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that_o a_o man_n at_o the_o first_o hear_n whether_o geometrician_n or_o not_o geometrician_n must_v abhor_v they_o as_o for_o example_n 1._o that_o two_o equal_a proportion_n be_v not_o double_a to_o one_o of_o the_o same_o proportion_n 2._o that_o a_o proportion_n be_v double_a triple_a etc._n etc._n of_o a_o number_n but_o not_o of_o a_o proportion_n 3._o that_o the_o same_o body_n without_o add_v to_o it_o or_o take_v from_o it_o be_v sometime_o great_a and_o sometime_o less_o 4._o that_o a_o quantity_n may_v grow_v less_o and_o less_o eternal_o so_o as_o at_o last_o to_o be_v equal_a to_o another_o quantity_n or_o which_o be_v all_o one_o that_o there_o be_v a_o last_o in_o eternity_n 5._o that_o the_o nature_n of_o a_o angle_n consist_v in_o that_o which_o lie_v between_o the_o line_n that_o comprehend_v the_o angle_n in_o the_o very_a point_n of_o their_o concourse_n that_o be_v to_o say_v a_o angle_n be_v the_o superficies_n which_o lie_v between_o the_o two_o point_n which_o touch_n or_o as_o they_o understand_v a_o point_n the_o superficies_n that_o lie_v between_o the_o two_o nothing_o which_o touch_n 6._o that_o the_o quotient_a be_v the_o proportion_n of_o the_o division_n to_o the_o dividend_n upon_o these_o and_o some_o such_o other_o principle_n be_v ground_v all_o that_o doctor_n wallis_n have_v say_v not_o only_o in_o his_o elenchus_n of_o my_o geometry_n but_o also_o in_o his_o treatise_n of_o the_o angle_n of_o contact_n and_o in_o his_o arithmetica_fw-la infinitorum_fw-la which_o two_o last_v i_o have_v fere_n in_o two_o or_o three_o leaf_n whole_o and_o clear_o confute_v and_o i_o very_o believe_v that_o since_o the_o begin_n of_o the_o world_n there_o have_v not_o be_v nor_o ever_o shall_v be_v so_o much_o absurdity_n write_v in_o geometry_n as_o be_v to_o be_v find_v in_o those_o book_n of_o he_o with_o which_o there_o be_v so_o much_o presumption_n join_v that_o a_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d
body_n of_o this_o kind_n of_o magnitude_n and_o quantity_n the_o subject_n be_v body_n and_o because_o for_o the_o compute_v of_o the_o magnitude_n of_o body_n it_o be_v not_o necessary_a that_o the_o body_n themselves_o shall_v be_v present_a the_o idea_n and_o memory_n of_o they_o supply_v their_o presence_n we_o reckon_v upon_o those_o imaginary_a body_n which_o be_v the_o quantity_n themselves_o and_o say_v the_o length_n be_v so_o great_a the_o breadth_n so_o great_a etc._n etc._n which_o in_o truth_n be_v no_o better_a than_o to_o say_v the_o length_n be_v so_o long_o or_o the_o breadth_n so_o broad_a etc._n etc._n but_o in_o the_o mind_n of_o a_o inteligent_a man_n it_o breed_v no_o mistake_n beside_o the_o quantity_n of_o body_n there_o be_v a_o quantity_n of_o time_n for_o see_v man_n without_o absurdity_n do_v ask_v how_o much_o it_o be_v by_o answer_v tantum_n so_o much_o they_o make_v manifest_a there_o be_v a_o a_o quantity_n that_o belong_v unto_o time_n namely_o a_o length_n and_o because_o length_n can_v be_v a_o accident_n of_o time_n which_o be_v itself_o a_o accident_n it_o be_v the_o accident_n of_o a_o body_n and_o consequent_o the_o length_n of_o the_o time_n be_v the_o length_n of_o the_o body_n by_o which_o length_n or_o line_n we_o determine_v how_o much_o the_o time_n be_v suppose_v some_o body_n to_o be_v move_v over_o it_o also_o we_o not_o improper_o ask_v with_o how_o much_o swiftness_n a_o body_n be_v move_v and_o consequent_o there_o be_v a_o quantity_n of_o motion_n or_o swiftness_n and_o the_o measure_n of_o that_o quantity_n be_v also_o a_o line_n but_o then_o again_o