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of_o all_o the_o country_n and_o of_o all_o size_n make_v according_a to_o the_o late_a discovery_n extant_a may_v be_v have_v past_v upon_o cloth_n and_o colour_a also_o sea_n plait_n mathematical_a projection_n book_n and_o instrument_n whatsoever_o be_v make_v and_o sell_v by_o philip_n lea._n finis_fw-la euclid_n element_n with_o the_o use_v euclid_n london_n print_v for_o philip_n lea_n globe_n maker_n at_o the_o atlas_n and_o hercules_n in_o cheapside_n near_o friday_n street_n there_o all_o sort_n of_o globe_n sphere_n map_n sea-plat_n mathematical_a book_n and_o instrument_n be_v make_v and_o sell_v plate_n 2._o proposition_n and_o use_v of_o the_o first_o book_n see_v plate_n 3_o plate_n 1._o definition_n of_o the_o first_o book_n proposition_n and_o use_v of_o the_o first_o book_n see_v plate_n 2._o plate_n 3._o proposition_n &_o use_v of_o the_o first_o book_n definition_n proposition_n &_o use_v of_o the_o second_o book_n plate_n 4_o definition_n of_o the_o three_o book_n proposition_n proposition_n and_o use_v plate_n 5._o definition_n propsition_n of_o the_o fouth_z book_n plate_n 6._o definition_n of_o the_o six_o book_n proposition_n &_o use_v plate_n 7._o definition_n of_o the_o eleven_o book_n proposition_n plate_n 8._o proposi_fw-la of_o the_o twelve_o book_n
the_o double_a area_n of_o the_o triangle_n i_o make_v use_v thereof_o in_o several_a other_o proposition_n as_o in_o the_o seven_o proposition_n fourteen_o problem_n to_o describe_v a_o square_n equal_a to_o a_o right_n line_v figure_n give_v to_o make_v a_o square_a equal_a to_o a_o right_o line_v figure_n a_o make_v by_o the_o 45_o of_o the_o 1_o a_o rectangle_n bc_n de_fw-fr equal_a to_o the_o right_o line_v figure_n a_o if_o its_o side_n cd_o dc_o be_v equal_a we_o shall_v have_v already_o our_o desire_n if_o they_o be_v unequal_a continue_v the_o line_n bc_n until_o cf_n be_v equal_a to_o cd_o and_o divide_v the_o line_n bf_n in_o the_o middle_n in_o the_o point_n g_o describe_v the_o semicircle_n fhb_n then_o continue_v dc_o to_o h_n the_o square_a of_o the_o line_n ch_z be_v equal_a to_o the_o right_o line_v figure_n a_o draw_v the_o line_n gh_n demonstration_n the_o line_n bf_n be_v equal_o divide_v in_o g_o and_o unequal_o in_o c_z thence_o by_o the_o 5_o the_o rectangle_n comprehend_v under_o bc_n cf_n or_o cd_o that_o be_v to_o say_v the_o rectangle_n bd_o with_o the_o square_a cg_n be_v equal_a to_o the_o square_n of_o gb_n or_o to_o its_o equal_a gh_o now_o by_o the_o 47_o of_o the_o 1_o the_o square_a of_o gh_n be_v equal_a to_o the_o square_n of_o ch_z cg_n therefore_o the_o rectangle_n bd_o and_o the_o square_a of_o cg_n be_v equal_a to_o the_o square_n of_o cg_n and_o of_o ch_z and_o take_v away_o the_o square_n of_o cg_n which_o be_v common_a to_o both_o the_o rectangle_n bd_o or_o the_o right_o line_v figure_n a_o be_v equal_a to_o the_o square_n of_o ch._n use_v this_o proposition_n serve_v in_o the_o first_o place_n to_o reduce_v into_o a_o square_a any_o right_o line_v figure_n whatever_o and_o whereas_o a_o square_a be_v the_o first_o measure_n of_o all_o superficies_n because_o its_o length_n and_o breadth_n be_v equal_a we_o measure_v by_o this_o mean_v all_o right_n line_v figure_n in_o the_o second_o place_n this_o proposition_n teach_v we_o to_o find_v a_o mean_a proportion_n between_o two_o give_v line_n as_o we_o shall_v see_v in_o the_o thirteen_o proposition_n of_o the_o six_o book_n this_o proposition_n may_v also_o serve_v to_o square_v curve_v line_v figure_n and_o even_a circle_n themselves_o for_o any_o crooked_a or_o curve_v line_v figure_n may_v to_o sense_n be_v reduce_v to_o a_o right_o line_v figure_n as_o if_o we_o inscribe_v in_o a_o circle_n a_o polygon_n have_v a_o thousand_o side_n it_o shall_v not_o be_v sensible_o different_a from_o a_o circle_n and_o reduce_v the_o polygone_a into_o a_o square_a we_o square_v near_o the_o circle_n the_o three_o book_n of_o euclid_n element_n this_o three_o book_n explain_v the_o propriety_n of_o a_o circle_n and_o compare_v the_o divers_a line_n which_o may_v be_v draw_v within_o and_o without_o its_o circumference_n it_o far_o consider_v the_o circumstance_n of_o circle_n which_o cut_v each_o other_o or_o which_o touch_n a_o straight_a line_n and_o the_o different_a angle_n which_o be_v make_v as_o well_o those_o in_o their_o centre_n as_o in_o their_o circumference_n in_o fine_a it_o give_v the_o first_o principle_n for_o establish_v the_o practice_n of_o geometry_n by_o the_o which_o we_o make_v use_v and_o that_o very_o commodious_o of_o a_o circle_n in_o almost_o all_o treatise_n in_o the_o mathematics_n definition_n 1._o book_n def._n of_o the_o 3_o book_n those_o circle_n be_v equal_a who_o diameter_n or_o semidiameter_n be_v equal_a 2._o 1._o fig._n 1._o a_o line_n touch_v a_o circle_n when_o meeting_n with_o the_o circumference_n thereof_o it_o cut_v not_o the_o same_o as_o the_o line_n ab_fw-la 3._o 2._o fig._n 2._o circle_n touch_v each_o other_o when_o meeting_n they_o cut_v not_o each_o other_o as_o the_o circle_n ab_fw-la and_o c._n 4._o 3._o fig._n 3._o right_o line_n in_o a_o circle_n be_v equal_o distant_a from_o the_o centre_n when_o perpendicular_o draw_v from_o the_o centre_n to_o those_o line_n be_v equal_a as_o if_o the_o line_n of_o eglantine_n be_v perpendicular_o to_o the_o line_n ab_fw-la cd_o be_v equal_a ab_fw-la cd_o shall_v be_v equal_o distant_a from_o the_o centre_n because_o the_o distance_n ought_v always_o to_o be_v take_v or_o measure_v by_o perpendicular_a line_n 5._o 4._o fig._n 4._o a_o segment_n of_o a_o circle_n be_v a_o figure_n