o star_n at_o pleasure_n by_o the_o name_n either_o of_o comet_n or_o new-star_n or_o rather_o seldom_o appear_v star_n comet_n upon_o the_o first_o supposition_n seem_v not_o to_o want_v some_o affinity_n with_o several_a especial_o of_o the_o more_o eminent_a meteor_n of_o our_o atmosphaere_n yet_o allow_v a_o vast_a difference_n between_o they_o in_o place_n proportion_n duration_n motion_n and_o the_o like_a circumstantial_o and_o as_o our_o globe_n though_o in_o its_o self_n large_a be_v but_o a_o minute_n thing_n compare_v with_o many_o
angle_n be_v infinite_o divisible_a into_o less_o and_o less_o part_n and_o therefore_o must_v at_o last_o be_v less_o than_o the_o recto-convexe_a angle_n of_o contact_n if_o the_o recto-convexe_a angle_n of_o contact_n be_v a_o true_a angle_n have_v truth_n of_o quantity_n i_o answer_v after_o all_o possible_a division_n of_o a_o right-lined_n angle_v into_o part_n of_o its_o own_o kind_a i_o mean_v such_o as_o be_v make_v by_o right_a line_n the_o least_o part_n be_v still_o a_o right-lined_n angle_n than_o which_o the_o recto-convexe_a angle_n of_o contact_n be_v most_o fair_o demonstrate_v to_o be_v ever_o less_o i_o e._n the_o inclination_n of_o its_o side_n be_v eveâ—_n less_o and_o the_o convexe_a arch_n will_v still_o at_o the_o angle_n fall_v within_o so_o as_o true_o from_o thence_o may_v be_v infer_v that_o the_o recto-convexe_a angle_n of_o contact_n can_v never_o be_v either_o a_o right-lined_n angle_n or_o equal_a to_o it_o or_o great_a than_o a_o right-lined_a angle_n but_o that_o therefore_o it_o be_v no_o angle_n and_o have_v no_o true_a quantity_n at_o all_o because_o it_o have_v not_o the_o quantity_n of_o a_o right-lined_n angle_n be_v a_o wild_a and_o perverse_a inference_n and_o elsewhere_o disprove_v and_o why_o shall_v the_o quantitativeness_n of_o the_o recto-convexe_a angle_n of_o contact_n be_v call_v in_o question_n because_o it_o be_v demonstrate_v less_o than_o the_o least_o right-lined_n angle_n if_o between_o the_o right-lined_n tangent_fw-la and_o the_o arch_a a_o right-line_n can_v be_v draw_v than_o you_o will_v confess_v the_o quantitativeness_n of_o it_o as_o undeniable_a and_o why_o do_v not_o the_o pass_n of_o a_o thousand_o crooked-line_n from_o the_o angular_a point_n between_o the_o side_n of_o the_o recto-convexe_a angle_n of_o contact_n as_o well_o prove_v its_o quantity_n and_o divisibility_n as_o the_o pass_n of_o one_o right-line_n between_o they_o can_v there_o be_v equal_a force_n of_o proof_n from_o the_o one_o as_o from_o the_o other_o and_o when_o a_o right-line_n &_o two_o equal_a circle_n all_o three_o touch_n in_o the_o same_o point_n there_o be_v two_o equal_a recto-convexe_a angle_n of_o contact_n adjacent_a the_o one_o continued_o to_o the_o other_o and_o situate_v the_o one_o without_o the_o other_o which_o in_o indivisibles_n be_v impossible_a and_o when_o the_o question_n be_v of_o the_o quantity_n of_o angle_n what_o be_v it_o we_o inquire_v but_o only_o what_o be_v the_o inclination_n of_o the_o side_n especial_o at_o the_o angular_a point_n and_o in_o recto-convexe_a angle_n of_o contact_n the_o answer_n be_v there_o be_v no_o inclination_n at_o all_o as_o of_o a_o right-line_n to_o a_o right-line_n but_o only_o as_o of_o a_o crooked-line_n to_o a_o right-line_n that_o it_o be_v wildness_n to_o say_v because_o in_o recto-convexe_a angle_n of_o contact_n there_o can_v be_v the_o inclination_n of_o a_o right-line_n to_o a_o right-line_n that_o therefore_o the_o side_n meet_v and_o part_v one_o from_o another_o and_o not_o lie_v both_o in_o a_o right-line_n do_v make_v no_o inclination_n one_o to_o another_o and_o see_v the_o convexo-convexe_a angle_n of_o contact_n contain_v under_o two_o arch_n