we_o must_v suppose_v another_o motion_n which_o determine_v the_o time_n of_o the_o former_a also_o of_o force_n there_o be_v a_o question_n of_o how_o much_o which_o be_v to_o be_v answer_v by_o so_o much_o that_o be_v by_o quantity_n if_o the_o force_n consist_v in_o swiftness_n the_o determination_n be_v the_o same_o with_o that_o of_o swiftness_n namely_o by_o a_o line_n if_o in_o swiftness_n and_o quantity_n of_o body_n joint_o then_o by_o a_o line_n and_o a_o solid_a or_o if_o in_o quantity_n of_o body_n only_o as_o weight_n by_o a_o solid_a only_o so_o also_o be_v number_n quantity_n but_o in_o no_o other_o sense_n then_o as_o a_o line_n be_v quantity_n divide_v into_o equal_a part_n of_o a_o angle_n which_o be_v of_o two_o line_n whatsoever_o they_o be_v meet_v in_o one_o point_n the_o digression_n or_o openess_n in_o other_o point_n it_o may_v be_v ask_v how_o great_a be_v that_o digression_n this_o question_n be_v answer_v also_o by_o quantity_n a_o angle_n therefore_o have_v quantity_n though_o it_o be_v not_o the_o subject_n of_o quantity_n for_o the_o body_n only_o can_v be_v the_o subject_n in_o which_o body_n those_o â—…ing_a line_n be_v mark_v and_o because_o two_o line_n may_v be_v make_v to_o divaricate_v by_o two_o cause_n one_o when_o have_v one_o end_n common_a and_o immovable_a they_o depart_v one_o from_o another_o at_o the_o other_o end_v circular_o and_o this_o be_v call_v surp_o a_o angle_n and_o the_o quantity_n thereof_o be_v the_o quantity_n of_o the_o arch_n which_o the_o two_o line_n intercept_v the_o other_o cause_n be_v the_o bend_n of_o a_o straight_a line_n into_o a_o circular_a or_o other_o crooked_a line_n till_o it_o touch_v the_o place_n of_o the_o same_o line_n whilst_o it_o be_v straight_o in_o one_o only_a point_n and_o this_o be_v call_v a_o angle_n of_o contingence_n and_o because_o the_o more_o it_o be_v bend_v the_o more_o it_o digress_v from_o the_o tangent_fw-la it_o may_v be_v ask_v how_o much_o more_o and_o therefore_o the_o answer_n must_v be_v make_v by_o quantity_n and_o consequent_o a_o angle_n of_o contingence_n have_v its_o quantity_n as_o well_o as_o that_o which_o be_v call_v simple_o a_o angle_n and_o in_o case_n the_o digression_n of_o two_o such_o crooked_a line_n from_o the_o tangent_fw-la be_v uniform_a as_o in_o circle_n the_o quantity_n of_o their_o digression_n may_v be_v determine_v for_o if_o a_o straight_a line_n be_v draw_v from_o the_o point_n of_o contact_n the_o digression_n of_o the_o lesser_a circle_n will_v be_v to_o the_o digression_n of_o the_o great_a circle_n as_o the_o part_n of_o the_o line_n draw_v from_o the_o point_n of_o contact_n and_o intercept_v by_o the_o circumference_n of_o the_o great_a circle_n be_v to_o the_o part_n of_o the_o same_o line_n intercept_v by_o the_o circumference_n of_o the_o lesser_a circle_n or_o which_o be_v all_o one_o as_o the_o great_a radius_fw-la be_v to_o the_o lesser_a radius_fw-la you_o may_v guess_v by_o this_o what_o will_v become_v of_o that_o part_n of_o your_o last_o book_n where_o you_o handle_v the_o question_n of_o the_o quantity_n of_o a_o angle_n of_o contingence_n also_o there_o lie_v a_o question_n of_o how_o much_o comparative_o one_o magnitude_n be_v to_o another_o magnitude_n as_o how_o much_o water_n be_v in_o a_o tun_n in_o respect_n of_o the_o ocean_n how_o much_o in_o respect_n of_o a_o piâ—_n little_a in_o the_o first_o respect_n much_o in_o the_o latter_a therefore_o the_o answer_n must_v be_v make_v by_o some_o respective_a quantity_n this_o respective_a quantity_n be_v call_v ratio_fw-la and_o proportion_n and_o be_v determine_v by_o the_o quantity_n