terminate_v on_o the_o one_o side_n by_o a_o straight_a line_n and_o on_o the_o other_o by_o the_o circumference_n of_o a_o circle_n as_o lon_n lmn_n 6._o the_o angle_n of_o a_o segment_n be_v a_o angle_n which_o the_o circumference_n make_v with_o a_o straight_a line_n as_o the_o angle_n oln_n lmn_n 7._o 5._o fig._n 5._o a_o angle_n be_v say_v to_o be_v in_o a_o segment_n of_o a_o circle_n when_o the_o line_n which_o form_n the_o same_o be_v therein_o as_o the_o angle_n fgh_o be_v in_o the_o segment_n fgh_o 8._o 6._o fig_n 6._o a_o angle_n be_v upon_o that_o arch_n to_o which_o it_o be_v opposite_a or_o to_o which_o it_o serve_v for_o a_o base_a as_o the_o angle_n fgh_o be_v upon_o the_o arch_n fih_o which_o may_v be_v say_v to_o be_v its_o base_a 9_o 6._o fig._n 6._o a_o sector_n be_v a_o figure_n comprehend_v under_o two_o semidiameter_n and_o under_o the_o arch_n which_o serve_v to_o they_o for_o a_o base_a as_o the_o figure_n sigh_o proposition_n i._o problem_n to_o find_v the_o centre_n of_o a_o circle_n if_o you_o will_v find_v the_o centre_n of_o the_o circle_n aebd_a draw_v the_o line_n ab_fw-la and_o divide_v the_o same_o in_o the_o middle_n in_o the_o point_n c_o at_o which_o point_n erect_v a_o perpendicular_a ed_z which_o you_o shall_v divide_v also_o equal_o in_o the_o point_n f._n this_o point_n f_o shall_v be_v the_o centre_n of_o the_o circle_n for_o if_o it_o be_v not_o imagine_v if_o you_o please_v that_o the_o point_n g_z be_v the_o centre_n draw_v the_o line_n give_v gb_n gc_fw-la demonstration_n if_o the_o point_n g_z be_v the_o centre_n the_o triangle_n gac_n gbc_n will_v have_v the_o side_n give_v gb_n equal_a by_o the_o definition_n of_o a_o circle_n ac_fw-la cb_n be_v equal_a to_o the_o line_n ab_fw-la have_v be_v divide_v in_o the_o middle_n in_o the_o point_n c._n and_o cg_n be_v common_a the_o angles_n gcb_n gca_n will_v then_o be_v equal_a by_o the_o 8_o of_o the_o 1_o and_o cg_n will_v be_v then_o a_o perpendicular_a and_o not_o cd_o which_o will_v be_v contrary_a to_o the_o hypothesis_n therefore_o the_o centre_n can_v be_v out_o of_o the_o line_n cd_o i_o further_o add_v that_o it_o must_v be_v in_o the_o point_n f_o which_o divide_v the_o same_o into_o two_o equal_a part_n otherwise_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n will_v not_o be_v equal_a corollary_n the_o centre_n of_o a_o circle_n be_v in_o a_o line_n which_o divide_v another_o line_n in_o the_o middle_n and_o that_o perpendicular_o use_v this_o first_o proposition_n be_v necessary_a to_o demonstrate_v those_o which_o follow_v proposition_n ii_o theorem_fw-la a_o straight_a line_n draw_v from_o one_o point_n of_o the_o circumference_n of_o a_o circle_n to_o another_o shall_v fall_v within_o the_o same_o let_v there_o be_v draw_v from_o the_o point_n b_o in_o the_o circumference_n a_o line_n to_o the_o point_n c._n i_o say_v that_o it_o shall_v fall_v whole_o within_o the_o circle_n to_o prove_v that_o it_o can_v fall_v without_o the_o circle_n as_o buc_n have_v find_v the_o centre_n thereof_o which_o be_v a_o draw_v the_o line_n ab_fw-la ac_fw-la au._n demonstration_n the_o sides_n ab_fw-la ac_fw-la of_o the_o triangle_n abc_n be_v equal_a whence_o by_o the_o 5_o of_o the_o 1_o the_o angel_n abc_n acb_n be_v equal_a and_o see_v the_o angle_n auc_n be_v exterior_a in_o respect_n of_o the_o triangle_n aub_n it_o be_v great_a than_o abc_n by_o the_o 16_o of_o the_o 1_o it_o shall_v be_v also_o great_a than_o the_o angle_n acb_n thence_o by_o the_o 19_o of_o the_o 1_o in_o the_o triangle_n acv_n the_o side_n ac_fw-la opposite_a to_o the_o great_a angle_n auc_n be_v great_a than_o ave_fw-la and_o by_o consequence_n hence_v not_o reach_v the_o circumference_n of_o the_o circle_n see_v it_o be_v short_a than_o ac_fw-la which_o do_v but_o reach_v the_o same_o wherefore_o the_o point_n v_n be_v within_o the_o circle_n the_o same_o may_v be_v prove_v of_o any_o point_n in_o the_o line_n ab_fw-la and_o therefore_o the_o whole_a line_n ab_fw-la fall_v within_o the_o circle_n use_v it_o be_v on_o this_o proposition_n that_o be_v ground_v those_o which_o demonstrate_v that_o a_o circle_n touch_v a_o straight_a line_n but_o only_o in_o one_o point_n for_o if_o the_o line_n shall_v touch_v two_o point_n of_o its_o circumference_n it_o will_v be_v then_o draw_v from_o one_o
when_o the_o term_n between_o they_o be_v take_v twice_o that_o be_v to_o say_v as_o antecedent_n and_o as_o consequent_a as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o b_o to_z c_z and_z of_o c_z to_z d._n 11._o then_o a_o to_o c_o shall_v be_v in_o duplicate_v ratio_fw-la of_o a_o to_z b_o and_o the_o ratio_fw-la of_o a_o to_z d_o shall_v be_v in_o triplicate_v ratio_fw-la to_o that_o of_o a_o to_o b._n it_o be_v to_o be_v take_v notice_n that_o there_o be_v a_o great_a deal_n of_o difference_n between_o double_a ratio_fw-la and_o duplicate_v ratio_fw-la we_o say_v that_o the_o ratio_fw-la of_o four_o to_o two_o be_v double_a that_o be_v to_o say_v four_z be_v the_o double_a of_o two_o whence_o it_o follow_v that_o the_o number_n two_o be_v that_o which_o give_v the_o name_n to_o this_o ratio_fw-la or_o rather_o to_o the_o antecedent_n of_o this_o ratio_fw-la so_o we_o we_o say_v double_a triple_a quadruple_a quintuple_a which_o be_v denomination_n take_v from_o those_o number_n duo_fw-la tres_fw-la quatuor_fw-la quinque_fw-la compare_v with_o unity_n for_o we_o better_o conceive_v a_o reason_n when_o its_o term_n be_v small_a but_o as_o i_o have_v already_o take_v notice_n those_o denomination_n fall_v rather_o on_o the_o antecedent_n than_o on_o the_o reason_n itself_o we_o call_v that_o double_a