of_o equal_a curvature_n be_v dividable_a into_o two_o equal_a part_n by_o a_o common_a right-line_n tangent_fw-la to_o they_o both_o angle_v of_o contact_n and_o of_o both_o the_o recto-convexe_a angle_n of_o contact_n clear_o appear_v though_o the_o quantitativenes_n of_o every_o one_o of_o they_o be_v demonstrable_a to_o be_v less_o than_o the_o quantity_n of_o any_o the_o least_o right-lined_n angle_v whatsoever_o because_o right-line_n can_v contain_v a_o less_o angle_n than_o be_v agreeable_a to_o the_o inclination_n they_o possible_o can_v have_v one_o to_o another_o whereas_o in_o crooked_a line_n general_o these_o be_v the_o constant_a property_n of_o curvature_n though_o the_o angle_n of_o coincidence_n or_o curvature_n as_o we_o for_o method_n brevity_n and_o distinction_n sake_n name_v they_o may_v be_v very_o unequal_a one_o to_o another_o according_a to_o the_o degree_n of_o the_o inflexion_n inclination_n and_o curvature_n yet_o constant_o the_o angle_n on_o the_o concave_a side_n from_o any_o point_n of_o the_o curvature_n manifest_o if_o it_o be_v equal_a uniform_a and_o regular_a be_v still_o great_a than_o any_o the_o great_a acute_a right_a or_o obtuse_a right-lined_n angle_n and_o the_o angle_n between_o the_o convexe_a arch_n and_o a_o right-line_n tangent_fw-la at_o the_o same_o point_n be_v constant_o less_o than_o any_o right-lined_n angle_n and_o yet_o either_o may_v be_v make_v still_o infinite_o less_o or_o great_a the_o recto-convexe_a angle_n of_o contact_n still_o remain_v less_o than_o the_o least_o right-lined_n angle_n and_o the_o angle_n of_o curvature_n or_o coincidence_n great_a still_o than_o the_o great_a right-lined_n angle_n which_o be_v as_o much_o as_o to_o say_v that_o in_o curvature_n the_o difference_n between_o the_o angle_n of_o curvature_n and_o two_o right_a right_n line_v angle_n can_v be_v a_o right-lined_n angle_n as_o in_o truth_n it_o can_v nor_o be_v in_o reason_n so_o to_o be_v expect_v but_o of_o necessity_n it_o must_v be_v a_o mixed-lined_n sc._n a_o recto-convexe_a angle_n and_o be_v the_o recto-convexe_a angle_n of_o contact_n at_o the_o same_o point_n and_o that_o a_o recto-convexe_a angle_n of_o contact_n by_o no_o multiplicity_n can_v equal_v or_o exceed_v a_o right-lined_n angle_n do_v not_o disprove_v either_o its_o angular_a nature_n or_o its_o quantitativeness_n both_o which_o be_v otherwise_o clear_v but_o it_o be_v rather_o a_o confirmation_n of_o the_o heterogeneal_a difference_n which_o be_v between_o the_o angle_n of_o the_o one_o sort_n and_o the_o other_o and_o in_o my_o judgement_n there_o need_v no_o great_a argument_n of_o the_o quantitativeness_n of_o recto-convexe_a angle_n of_o contact_n than_o the_o absurdity_n follow_v upon_o the_o contrary_a doctrine_n that_o the_o angle_n of_o a_o semicircle_n and_o a_o right_a right-lined_n angle_n be_v equal_a viz._n the_o whole_a and_o a_o part_n the_o one_o be_v a_o mixed-lined_n and_o the_o other_o a_o right-lined_n angle_n and_o in_o the_o endeavour_n of_o coaptation_n and_o be_v coapt_v on_o one_o side_n the_o other_o side_n all_o the_o way_n fall_v within_o or_o without_o the_o other_o so_o as_o both_o the_o side_n of_o the_o one_o angle_n be_v impossible_a to_o be_v coapt_v to_o both_o the_o side_n of_o the_o other_o but_o will_v both_o still_a lie_n within_o both_o the_o side_n of_o the_o other_o and_o the_o angle_n of_o semicircle_n must_v either_o be_v confess_v unequal_a in_o unequal_a circle_n or_o the_o curvature_n of_o unequal_a circumference_n be_v manifest_o against_o the_o truth_n assert_v to_o be_v equal_a and_o if_o still_o they_o be_v aver_v to_o be_v equal_a its_o desire_a their_o equality_n shall_v be_v demonstrate_v and_o the_o way_n of_o admeasure_v their_o