of_o their_o difference_n and_o if_o their_o difference_n be_v compare_v also_o with_o the_o quantity_n themselves_o that_o differ_v it_o be_v call_v simple_o proportion_n or_o proportion_n geometrical_a but_o if_o the_o difference_n be_v not_o so_o compare_v than_o it_o be_v call_v proportion_n arithmetical_a and_o where_o the_o difference_n be_v none_o there_o be_v no_o quantity_n of_o the_o proportion_n which_o in_o this_o case_n be_v but_o a_o bare_a comparison_n also_o concern_v heat_n light_n and_o divers_a other_o quality_n which_o have_v degree_n there_o lie_v a_o question_n of_o how_o much_o to_o be_v answer_v by_o a_o so_o much_o and_o consequent_o they_o have_v their_o quantity_n though_o the_o same_o with_o the_o quantity_n of_o swiftness_n because_o the_o intention_n and_o remission_n of_o such_o quality_n be_v but_o the_o intention_n and_o remission_n of_o the_o swiftness_n of_o that_o motion_n by_o which_o the_o agent_n produce_v such_o a_o quality_n and_o as_o quantity_n may_v be_v consider_v in_o all_o the_o operation_n of_o nature_n so_o also_o do_v geometry_n run_v quite_o through_o the_o whole_a body_n of_o natural_a philosophy_n to_o the_o principle_n of_o geometry_n the_o definition_n appertain_v also_o of_o a_o measure_n which_o be_v this_o one_o quantity_n be_v the_o measure_n of_o another_o quantity_n when_o it_o or_o the_o multiple_n of_o it_o be_v coincident_a in_o all_o point_n with_o the_o other_o quantity_n in_o which_o definition_n in_o stead_n of_o that_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d of_o euclid_n i_o put_v coincidence_n for_o that_o superposition_n of_o quantity_n by_o which_o they_o render_v the_o word_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d can_v be_v understand_v of_o body_n but_o only_o of_o line_n and_o superficies_n nevertheless_o many_o body_n may_v be_v coincident_a successive_o with_o one_o and_o the_o same_o place_n and_o that_o place_n will_v be_v their_o measure_n as_o we_o see_v practise_v continual_o in_o the_o measure_n of_o liquid_a body_n which_o art_n of_o measure_v may_v proper_o be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d but_o not_o superposition_n also_o the_o definition_n of_o great_a lesle_n and_o equal_a be_v necessary_a principle_n of_o geometry_n for_o it_o can_v be_v imagine_v that_o any_o geometrician_n shall_v demonstrate_v to_o we_o the_o equality_n and_o inequality_n of_o magnitude_n except_o he_o tell_v we_o first_o what_o those_o word_n do_v signify_v and_o it_o be_v a_o wonder_n to_o i_o '_o that_o euclid_n have_v not_o any_o where_o define_v what_o be_v equal_n or_o at_o least_o what_o be_v equal_a body_n but_o serve_v his_o turn_n throughout_o with_o that_o forementioned_a 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d which_o have_v no_o place_n in_o solid_n nor_o in_o time_n nor_o in_o swiftness_n nor_o in_o circular_a or_o other_o crooked_a line_n and_o therefore_o no_o wonder_n to_o i_o why_o geometry_n have_v not_o proceed_v to_o the_o calculation_n neither_o of_o crooked_a line_n nor_o sufficient_o of_o motion_n nor_o of_o many_o other_o thing_n that_o have_v proportion_n to_o one_o another_o equal_a body_n superficies_n and_o line_n be_v those_o of_o which_o every_o one_o be_v capable_a of_o be_v co-incident_a with_o the_o place_n of_o every_o one_o of_o the_o rest_n and_o equal_a time_n wherein_o with_o one_o and_o the_o same_o motion_n equal_a line_n be_v describe_v and_o equal_o swift_a be_v those_o motion_n by_o which_o we_o run_v over_o equal_a space_n in_o any_o time_n determine_v by_o any_o other_o motion_n and_o universal_o all_o quantity_n be_v