triple_a reason_n or_o ratio_fw-la when_o the_o antecedent_n be_v double_a or_o triple_a to_o the_o consequent_a but_o when_o we_o say_v the_o reason_n be_v duplicate_v we_o mean_v a_o reason_n compound_v of_o two_o like_a reason_n as_o if_o there_o be_v the_o same_o reason_n of_o two_o to_o four_o as_o of_o four_o to_o eight_o the_o reason_n of_o two_o and_o eight_o be_v compound_v of_o the_o reason_n of_o two_o and_o four_o and_o of_o that_o of_o four_o and_o eight_o which_o be_v alike_o and_o as_o equal_v the_o reason_n or_o ratio_fw-la of_o two_o to_o eight_o shall_v be_v duplicate_v by_o each_o three_o to_o twenty_o seven_o be_v a_o duplicate_v reason_n of_o that_o of_o three_o to_o nine_o the_o reason_n of_o two_o to_o four_o be_v call_v subduple_n that_o be_v to_o say_v two_o be_v the_o half_a of_o four_o but_o the_o reason_n of_o two_o to_o eight_o be_v duple_n of_o the_o sub-duple_a that_o be_v to_o say_v that_o two_o be_v the_o half_a of_o the_o half_a of_o eight_o as_o three_o be_v the_o three_o of_o the_o three_o of_o twenty_o seven_o where_o you_o see_v there_o be_v take_v twice_o the_o denominator_fw-la ½_n and_o ⅓_n in_o like_a manner_n eight_o to_o two_o be_v a_o duplicate_v reason_n of_o eight_o to_o four_o because_o eight_o be_v double_a to_o four_o but_o eight_o be_v the_o double_a of_o the_o double_a of_o two_o if_o there_o be_v four_o term_n in_o the_o same_o continue_a reason_n that_o of_o the_o first_o and_o last_o be_v triple_a to_o that_o of_o the_o first_o and_o second_o as_o if_o one_o put_v these_o four_o number_n two_o four_o eight_o sixteen_o the_o reason_n of_o two_o to_o sixteen_o be_v triple_a of_o two_o to_o four_o for_o two_o be_v the_o half_a of_o the_o half_a of_o the_o half_a of_o sixteen_o as_o the_o reason_n of_o sixteen_o to_o two_o be_v triple_a of_o sixteen_o to_o eight_o for_o sixteen_o be_v the_o double_a of_o eight_o and_o it_o be_v the_o double_a of_o the_o double_a of_o the_o double_a of_o two_o 12._o magnitude_n be_v homologous_a the_o antecedent_n to_o the_o antecedent_n and_o the_o consequent_n to_o the_o consequent_n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o c_z to_z d_o a_o and_z c_o be_v homologous_a or_o magnitude_n of_o a_o like_o ratio_fw-la the_o follow_a definition_n be_v way_n of_o argue_v by_o proportion_n and_o it_o be_v principal_o to_o demonstrate_v the_o same_o that_o this_o book_n be_v compose_v 13._o alternate_a reason_n or_o by_o permutation_n or_o exchange_n be_v when_o we_o compare_v the_o antecedent_n one_o with_o the_o other_o as_o also_o the_o consequent_n for_o example_n if_o because_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v there_o be_v the_o same_o reason_n of_o a_o to_o c_o as_o of_o b_o to_z d_o this_o way_n of_o reason_v can_v take_v place_n but_o when_o the_o four_o term_n be_v of_o the_o same_o specie_fw-la that_o be_v to_o say_v either_o all_o four_o line_n or_o superficies_n or_o solid_n proposition_n 16._o 14._o converse_v or_o inverse_v reason_n be_v a_o comparison_n of_o the_o consequent_n to_o the_o antecedent_n as_o if_o because_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v there_o be_v the_o same_o reason_n of_o b_o to_o a_o as_o there_o be_v of_o d_o to_z c._n proposition_n 16._o 15._o composition_n of_o reason_n be_v a_o comparison_n of_o the_o antecedent_n and_o consequent_a take_v together_o to_o the_o consequent_a alone_o as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o i_o conclude_v also_o that_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d._n prop._n 18._o 16._o division_n of_o reason_n be_v a_o comparison_n of_o the_o excess_n of_o the_o antecedent_n above_o the_o consequent_a to_o the_o same_o consequent_a as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_z b_o as_o of_o cd_o to_o d_o i_o conclude_v that_o there_o be_v the_o same_o reason_n of_o a_o to_o b_o as_o of_o cd_o prop._n 17._o 17._o conversion_n of_o reason_n be_v the_o comparison_n of_o the_o antecedent_n to_o the_o difference_n of_o the_o term_n as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_z b_o as_o of_o cd_o to_o d_o i_o conclude_v that_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o a_o as_o of_o cd_o to_o c._n proposition_n 18._o 18._o proportion_n of_o equality_n be_v a_o comparison_n of_o the_o extreme_a quantity_n in_o leave_v out_o those_o in_o the_o middle_n a_o b_o c_o d_o e_o f_o g_o h._n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o e_z to_o f_o and_o of_o b_o to_z c_z as_o of_o f_o to_o g_z and_z of_o c_z to_z d_o as_o of_o g_z to_z h._n i_o draw_v this_o consequence_n that_o there_o be_v therefore_o the_o same_o reason_n of_o a_o to_z d_o as_o of_o e_z to_z h._n 19_o proportion_n of_o equality_n well_o rank_v be_v that_o in_o which_o one_o compare_v the_o term_n in_o the_o same_o manner_n of_o order_n as_o in_o the_o precede_a example_n prop._n 22._o 20._o proportion_n of_o equality_n ill_o rank_v be_v that_o in_o which_o one_o compare_v the_o term_n with_o a_o different_a order_n as_o if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o g_z to_o h_n and_z of_o b_o to_z c_z as_o of_o f_o to_o g_z and_z of_o c_z to_z d_o as_o of_o e_z to_z f._n i_o draw_v this_o conclusion_n that_o there_o be_v the_o same_o reason_n of_o a_o to_z d_o as_o of_o e_z to_z h._n prop._n 28._o here_o be_v all_o the_o way_n of_o argue_v by_o proportion_n there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o therefore_o by_o alternate_a reason_n there_o be_v the_o same_o reason_n of_o a_o to_o c_o as_o of_o b_o to_z d_o and_o by_o inversed_a reason_n there_o be_v the_o same_o reason_n of_o b_o to_o a_o as_o of_o d_o to_z c_z and_o by_o