equality_n show_v but_o you_o will_v say_v if_o recto-convexe_a angle_n of_o contact_n be_v quantitative_a why_o can_v they_o not_o exhaust_v any_o other_o angle_n whatsoever_o contain_v under_o the_o very_a same_o side_n for_o possible_o you_o will_v urge_v that_o we_o shall_v not_o question_v the_o homogeneity_n between_o such_o angle_n to_o this_o i_o answer_v without_o examine_v what_o homogeneity_n may_v in_o other_o respect_n be_v between_o they_o that_o a_o quantitative_a and_o mathematical_a homogeneity_n can_v with_o no_o reason_n between_o they_o be_v imagine_v because_o the_o difference_n which_o be_v between_o they_o be_v a_o right-lined_n angle_n to_o which_o all_o angle_n of_o contact_n whatsoever_o be_v heterogeneal_a and_o yourself_o will_v not_o assert_v any_o mathematical_a homogeneity_n or_o at_o least_o proportionablenes_n which_o as_o to_o this_o purpose_n be_v all_o one_o between_o any_o recto-convexe_a angle_n of_o contact_n and_o a_o right-lined_n angle_n as_o in_o fig._n 18._o let_v kab_n be_v a_o recto-convexe_a angle_n of_o contact_n and_o kaf_n another_o recto-convexe_a angle_n under_o the_o very_a same_o side_n and_o let_v ad_fw-la be_v a_o right-line_n tangent_fw-la upon_o the_o arch_n of_o therefore_o the_o recto-convexe_a angle_n of_o contact_n d_o of_o and_o kab_n be_v equal_a the_o right-lined_n angle_v kad_a be_v the_o difference_n of_o the_o recto-convexe_a angle_n of_o contact_n kab_n and_o the_o other_o recto-convexe_a angle_n kaf_n under_o the_o divarication_n of_o the_o very_a same_o side_n so_o as_o it_o be_v impossible_a to_o divide_v this_o or_o any_o other_o angle_n whatsoever_o which_o be_v not_o isoclitical_a to_o divide_v i_o say_v all_o of_o it_o into_o any_o number_n at_o pleasure_n give_v of_o part_n which_o shall_v be_v homogeneal_a all_o of_o they_o one_o unto_o another_o for_o how_o many_o soever_o be_v homogeneal_a the_o angle_n of_o contact_n or_o that_o which_o be_v take_v out_o of_o it_o or_o that_o unto_o which_o it_o adhaeres_fw-la will_v have_v and_o make_v heterogeneity_n and_o
other_o for_o neither_o can_v the_o whole_a right_a right-lined_n angle_n nor_o the_o recto-concave_a angle_n of_o the_o semi_a circle_n ever_o be_v exhaust_v by_o any_o number_n whatsoever_o of_o such_o heterogeneal_a part_n as_o be_v the_o recto-convexe_a angle_n of_o contact_n nor_o ever_o any_o equality_n or_o other_o proportion_n can_v possible_o be_v show_v between_o the_o right_a right-lined_n angle_n and_o the_o recto-concave_a angle_n of_o the_o semi_a circle_n because_o there_o be_v no_o way_n possible_a in_o which_o their_o quantity_n can_v be_v proportionable_o mensurable_a for_o not_o without_o very_o good_a reason_n unto_o all_o magnitude_n be_v to_o be_v allow_v their_o special_a property_n as_o to_o all_o positure_n and_o figuration_n they_o to_o angle_v these_o thing_n be_v peculiar_a be_v otherwise_o in_o other_o magnitude_n viz._n in_o angle_n which_o be_v true_o and_o on_o all_o hand_n confess_o homogeneal_a you_o can_v to_o any_o give_a angle_n set_v forth_o 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this_o point_n the_o principal_a thing_n we_o have_v labour_v herein_o to_o dilucidate_n &_o as_o we_o doubt_v not_o have_v effect_v be_v that_o mathematical_a homogenealness_n be_v not_o a_o homogeneity_n of_o all_o the_o part_n whatsoever_o that_o be_v in_o the_o magnitude_n which_o be_v homogeneal_a in_o respect_n of_o some_o special_a way_n of_o measure_v their_o quantity_n or_o a_o undivideableness_n of_o such_o homogeneal_a magnitude_n into_o part_n otherwise_o heterogeneal_a according_a to_o which_o acceptation_n the_o word_n be_v chief_o take_v in_o other_o part_n of_o philosophy_n for_o there_o be_v no_o right-lined_n angle_v whatsoever_o nor_o any_o other_o angle_n whatsoever_o but_o as_o be_v up_o and_o down_o herein_o show_v may_v be_v separate_o divide_v into_o heterogeneal_a part_n but_o mathematical_a homogeneity_n be_v homogeneity_n in_o the_o way_n of_o measure_v the_o quantity_n of_o the_o compare_v magnitude_n sc._n in_o the_o same_o indefinite_a measure_n and_o 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all_o recto-convexe_a angle_n of_o semi_a circle_n all_o acute_a or_o right_a right-lined_n angle_n these_o may_v all_o measure_v themselves_o and_o in_o what_o order_n their_o side_n fall_v within_o or_o without_o in_o any_o obtuse_a right-lined_n angle_v whatsoever_o yet_o because_o this_o their_o homometry_n be_v only_o of_o the_o situation_n or_o order_n in_o which_o the_o side_n part_v from_o the_o angular_a point_n but_o not_o of_o their_o quantity_n in_o a_o indefinite_a measure_n and_o according_a to_o the_o denomination_n of_o the_o same_o quantitative_a measure_n so_o as_o to_o lodge_v a_o proportionality_n between_o the_o magnitude_n so_o compare_v together_o in_o their_o common_a way_n of_o measure_v they_o be_v not_o nor_o can_v thereby_o be_v vindicate_v from_o their_o otherwise_o innate_a mathematical_a heterogeneity_n which_o concern_v some_o of_o they_o be_v confess_v on_o all_o hand_n and_o be_v without_o the_o verge_n of_o the_o controversy_n and_o as_o follow_v angle_n be_v of_o a_o concrete_a nature_n have_v in_o they_o something_o quantitative_a and_o something_o not_o quantitative_a whereas_o that_o which_o be_v to_o be_v the_o indefinite_a measure_n of_o homogeneal_a quantity_n be_v to_o be_v consider_v abstract_o as_o quantity_n without_o heterogeneal_a concretion_n so_o it_o be_v the_o circumference_n of_o a_o circle_n that_o measure_v all_o right-lined_n angle_n and_o when_o all_o plane_n angle_n be_v say_v to_o be_v homogeneal_a it_o be_v not_o in_o respect_n of_o a_o common_a indefinite_a quantity_n by_o which_o they_o be_v all_o measure_a which_o the_o recto-convexe_a angle_n of_o contact_n do_v sufficient_o evince_v but_o as_o be_v manifest_v it_o be_v only_o because_o of_o the_o position_n and_o situation_n of_o the_o side_n in_o the_o same_o plane_n which_o homogeneity_n be_v of_o no_o concern_n unto_o quantity_n nor_o by_o any_o necessity_n can_v thereupon_o infer_v the_o consequent_a of_o proportionableness_n but_o to_o 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of_o have_v their_o contain_v side_n to_o lie_v still_o in_o the_o same_o plane_n whereby_o they_o distinguish_v themselves_o from_o all_o other_o superficial_a angle_n which_o be_v heterepipedal_n who_o contain_v line_n or_o side_n
special_a property_n may_v appear_v in_o that_o very_a instance_n of_o compel_v the_o parallelisme_n of_o curve_v line_n to_o answer_v the_o consectary_n and_o idiom_n of_o the_o parallelisme_n of_o right-line_n in_o which_o to_o omit_v the_o allege_v instance_n as_o by_o you_o unprove_v and_o for_o good_a reason_n by_o we_o to_o be_v deny_v a_o right-line_n tangent_fw-la of_o the_o lesser_a concentric_a circle_n cut_v the_o circumference_n of_o the_o great_a and_o infinite_a right-line_n cut_v the_o great_a neither_o cut_v nor_o touch_v the_o lesser_a which_o be_v repugnant_a to_o the_o nature_n of_o parallelisme_n in_o right_a line_n that_o that_o which_o be_v so_o much_o contend_v for_o that_o a_o crooked-line_n and_o a_o right-line_n be_v homogeneal_a as_o to_o length_n and_o their_o general_a lineariness_n be_v or_o ought_v never_o to_o be_v deny_v there_o be_v all_o possibility_n of_o equality_n and_o true_o proportionable_a inequality_n between_o they_o what_o kind_n of_o curvature_n soever_o the_o crooked-line_n bear_v but_o that_o they_o be_v homogeneal_a as_o