composition_n there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d._n by_o division_n of_o reason_n if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o b_o as_o of_o cd_o to_o d_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o and_o by_o conversion_n there_o be_v the_o same_o reason_n of_o ab_fw-la to_o a_o as_o of_o cd_o to_o c._n by_o reason_n of_o equality_n well_o rank_v if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o c_z to_z d_o and_o also_o the_o same_o reason_n of_o b_o to_z e_z as_o of_o d_o to_o f_o there_o will_v be_v the_o same_o reason_n of_o a_o to_z e_z as_o of_o c_z to_z f._n by_o reason_n of_o equality_n ill_o rank_v if_o there_o be_v the_o same_o reason_n of_o a_o to_z b_o as_o of_o d_o to_o f_o and_o also_o the_o same_o reason_n of_o b_o to_z e_z as_o of_o c_z to_z d_o there_o will_v be_v the_o same_o reason_n of_o a_o to_z e_z as_o of_o c_z to_z f._n this_o book_n contain_v twenty_o five_o proposition_n of_o euclid_n to_o which_o there_o have_v be_v add_v ten_o which_o be_v receive_v the_o first_o six_o of_o this_o book_n be_v useful_a only_o to_o prove_v the_o follow_a proposition_n by_o the_o method_n
whereon_o the_o head_n thereof_o you_o shall_v find_v the_o book_n it_o belong_v to_o and_o the_o proposition_n or_o use_n continue_v in_o their_o order_n eight_o book_n of_o euclid_n element_n with_o the_o use_v of_o each_o proposition_n the_o first_o book_n euclid_n design_n in_o this_o book_n be_v to_o give_v the_o first_o principle_n of_o geometry_n and_o to_o do_v the_o same_o methodical_o he_o begin_v with_o the_o definition_n and_o explication_n of_o the_o most_o ordinary_a term_n than_o he_o exhibit_v certain_a supposition_n and_o have_v propose_v those_o maxim_n which_o natural_a reason_n teach_v he_o pretend_v to_o put_v forward_o nothing_o without_o demonstration_n and_o to_o convince_v any_o one_o which_o will_v consent_v to_o nothing_o but_o what_o he_o shall_v be_v oblige_v to_o acknowledge_v in_o his_o first_o proposition_n he_o treat_v of_o line_n and_o of_o the_o several_a angle_n make_v by_o their_o intersect_v each_o other_o and_o have_v occasion_n to_o demonstrate_v their_o propriety_n and_o compare_v certain_a triangle_n he_o do_v the_o same_o in_o the_o first_o eight_o proposition_n then_o teach_v the_o practical_a way_n of_o divide_v a_o angle_n and_o a_o line_n into_o two_o equal_a part_n and_o to_o draw_v a_o perpendicular_a he_o pursue_v to_o the_o propriety_n of_o a_o triangle_n and_o have_v show_v those_o of_o parallel_n line_n he_o make_v a_o end_n of_o the_o explication_n of_o this_o first_o figure_n and_o pass_v forward_o to_o parallellogram_n give_v a_o way_n to_o reduce_v all_o sort_n of_o polygon_n into_o a_o more_o regular_a figure_n he_o end_v this_o book_n with_o that_o celebrate_a proposition_n of_o pythagoras_n and_o demonstrate_v that_o in_o a_o rectangular_a triangle_n the_o square_a of_o the_o base_a be_v equal_a to_o the_o sum_n of_o the_o square_n of_o the_o side_n include_v the_o right_a angle_n definition_n 1._o a_o point_n be_v that_o which_o have_v no_o part_n this_o definition_n be_v to_o be_v understand_v in_o this_o sense_n the_o quantity_n which_o we_o conceive_v without_o distinguish_v its_o part_n or_o without_o think_v that_o it_o have_v any_o be_v a_o mathematical_a point_n far_o differ_v from_o those_o of_o zeno_n which_o be_v altogether_o indivisible_a since_o one_o may_v doubt_v with_o a_o great_a deal_n of_o reason_n if_o those_o last_o be_v possible_a which_o yet_o we_o can_v of_o the_o first_o if_o we_o conceive_v they_o as_o we_o ought_v 2._o a_o line_n be_v a_o length_n without_o breadth_n the_o sense_n of_o this_o definition_n be_v the_o same_o with_o that_o of_o the_o forego_n the_o quantity_n which_o we_o consider_v have_v length_n without_o make_v any_o reflection_n on_o its_o breadth_n be_v that_o we_o understand_v by_o the_o word_n line_n although_o one_o can_v draw_v a_o real_a line_n which_o have_v not_o a_o determinate_a breadth_n it_o be_v general_o say_v that_o a_o line_n be_v produce_v by_o the_o motion_n of_o a_o point_n which_o we_o ought_v well_o to_o take_v notice_n of_o see_v that_o by_o a_o motion_n after_o that_o manner_n may_v be_v produce_v all_o sort_n of_o quantity_n imagine_v then_o that_o a_o point_n move_v and_o that_o it_o leave_v a_o trace_n in_o the_o middle_n of_o the_o way_n which_o it_o pass_v the_o trace_v be_v a_o line_n 3._o the_o two_o end_n of_o a_o line_n be_v point_n 4._o a_o straight_a line_n be_v that_o who_o point_n be_v place_v exact_o in_o the_o midst_n or_o if_o you_o will_v rather_o have_v it_o a_o straight_a line_n be_v the_o short_a of_o all_o the_o line_n which_o may_v be_v draw_v from_o one_o point_n to_o another_o 5._o a_o superficies_n be_v a_o quantity_n to_o which_o be_v give_v length_n and_o breadth_n without_o consider_v the_o thickness_n 6._o the_o extremity_n of_o a_o superficies_n be_v line_n 7._o a_o plain_a or_o straight_o superficies_n be_v that_o who_o line_n be_v place_v equal_o between_o the_o extremity_n or_o that_o to_o which_o a_o straight_a line_n may_v be_v apply_v any_o manner_n of_o way_n 1._o plate_n i._n fig._n 1._o i_o have_v already_o take_v notice_n that_o motion_n be_v capable_a of_o produce_v all_o sort_n of_o quantity_n whence_o we_o say_v that_o when_o a_o line_n pass_v over_o another_o it_o produce_v a_o superficies_n or_o a_o plain_n and_o that_o that_o motion_n have_v a_o likeness_n to_o arithmetical_a multiplication_n imagine_v that_o the_o line_n ab_fw-la move_v along_o the_o line_n bc_n keep_v the_o same_o situation_n without_o incline_v one_o way_n or_o the_o other_o the_o point_n a_o