to_o the_o positure_n of_o their_o longitude_n the_o site_n and_o manner_n of_o their_o extension_n have_v unto_o i_o be_v always_o unconceivable_a whence_o the_o truth_n on_o both_o hand_n clear_o follow_v viz._n that_o a_o arch_n and_o a_o right-line_n may_v be_v equal_a and_o hold_v always_o a_o true_a limit_v exact_a proportion_n one_o to_o another_o but_o the_o arch_n and_o it_o be_v chorde_v never_o can_v be_v equal_a i._n e._n there_o never_o can_v be_v equality_n between_o a_o right-line_n and_o a_o crooked-line_n both_o posit_v between_o the_o same_o two_o terminate_v point_n nor_o any_o analogy_n between_o the_o rectitude_n of_o the_o one_o and_o the_o curvature_n of_o the_o other_o and_o the_o seek_v to_o prove_v the_o equality_n of_o angle_n contain_v under_o homologal_a line_n the_o one_o curve_v the_o other_o a_o right-line_n from_o your_o usual_a fancy_n of_o a_o regular_a polygone_a of_o infinite_a angle_n in_o every_o circle_n be_v too_o wild_a to_o be_v persuasive_a for_o though_o at_o any_o mean_a point_n in_o curve_v line_n the_o two_o part_n of_o the_o curve-line_n may_v be_v conceive_v special_o to_o meet_v as_o several_a part_n and_o line_n and_o so_o to_o have_v inclination_n the_o one_o to_o the_o other_o and_o so_o to_o constitute_v a_o angle_n which_o we_o call_v the_o angle_n of_o curvature_n and_o coincidence_n not_o reasonable_o to_o be_v deny_v by_o those_o with_o who_o it_o be_v so_o ordinary_a to_o make_v such_o supposition_n and_o especial_o such_o as_o can_v so_o usual_o against_o possibility_n imagine_v angle_n in_o a_o right-line_n remain_v a_o right-line_n yet_o that_o angle_n shall_v be_v without_o side_n and_o a_o perimeter_n of_o any_o figure_n conceive_v at_o once_o to_o be_v all_o angular_a point_n and_o no_o lineary_a side_n clear_o â—â—â—â—stes_v the_o perimeter_n of_o the_o nature_n of_o a_o line_n 〈◊〉_d to_o i_o it_o seem_v far_o from_o the_o nature_n of_o a_o â—â—â—ular_a figure_n that_o have_v nothing_o but_o point_n â—â—stead_o of_o line_n to_o bound_v it_o but_o which_o be_v most_o material_a that_o a_o number_n actual_o infinite_a shall_v be_v so_o easy_o give_v be_v hard_o to_o allow_v and_o that_o indivisibles_n as_o point_n shall_v be_v so_o adjacent_a 〈◊〉_d to_o another_o one_o without_o another_o withâ—â—â—_n coincidence_n identity_n and_o unity_n be_v new_a â—â—â—â—osophy_n and_o not_o easy_o capable_a of_o any_o in_o 〈◊〉_d defence_n therefore_o that_o argumentation_n 〈◊〉_d that_o such_o a_o regular_a polygone_a of_o in_o 〈◊〉_d whether_o side_n or_o angle_n be_v either_o a_o circle_n 〈◊〉_d inscribable_a in_o a_o circle_n be_v too_o vain_a for_o it_o can_v be_v neither_o be_v nothing_o because_o there_o neither_o be_v nor_o can_v be_v any_o such_o thing_n for_o if_o any_o such_o be_v allow_v they_o must_v of_o necessity_n have_v equal_a and_o infinite_a perimeter_n which_o be_v too_o gross_a to_o be_v admit_v in_o itself_o and_o beside_o render_v the_o whole_a matter_n unapplyable_a to_o circle_n which_o be_v acknowledge_v to_o be_v some_o less_o than_o other_o so_o as_o all_o discourse_n of_o a_o regular_a polygone_a of_o infinite_a angle_n be_v discourse_n not_o only_o of_o a_o nonentity_n but_o a_o absolute_a impossibility_n which_o render_v all_o supposition_n thereof_o unjustifiable_a and_o of_o the_o same_o fineness_n be_v those_o say_n that_o the_o magnitude_n of_o a_o angle_n be_v not_o to_o be_v judge_v of_o from_o the_o divarication_n which_o the_o side_n have_v without_o the_o angular_a point_n or_o point_n of_o concurrence_n but_o from_o the_o divarication_n which_o they_o have_v in_o the_o point_n of_o concurrence_n as_o if_o in_o a_o indivisible_a point_n they_o can_v have_v any_o divarication_n at_o all_o but_o as_o if_o it_o