shall_v describe_v the_o line_n ad_fw-la the_o point_n b_o the_o line_n bc_n and_o the_o other_o point_n between_o other_o parallel_n line_n which_o shall_v compose_v the_o superficies_n abcd._n i_o add_v that_o this_o motion_n correspond_v with_o arithmetical_a multiplication_n for_o if_o i_o know_v the_o number_n of_o point_n which_o be_v in_o the_o line_n ab_fw-la bc_n multiply_v of_o they_o one_o by_o the_o other_o i_o shall_v have_v the_o number_n of_o point_n which_o compose_v the_o superficies_n abcd_v as_o if_o ab_fw-la contain_v four_o point_n and_o bc_n six_o say_v four_o time_n six_o be_v twenty_o four_o the_o superficies_n ab_fw-la cd_o shall_v be_v compose_v of_o twenty_o four_o point_n now_o i_o may_v take_v for_o a_o mathematical_a point_n any_o quantity_n whatsoever_o for_o example_n a_o foot_n provide_v i_o do_v not_o subdivide_v the_o same_o into_o part_n 8._o a_o plain_a angle_n be_v the_o open_n of_o two_o line_n which_o intersect_v each_o other_o and_o which_o compose_v not_o one_o single_a line_n 2._o fig._n 2._o as_o the_o opening_z d_o of_o the_o line_n ab_fw-la cb_n which_o be_v not_o part_n of_o the_o same_o line_n a_o right_a line_a angle_n be_v the_o open_n of_o two_o straight_a line_n it_o be_v principal_o of_o this_o sort_n of_o angle_n which_o i_o intend_v to_o treat_v of_o at_o present_a because_o experience_n do_v make_v i_o perceive_v that_o the_o most_o part_n of_o those_o who_o begin_v do_v mistake_v the_o measure_v the_o quantity_n of_o a_o angle_n by_o the_o length_n of_o the_o line_n which_o compose_v the_o same_o 4._o fig._n 3_o 4._o the_o most_o open_a angle_n be_v the_o great_a that_o be_v to_o say_v when_o the_o line_n include_v a_o angle_n be_v far_o asunder_o than_o those_o of_o another_o angle_n take_v they_o at_o the_o same_o distance_n from_o the_o point_n of_o intersection_n of_o their_o line_n the_o first_o be_v great_a than_o the_o second_o so_o the_o angle_n a_o be_v great_a than_o e_z because_o if_o we_o take_v the_o point_n b_o and_o d_o as_o far_o distant_a from_o the_o point_n a_o as_o the_o point_n g_o and_o l_o be_v from_o the_o point_n e_o the_o point_n b_o and_o d_o be_v far_o asunder_o than_o the_o point_n g_o and_o l_o from_o whence_o i_o conclude_v that_o if_o eglantine_n el_fw-es be_v continue_v the_o angle_n e_o will_v be_v of_o the_o same_o measure_n and_o less_o than_o the_o angle_n a._n we_o make_v use_v of_o three_o letter_n to_o express_v a_o angle_n and_o the_o second_o letter_n denote_v the_o angular_a point_n as_o the_o angle_n bad_a be_v the_o angle_n which_o the_o line_n ba_o ad_fw-la do_v form_n at_o the_o point_n a_o the_o angle_n bac_n be_v that_o which_o be_v form_v by_o the_o line_n ba_o ac_fw-la the_o angle_n god_n be_v comprehend_v under_o the_o line_n ca_n ad._n 3._o fig._n 3._o the_o arch_n of_o a_o circle_n be_v the_o measure_n of_o a_o angle_n thus_o design_v to_o measure_v the_o quantity_n of_o the_o angle_n bad_a i_o put_v one_o foot_n of_o the_o compass_n on_o the_o point_v a_o and_o with_o the_o other_o i_o describe_v a_o arch_n of_o a_o circle_n bcd_v the_o angle_n shall_v be_v the_o great_a by_o how_o much_o the_o arch_n bcd_v which_o be_v the_o measure_n thereof_o shall_v contain_v a_o great_a portion_n of_o a_o circle_n and_o because_o that_o common_o a_o arch_n of_o a_o circle_n be_v divide_v into_o three_o hundred_o and_o sixty_o equal_a part_n call_v degree_n it_o be_v say_v that_o a_o angle_n contain_v twenty_o thirty_o forty_o degree_n when_o the_o arch_n include_v betwixt_o its_o line_n contain_v twenty_o thirty_o forty_o degree_n so_o the_o angle_n be_v great_a which_o contain_v the_o great_a number_n of_o degree_n as_o the_o angle_n bad_a be_v great_a than_o gel_n the_o line_n ca_n divide_v the_o angle_n bad_a in_o the_o middle_n because_o the_o arch_n bc_n cd_o be_v equal_a and_o the_o angle_n bac_n be_v a_o part_n of_o bad_a because_o the_o arch_a bc_n be_v part_n of_o the_o arch_n bd._n 10._o when_o a_o line_n fall_v on_o another_o line_n make_v the_o angle_n on_o each_o side_n thereof_o equal_a those_o angles_n be_v right_a angle_n and_o the_o line_n so_o fall_v be_v a_o perpendicular_a 5._o fig._n 5._o as_o if_o the_o line_n ab_fw-la fall_v on_o cd_o
side_n 13._o the_o straight_a line_n have_v not_o the_o same_o common_a segment_n 20._o fig._n 20._o i_o will_v say_v that_o of_o two_o straight_a line_n ab_fw-la cd_o which_o meet_v each_o other_o in_o the_o point_n b_o be_v not_o make_v one_o single_a line_n bd_o but_o that_o they_o cut_v each_o other_o and_o separate_v after_o their_o so_o meet_v for_o if_o a_o circle_n be_v describe_v on_o the_o centre_n b_o afb_n shall_v be_v a_o semicircle_n see_v the_o line_n abdella_n pass_v through_o the_o centre_n b_o divide_v the_o circle_n into_o two_o equal_o the_o segment_n cfd_n shall_v be_v also_o a_o semicircle_n if_o cbd_v be_v a_o straight_a line_n because_o it_o pass_v through_o the_o centre_n b_o therefore_o the_o segment_n cfd_n shall_v be_v equal_a to_o the_o segment_n afd_v a_o part_n as_o great_a as_o the_o whole_a which_o will_v be_v contrary_a to_o the_o nine_o axiom_n advertisement_n we_o have_v two_o sort_n of_o proposition_n some_o whereof_o consider_v only_o a_o truth_n without_o descend_v to_o the_o practice_n thereof_o and_o we_o call_v those_o theorm_n the_o other_o propose_v something_o to_o be_v do_v or_o make_v and_o be_v call_v problem_n the_o first_o number_n of_o citation_n be_v that_o of_o the_o proposition_n the_o second_o that_o of_o the_o book_n as_o by_o the_o 2_o of_o the_o 3_o be_v signify_v the_o second_o proposition_n of_o the_o three_o book_n but_o if_o one_o meet_v with_o one_o number_n