be_v resolve_v that_o even_o this_o shall_v be_v transcend_v in_o monstrosity_n for_o the_o justify_n of_o the_o equality_n 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objection_n can_v be_v make_v out_o of_o that_o if_o other_o thing_n of_o a_o less_o veritable_a nature_n have_v not_o be_v take_v in_o as_o in_o most_o of_o they_o the_o fancy_v possibility_n of_o a_o regular_a polygone_a of_o infinite_a angle_n and_o frequent_o that_o a_o circle_n be_v that_o regular_a polygone_a but_o beside_o though_o what_o be_v less_o than_o any_o positive_a quantity_n whatsoever_o be_v not_o any_o quantity_n at_o all_o yet_o this_o hinder_v not_o but_o quantity_n may_v be_v mathematical_o heterogeneal_a and_o improportionable_a one_o to_o another_o so_o every_o surface_n be_v less_o than_o any_o solid_a and_o angle_n of_o contact_n be_v not_o less_o than_o any_o quantity_n whatsoever_o for_o there_o be_v in_o the_o least_o of_o they_o a_o endless_a unexhausted_a divisibility_n which_o how_o it_o can_v consist_v with_o a_o nonquantitativenes_n let_v those_o that_o have_v a_o mind_n to_o be_v serious_a solemn_o consider_v to_o the_o objection_n that_o will_v prove_v neither_o semicircumference_n to_o contain_v a_o angle_n with_o the_o right-lined_n tangent_fw-la of_o it_o in_o its_o extreem_n point_n because_o the_o two_o semi_a circumference_n contain_v no_o angle_n at_o that_o point_n but_o be_v one_o regular_o continue_a line_n and_o the_o circumference_n and_o right-lined_n tangent_fw-la be_v line_n coincident_a at_o least_o as_o to_o the_o point_n of_o contact_n manifest_a and_o reasonable_a answer_n can_v be_v to_o seek_v out_o of_o what_o have_v already_o be_v say_v for_o first_o what_o hinder_v the_o reasonable_a conceive_n of_o angularity_n at_o any_o point_n of_o a_o curve-line_n where_o be_v both_o concurrence_n inclination_n and_o divisibility_n more_o than_o the_o notion_n of_o divisibility_n at_o any_o mean_a point_n of_o a_o right-line_n and_o not_o to_o doubt_v but_o a_o curve_v line_n may_v be_v conceive_v reasonable_o as_o one_o continue_a line_n as_o well_o as_o two_o or_o more_o incline_v and_o concur_v right-line_n 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rect-angle_n under_o the_o semidiameter_n and_o a_o right-line_n
the_o production_n and_o figuration_n of_o the_o contain_v side_n it_o will_v not_o only_o be_v necessary_a for_o we_o to_o yield_v unto_o you_o that_o recto-convexe_a angle_n of_o contact_n be_v not_o quantitative_a but_o beside_o both_o you_o and_o we_o contrary_a to_o what_o we_o have_v always_o hitherto_o judge_v shall_v be_v constrain_v to_o acknowledge_v that_o there_o be_v no_o quantitativenes_n neither_o in_o crooked-lined_n nor_o right-lined_n nor_o any_o other_o angle_v whatsoew_a whether_o superficial_a in_o one_o or_o several_a plain_n or_o solid_a and_o can_v any_o thing_n be_v more_o horrid_a then_o to_o say_v the_o quantity_n of_o angle_n be_v not_o to_o be_v measure_v by_o the_o divarication_n of_o the_o side_n at_o the_o angular_a point_n but_o by_o their_o divarication_n in_o the_o angular_a point_n where_o they_o have_v none_o at_o all_o but_o yet_o though_o it_o be_v thus_o evident_a that_o the_o inclination_n of_o the_o side_n at_o the_o angular_a point_n may_v and_o frequent_o be_v much_o less_o or_o great_a than_o the_o inclination_n of_o the_o same_o side_n at_o other_o point_n which_o as_o be_v above_o hint_v be_v not_o to_o be_v leave_v out_o in_o the_o full_a genuine_a and_o clear_a consideration_n of_o the_o nature_n of_o angle_n their_o kind_n figuration_n and_o quantity_n however_o the_o inclination_n of_o the_o side_n at_o the_o angular_a point_n be_v that_o which_o be_v most_o usual_o inquire_v after_o and_o most_o useful_a