thereby_o be_v mean_v the_o proposition_n of_o the_o book_n who_o explication_n be_v in_o hand_n proposition_n i._o problem_n upon_o a_o finite_a right_a line_n ab_fw-la to_o describe_v a_o equilateral_a triangle_n acb_n from_o the_o centre_n a_o and_o b_o at_o the_o distance_n of_o ab_fw-la describe_v the_o circle_n cut_v each_o other_o in_o the_o point_n c_o from_o whence_o draw_v two_o right_a line_n ca_n cb_n thence_o be_v ac_fw-la ab_fw-la bc_n ac_fw-la equal_a wherefore_o the_o triangle_n acb_n be_v equilateral_a which_o be_v to_o be_v do_v use_v euclid_n have_v not_o apply_v this_o proposition_n to_o any_o other_o use_n but_o to_o demonstrate_v the_o two_o follow_a proposition_n but_o we_o may_v apply_v it_o to_o the_o measure_n of_o a_o inaccessible_a line_n 1._o use_v 1._o as_o for_o example_n let_v ab_fw-la be_v a_o inaccessible_a line_n which_o be_v so_o by_o reason_n of_o a_o river_n or_o some_o other_o impediment_n make_v a_o equaliteral_a triangle_n as_o bde_v on_o wood_n or_o brass_n or_o on_o some_o other_o convenient_a thing_n which_o have_v place_v horizontal_o at_o a_o station_n at_o b_o look_v to_o the_o point_n a_o along_o the_o side_n bd_o and_o to_o some_o other_o point_n c_o along_o the_o side_n be_v then_o carry_v your_o triangle_n along_o the_o line_n bc_n so_o far_o that_o be_v until_o such_o time_n as_o you_o can_v see_v the_o point_n b_o your_o first_o station_n by_o the_o side_n cg_n and_o the_o point_v a_o by_o the_o side_n ce_fw-fr i_o say_v that_o then_o the_o line_n cb_n and_o ca_n be_v equal_a wherefore_o if_o you_o measure_v the_o line_n bc_n you_o will_v likewise_o know_v the_o length_n of_o the_o line_n ab_fw-la proposition_n ii_o problem_n at_o a_o point_n give_v a_o to_o make_v a_o right_a line_n ac_fw-la equal_a to_o a_o right_a line_n give_v bc._n from_o the_o centre_n c_o at_o the_o distance_n cb_n describe_v the_o circle_n cbe_n join_v ac_fw-la upon_o which_o raise_v the_o equilateral_a triangle_n adc_n produce_v dc_o to_o e_o from_o the_o centre_n d_o and_o the_o distance_n de_fw-fr describe_v the_o circle_n deh_n let_v da_fw-la be_v produce_v to_o the_o point_n g_o in_o the_o circumference_n thereof_o then_o agnostus_n be_v equal_a to_o cb_n for_o dg_n be_v equal_a to_o de_fw-fr and_o da_fw-la to_o dc_o wherefore_o agnostus_n ce_fw-fr bc_n agnostus_n be_v equal_a which_o be_v to_o be_v do_v schol_n the_o line_n agnostus_n may_v be_v take_v with_o a_o pair_n of_o compass_n but_o the_o so_o do_v answer_v not_o postulate_v as_o proclus_n well_o intimate_v proposition_n iii_o problem_n two_o right_a line_n a_o and_o bc_n be_v give_v from_o the_o great_a bc_n to_o take_v away_o the_o right_a line_n be_v equal_a to_o the_o lesser_a a._n at_o the_o point_n b_o draw_v the_o right_a line_n bd_o equal_a to_o a_o the_o circle_n describe_v from_o the_o centre_n b_o at_o the_o distance_n bd_o shall_v cut_v off_o be_v equal_a to_o bd_o equal_a to_o a_o equal_a to_o be_v which_o be_v to_o be_v do_v the_o use_n of_o the_o two_o precede_a proposition_n be_v very_o evident_a since_o we_o be_v oblige_v very_o often_o in_o our_o geometrical_a practice_n to_o draw_v a_o line_n equal_a to_o a_o line_n give_v or_o to_o take_v away_o from_o a_o great_a line_n give_v a_o part_n equal_a to_o a_o lesser_a proposition_n iu_o theorem_fw-la if_o two_o triangle_n abc_n edf_n have_v two_o side_n of_o the_o one_o basilius_n ac_fw-la equal_a to_o two_o side_n of_o the_o other_o ed_z df_n each_o to_o his_o correspondent_a side_n that_o be_v ba_z to_z ed_z and_o ac_fw-la to_o df_n and_o have_v the_o angle_n a_o equal_a to_o the_o angle_n d_o contain_v under_o the_o equal_a right_a line_n they_o shall_v have_v the_o base_a bc_n equal_a to_o the_o base_a of_o and_o the_o triangle_n bac_n shall_v be_v equal_a to_o the_o triangle_n edf_n and_o the_o remain_a angle_n b_o c_o shall_v be_v equal_a to_o the_o remain_a angle_n e_z f_o each_o to_o each_o which_o be_v subtend_v by_o the_o equal_a side_n if_o the_o point_n d_o be_v apply_v to_o the_o point_n a_o and_o the_o right_a line_n de_fw-fr place_v on_o the_o right_a line_n ab_fw-la the_o point_n e_o shall_v fall_v upon_o b_o because_o de_fw-fr be_v equal_a to_o ab_fw-la also_o the_o right_a line_n df_n shall_v fall_v upon_o ac_fw-la because_o the_o angle_n a_o be_v equal_a to_o d_o moreover_o the_o point_n f_o shall_v fall_v on_o the_o point_n c_o because_o ac_fw-la be_v equal_a to_o df_n therefore_o the_o right_a line_n of_o bc_n shall_v agree_v because_o they_o have_v the_o same_o term_n and_o so_o consequent_o be_v equal_a wherefore_o the_o triangle_n bac_n be_v equal_a to_o def_n and_o the_o angle_n b_o e_o as_o also_o the_o angle_n c_o f_o do_v agree_v and_o be_v equal_a which_o be_v to_o be_v demonstrate_v use_v 4._o use_v 4._o svppose_v i_o be_v to_o measure_v the_o inaccessible_a line_n ab_fw-la i_o look_v from_o the_o point_n c_o to_o the_o point_n a_o and_o b_o than_o i_o measure_v the_o angle_n c_o thus_o i_o place_v a_o board_n or_o table_n horizontal_o and_o look_v successive_o with_o a_o ruler_n towards_o the_o point_n a_o and_o b_o i_o draw_v two_o line_n make_v the_o angle_n acb_n than_o i_o measure_v the_o line_n ac_fw-la and_o bc_n which_o i_o suppose_v to_o be_v accessible_a i_o turn_v about_o my_o board_n or_o table_n towards_o some_o other_o place_n in_o the_o field_n place_v it_o again_o horizontal_o at_o the_o point_n f_o and_o look_v along_o those_o line_n i_o have_v draw_v on_o my_o table_n i_o make_v the_o angle_n def_n equal_a to_o the_o angle_n c_o ay_o make_v also_o fd_v fe_o equal_a to_o ca_n cb_n now_o according_a to_o this_o proposition_n the_o line_n ab_fw-la de_fw-fr be_v equal_a wherefore_o