to_o be_v search_v and_o observe_v in_o geometry_n for_o the_o discovery_n which_o be_v from_o thence_o make_v of_o line_n how_o they_o fall_v coincident_o or_o within_o or_o without_o other_o to_o the_o objection_n that_o in_o fig._n 12._o the_o right-lined_n tangent_fw-la ab_fw-la and_o the_o arch_a all_o make_v all_o one_o and_o the_o same_o equal_a inclination_n to_o the_o right-line_n secant_fw-la ac_fw-la in_o the_o common_a angular_a point_n a_o and_o that_o therefore_o the_o right-lined_n angle_n bac_n under_o the_o right-lined_n secant_fw-la ac_fw-la and_o the_o right-line_n tangent_fw-la ab_fw-la be_v equal_a to_o the_o mixed-lined_n recto-concave_a angle_n eal_n under_o the_o right-line_n secant_fw-la ac_fw-la and_o the_o arch_a all_o i_o answer_v as_o before_o inclination_n be_v not_o in_o the_o angular_a point_n abstract_o consider_v without_o regard_n to_o the_o side_n pass_v out_o of_o it_o but_o inclination_n be_v the_o relative_a situation_n which_o the_o concur_v side_n have_v at_o the_o angular_a point_n at_o least_o that_o be_v their_o inclination_n there_o for_o a_o point_n to_o a_o line_n can_v have_v no_o inclination_n it_o may_v have_v distance_n from_o the_o line_n but_o can_v be_v incline_v unto_o it_o because_o of_o its_o indivisibility_n and_o have_v already_o show_v the_o inclination_n of_o side_n to_o side_n to_o be_v of_o the_o essence_n notion_n and_o nature_n of_o a_o angle_n a_o little_a may_v be_v reply_n enough_o to_o all_o those_o hypersceptical_a objection_n which_o be_v sound_v upon_o the_o imagination_n of_o a_o angle_n in_o a_o right-line_n or_o any_o inclination_n or_o angularity_n imagine_v between_o a_o right_a line_n and_o a_o point_n especial_o the_o point_n be_v in_o the_o right-line_n and_o equality_n or_o inequality_n of_o angle_n be_v not_o nor_o can_v be_v judge_v of_o by_o the_o so_o abstract_o consider_v angular_a point_n in_o which_o a_o thousand_o several_a side_n of_o several_a and_o unequal_a angle_n may_v meet_v indifferent_o but_o the_o judgement_n of_o the_o magnitude_n and_o equality_n and_o inequality_n of_o angle_n be_v from_o the_o side_n and_o the_o order_n of_o the_o divarication_n in_o which_o they_o pass_v especial_o first_fw-mi of_o all_o from_o the_o angular_a point_n beside_o how_o strange_o be_v it_o take_v for_o grant_v without_o prove_v thâ—â—_n the_o right-line_n tangent_fw-la ab_fw-la and_o the_o arch_a all_o be_v equal_o in_o the_o angular_a point_n a_o incline_v unto_o the_o right-line_n secant_fw-la ac_fw-la if_o that_o can_v be_v once_o prove_v the_o concern_v of_o it_o will_v turn_v the_o scale_n of_o the_o controversy_n but_o demonstration_n be_v so_o clear_a to_o the_o contrary_a that_o as_o without_o proof_n it_o be_v not_o fit_a to_o be_v admit_v so_o for_o the_o proof_n of_o it_o i_o know_v nothing_o can_v be_v produce_v beside_o a_o utter_a despair_n of_o ever_o make_v it_o out_o for_o if_o the_o congruency_n of_o the_o side_n terminative_o in_o the_o angular_a point_n be_v sufficient_a to_o constitute_v equality_n in_o angle_n it_o appear_v not_o how_o any_o angle_n meet_v in_o the_o same_o or_o different_a angular_a point_n can_v be_v unequal_a every_o point_n by_o reason_n of_o its_o indivisibility_n be_v incapable_a of_o inequality_n as_o well_o as_o inclination_n and_o if_o all_o such_o angle_n so_o constitute_v by_o the_o fall_n within_o or_o without_o of_o the_o side_n shall_v be_v doubt_v and_o question_v whether_o they_o be_v true_a and_o quantitative_a angle_n and_o whether_o the_o addition_n or_o subduction_n of_o they_o be_v able_a to_o diversify_v other_o angle_n and_o their_o quantity_n all_o the_o pain_n of_o the_o geometrician_n to_o prove_v the_o intracadency_n and_o extracadency_n of_o the_o angular_a side_n from_o the_o same_o angular_a point_n be_v vain_a and_o to_o no_o purpose_n the_o angle_n remain_v altogether_o the_o same_o and_o equal_a whether_o such_o angle_n of_o contact_n be_v add_v to_o they_o or_o take_v from_o they_o but_o yet_o though_o the_o true_a 〈◊〉_d and_o genuine_a nature_n of_o a_o angle_n consist_v in_o the_o mutual_a habitude_n and_o inclination_n of_o the_o contain_n and_o concur_v side_n it_o be_v not_o ever_o necessary_a to_o consider_v it_o with_o such_o a_o largeness_n in_o geometry_n the_o inclination_n which_o the_o side_n bear_v mutual_o each_o to_o other_o at_o the_o angular_a point_n be_v out_o of_o doubt_n that_o which_o be_v of_o most_o constant_a necessity_n high_a concern_v and_o usefulnes_n in_o all_o angle_n to_o be_v observe_v it_o be_v true_a in_o isoclitical_a angle_n the_o inclination_n of_o the_o two_o contain_v side_n be_v every_o where_o the_o same_o and_o equal_a it_o be_v indifferent_o by_o geometrician_n take_v by_o a_o circle_n who_o centre_n be_v in_o the_o angular_a point_n of_o what_o diameter_n soever_o thereunto_o applyable_a and_o at_o what_o same_o distance_n soever_o from_o the_o angle_n or_o at_o what_o same_o longitude_n soever_o from_o thence_o in_o the_o side_n the_o point_n be_v at_o which_o their_o inclination_n be_v observe_v by_o intercept_a arch_n and_o the_o same_o two_o point_n terminate_a the_o intercept_v arch_n at_o which_o their_o mutual_a inclination_n be_v observe_v constant_o offer_v themselves_o together_o whether_o you_o take_v point_n at_o equal_a distance_n from_o the_o angular_a point_n or_o intercept_n in_o the_o side_n equal_a longitude_n between_o they_o and_o the_o same_o angular_a point_n but_o in_o anisoclitical_a angle_n the_o inclination_n of_o the_o anisoclitical_a side_n vary_v still_o in_o the_o continuity_n of_o their_o production_n if_o we_o use_v the_o former_a method_n of_o measure_v the_o inclination_n of_o the_o side_n by_o intercept_a arch_n of_o a_o circle_n draw_v upon_o the_o angular_a point_n as_o centre_n in_o they_o therefore_o geometrician_n concern_v themselves_o little_a further_o than_o to_o observe_v the_o mutual_a inclination_n of_o the_o side_n at_o the_o very_a point_n of_o their_o angle_n and_o not_o at_o any_o other_o point_n in_o the_o anisoclitical_a side_n save_v only_o the_o point_n of_o their_o concurrence_n because_o of_o the_o constant_a variation_n of_o their_o inclination_n both_o in_o respect_n of_o such_o and_o every_o other_o method_n and_o way_n of_o measure_v according_a to_o the_o continuity_n of_o their_o production_n and_o by_o arch_n of_o circle_n draw_v upon_o the_o angular_a point_n as_o centre_n it_o be_v impossible_a to_o measure_v the_o inclination_n of_o the_o anisoclitical_a line_n at_o the_o point_n of_o their_o concurrence_n the_o only_a way_n therefore_o which_o remain_v unto_o geometrician_n to_o measure_v such_o anisoclitical_a angle_n i._n e_o the_o inclination_n of_o their_o side_n at_o the_o point_n of_o their_o concurrence_n be_v by_o observe_v the_o line_n in_o what_o order_n they_o depart_v from_o the_o angular_a point_n viz._n which_o line_n fall_v within_o which_o without_o and_o which_o be_v coincident_a with_o that_o unto_o which_o it_o be_v compare_v so_o in_o fig._n 12._o if_o ahf_n and_o adk_n be_v equal_a circle_n touch_v in_o the_o point_n a_o and_o g_z the_o centre_n and_o agk_n the_o diameter_n of_o the_o circle_n adk_n and_o ael_n the_o arch_n of_o a_o great_a circle_n touch_v both_o the_o former_a circle_n in_o the_o same_o point_n a_o and_o with_o its_o concave_a side_n at_o a_o respect_v the_o centre_n g_o and_o with_o its_o convexe_a side_n at_o a_o respect_v the_o circle_n haf_n also_o if_o ab_fw-la be_v a_o right-line_n