whatsoever_o the_o length_n of_o the_o line_n de_fw-fr be_v find_v to_o be_v the_o same_o be_v the_o measure_n of_o the_o inaccessible_a line_n ab_fw-la another_o use_n may_v be_v this_o 4._o use_v 4._o svppose_v you_o be_v at_o a_o billiard_n table_n and_o you_o will_v strike_v a_o ball_n b_o by_o reflection_n with_o another_o ball_n a_o admit_v cd_o be_v one_o side_n of_o the_o table_n now_o imagine_v a_o perpendicular_a line_n bde_v i_o take_v the_o line_n de_fw-fr equal_a to_o db_v i_o say_v that_o if_o you_o aim_v and_o strike_v your_o ball_n a_o direct_o towards_o the_o point_n e_o the_o ball_n a_o meeting_n the_o side_n of_o the_o table_n at_o f_o shall_v reflect_v from_o thence_o to_o b_o for_o in_o the_o triangle_n bfd_n efd_n the_o side_n fd_n be_v common_a and_o the_o side_n bd_o de_fw-fr equal_a the_o angles_n bfd_v efd_v equal_a by_o the_o proposition_n the_o angel_n afc_n dfe_n be_v opposite_a be_v also_o equal_a as_o i_o shall_v demonstrate_v hereafter_o therefore_o the_o angle_n of_o incidence_n afc_n be_v equal_a to_o the_o angle_n of_o reflection_n bfd_n and_o by_o consequence_n the_o reflection_n will_v be_v from_o of_o to_o fb_v proposition_n v._o theorem_fw-la in_o every_o isosceles_a triangle_n the_o angle_n which_o be_v above_o the_o base_a be_v equal_a as_o also_o those_o which_o be_v underneath_o let_v abc_n be_v a_o isosceles_a triangle_n viz_o that_o the_o side_n ab_fw-la ac_fw-la be_v equal_a i_o say_v that_o the_o angle_n abc_n acb_n be_v equal_a as_o also_o the_o angel_n gbc_n hcb_n which_o lie_v under_o
under_o ab_fw-la and_o ac_fw-la shall_v be_v three_o time_n 8_o or_o 24_o the_o square_a of_o ac_fw-la 3_o be_v 9_o the_o rectangle_n comprehend_v under_o ac_fw-la 3_o and_o cb_n 5_o be_v 3_o time_n 5_o or_o 15._o it_o be_v evident_a that_o 15_o and_o 9_o be_v 24._o use_v at_fw-fr  _fw-fr 43_o c_z 40._o 3_o b_o  _fw-fr 3_o 120._o  _fw-fr 9_o 129_o  _fw-fr  _fw-fr this_o proposition_n serve_v likewise_o to_o demonstrate_v the_o ordinary_a practice_n of_o multiplication_n for_o example_n if_o one_o will_v multiply_v the_o number_n 43_o by_o 3_o have_v separate_v the_o number_n of_o 43_o into_o two_o part_n in_o 40_o and_o 3_o three_o time_n 43_o shall_v be_v as_o much_o as_o three_o time_n 3_o which_o be_v nine_o the_o square_a of_o three_o and_o three_o time_n forty_o which_o be_v 120_o for_o 129_o be_v three_o time_n 43._o those_o which_o be_v young_a beginner_n ought_v not_o to_o be_v discourage_v if_o they_o do_v not_o conceive_v immediate_o these_o proposition_n for_o they_o be_v not_o difficult_a but_o because_o they_o do_v imagine_v they_o contain_v some_o great_a mystery_n proposition_n iu_o theorem_fw-la if_o a_o line_n be_v divide_v into_o two_o part_n the_o square_a of_o the_o whole_a line_n shall_v be_v equal_a to_o the_o two_o square_n make_v of_o its_o part_n and_o to_o two_o rectangle_v comprehend_v under_o the_o same_o part_n let_v the_o line_n ab_fw-la be_v divide_v in_o c_o and_o let_v the_o square_a thereof_o abde_n be_v make_v let_v the_o diagonal_a ebb_n be_v draw_v and_o the_o perpendicular_a cf_n cut_v the_o same_o and_o through_o that_o point_n let_v there_o be_v draw_v gl_n parallel_n to_o ab_fw-la it_o be_v evident_a that_o the_o square_a abde_n be_v equal_a to_o the_o four_o rectangle_v gf_n cl_n cg_n lf_n the_o two_o first_o be_v the_o square_a of_o ac_fw-la and_o of_o cb_n the_o two_o compliment_n be_v comprehend_v under_o ac_fw-la cb._n demonstration_n the_o side_n ae_n ab_fw-la be_v equal_a thence_o the_o angle_n aeb_fw-mi abe_n be_v half_a right_n and_o because_o of_o the_o parallel_n gl_n ab_fw-la the_o angle_n of_o the_o triangle_n of_o the_o square_a ge_z by_o the_o 29_o shall_v be_v equal_a as_o also_o the_o side_n by_o the_o 6_o of_o the_o 1._o thence_o gf_n be_v the_o square_a of_o ac_fw-la in_o like_a manner_n the_o rectangle_n cl_n be_v the_o square_a of_o cb_n the_o rectangle_n gc_n be_v comprehend_v under_o ac_fw-la and_o agnostus_n equal_a to_o bl_v or_o bc_n the_o rectangle_n lf_o be_v comprehend_v under_o ld_n equal_a to_o ac_fw-la and_o under_o fd_n equal_a to_o bc._n coral_n if_o a_o diagonal_a be_v draw_v in_o a_o square_a the_o rectangle_v through_o which_o it_o pass_v be_v square_n use_v a_o 144_o b_o 22_o c_o 12_o this_o proposition_n give_v we_o the_o practical_a way_n of_o find_v or_o extract_v the_o square_a root_n of_o a_o number_n propound_v let_v the_o same_o be_v the_o number_n a_o 144_o represent_v by_o the_o square_a ad_fw-la and_o its_o root_n by_o the_o line_n ab_fw-la moreover_o i_o know_v that_o the_o line_n require_v ab_fw-la must_v have_v two_o figure_n i_o therefore_o imagine_v that_o the_o line_n ab_fw-la be_v divide_v in_o c_o and_o that_o ac_fw-la represent_v the_o first_o figure_n and_o bc_n the_o second_o i_o seek_v the_o root_n of_o the_o first_o figure_n of_o the_o number_n 144_o which_o be_v 100_o and_o i_o find_v that_o it_o be_v 10_o and_o make_v its_o square_a 100_o represent_v by_o the_o square_a gf_n i_o subtract_v the_o same_o from_o 144_o and_o there_o remain_v 44_o for_o the_o rectangle_v gc_a fl_fw-mi and_o the_o square_a cl._n but_o because_o this_o gnomonicall_a figure_n be_v not_o proper_a i_o transport_v the_o rectangle_n fl_fw-mi in_o kg_v and_o so_o i_o have_v the_o rectangle_n kl_n contain_v 44._o i_o know_v also_o almost_o all_o the_o length_n of_o the_o side_n kb_n for_o ac_fw-la be_v 10_o therefore_o kc_n be_v 20_o i_o must_v then_o divide_v 44_o by_o 20_o that_o be_v to_o say_v to_o find_v the_o divisor_n i_o double_v the_o root_n find_v and_o i_o say_v how_o many_o time_n 20_o in_o 44_o i_o find_v it_o 2_o time_n for_o the_o side_n bl_n but_o because_o 20_o be_v not_o the_o whole_a side_n kb_n but_o only_a kc_n this_o 2_o which_o come_v in_o the_o quotient_n be_v to_o be_v add_v to_o the_o divisor_n which_o then_o will_v be_v 22._o so_o i_o find_v the_o same_o 2_o time_n precise_o in_o 44_o the_o square_a root_n then_o shall_v be_v 12._o you_o see_v that_o the_o square_a of_o 144_o be_v equal_a to_o the_o square_n of_o 10_o to_o the_o square_n of_o 2_o which_o be_v 4_o and_o to_o twice_o 20_o which_o be_v two_o rectangle_v comprehend_v under_o 2_o and_o under_o 10._o proposition_n v._o theorem_fw-la if_o a_o right_a line_n be_v cut_v into_o equal_a part_n and_o into_o unequal_a part_n the_o rectangle_n comprehend_v under_o the_o unequal_a part_n together_o with_o the_o square_n which_o be_v of_o the_o middle_a part_n or_o difference_n of_o the_o part_n be_v equal_a to_o the_o square_n of_o half_a the_o line_n if_o the_o line_n ab_fw-la be_v divide_v equal_o in_o c_z and_o unequal_o in_o d_o the_o rectangle_n ah_o comprehend_v under_o the_o segment_n ad_fw-la db_fw-la together_o with_o the_o square_n of_o cd_o shall_v be_v equal_a to_o the_o square_a cf_n that_o be_v of_o half_a of_o ab_fw-la viz._n cb._n make_v a_o end_n of_o the_o figure_n as_o you_o see_v it_o the_o rectangle_v lg_n diego_n shall_v be_v square_n by_o the_o coral_n of_o the_o 4_o i_o prove_v that_o the_o rectangle_n ah_o comprehend_v under_o ad_fw-la and_o dh_n equal_a to_o db_v with_o the_o square_a lg_n be_v equal_a to_o the_o square_a cf._n demonstration_n the_o rectangle_n all_o be_v equal_a to_o the_o rectangle_n df_n the_o one_o and_o the_o other_o be_v comprehend_v under_o half_a the_o line_n ab_fw-la and_o under_o bd_o or_o dh_n equal_a thereto_o add_v to_o both_o the_o rectangle_n ch_z the_o rectangle_n ah_o shall_v be_v equal_a to_o the_o gnomon_n lbg_n again_o to_o both_o add_v the_o square_a lg_n the_o rectangle_n ah_o with_o the_o square_a lg_n shall_v be_v equal_a to_o the_o square_a cf._n arithmetical_o let_v ab_fw-la be_v 10_o ac_fw-la be_v 5_o as_o also_o cb._n let_v cd_o be_v 2_o and_o db_n 3_o the_o rectangle_n comprehend_v under_o ad_fw-la 7_o and_o db_n 3_o that_o be_v to_o say_v 21_o with_o the_o square_n of_o cd_o 2_o which_o be_v 4_o shall_v be_v equal_a to_o the_o square_n of_o cb_n 5_o which_o be_v 25._o use_v this_o proposition_n be_v very_o useful_a in_o the_o three_o book_n we_o make_v use_v thereof_o in_o algebra_n to_o demonstrate_v the_o way_n of_o find_v the_o root_n of_o a_o affect_a square_a or_o equation_n proposition_n vi_o theorem_fw-la if_o one_o add_v a_o line_n to_o another_o which_o be_v divide_v into_o two_o equal_a part_n the_o rectangle_n comprehend_v under_o the_o line_n compound_v of_o both_o and_o under_o the_o line_n add_v together_o with_o the_o square_n of_o half_a the_o divide_a line_n be_v equal_a to_o the_o square_n of_o a_o line_n compound_v of_o half_a the_o divide_a line_n and_o the_o line_n add_v if_o one_o add_v the_o line_n bd_o to_o the_o line_n ab_fw-la which_o be_v equal_o divide_v in_o c_o the_o rectangle_n a_fw-la comprehend_v under_o ad_fw-la and_o under_o dn_n or_o db_n with_o the_o square_n of_o cb_n be_v equal_a to_o the_o square_n of_o cd_o make_v the_o square_n of_o cd_o and_o have_v draw_v the_o diagonal_a fd_n draw_v bg_n parallel_v to_o fc_n which_o cut_v fd_v in_o the_o point_n h_n through_o which_o pass_v hn_n parallel_v to_o ab_fw-la kg_n shall_v be_v the_o square_n of_o bc_n and_o bn_n that_o of_o bd._n demonstration_n the_o rectangle_v ak_v ch_z on_o equal_a base_n ac_fw-la bc_n be_v equal_a by_o the_o 38_o of_o the_o 1_o the_o compliment_n ch_z he_o be_v equal_a by_o the_o 43_o of_o the_o 1_o therefore_o the_o rectangle_v ak_v he_o be_v equal_a add_v to_o both_o the_o rectangle_n cn_fw-la and_o the_o square_a kg_n the_o rectangle_v ak_v cn_fw-la that_o be_v to_o say_v the_o rectangle_n a_fw-la with_o the_o square_a kg_n shall_v be_v equal_a to_o the_o rectangle_v cn_fw-la he_o and_o to_o the_o square_a kg_n that_o be_v to_o say_v to_o the_o square_a ce._n arithmetical_o or_o by_o number_n let_v ab_fw-la be_v 8_o ac_fw-la 4_o cb_n 4_o bd_o 3_o than_o ad_fw-la shall_v be_v 11._o it_o be_v evident_a that_o the_o rectangle_n a_fw-la three_o time_n 11_o that_o be_v to_o say_v 33_o with_o the_o square_n of_o kg_v 16_o which_o together_o be_v 49_o be_v equal_a to_o the_o square_n of_o cd_o 7_o which_o be_v 49_o for_o 7_o time_n 7_o be_v 49._o use_v 6._o fig._n 6._o maurolycus_n measure_v the_o whole_a earth_n by_o one_o single_a
square_n of_o the_o line_n add_v be_v double_a to_o the_o square_n of_o half_a the_o line_n and_o to_o the_o square_n which_o be_v compose_v of_o the_o half_a line_n and_o the_o line_n add_v if_o one_o suppose_v ab_fw-la to_o be_v divide_v in_o the_o middle_n at_o the_o point_n c_o and_o if_o thereto_o be_v add_v the_o line_n bd_o the_o square_n of_o ab_fw-la and_o bd_o shall_v be_v double_a to_o the_o square_n of_o ac_fw-la and_o cd_o add_v together_o draw_v the_o perpendicular_n ce_fw-fr df_n equal_a to_o ac_fw-la then_o draw_v the_o line_n ae_n of_o agnostus_n ebg_n demonstration_n the_o line_n ac_fw-la ce_fw-fr cb_n be_v equal_a and_o the_o angle_n at_o the_o point_n c_o being_n right_n the_o angle_n aec_fw-la ceb_fw-mi cbe_n dbg_n dgb_n shall_v be_v half_a right_n and_o the_o line_n db_n dg_fw-mi and_o of_o fg_v cd_o shall_v be_v equal_a the_o square_a of_o ae_n be_v double_a to_o the_o square_n of_o ac_fw-la the_o square_a of_o eglantine_n be_v double_a to_o the_o square_n of_o of_o or_o cd_o by_o the_o 47_o of_o the_o 1_o now_o the_o square_n of_o agnostus_n be_v equal_a to_o the_o square_n of_o ae_n eglantine_n by_o the_o 47_o of_o the_o 1_o therefore_o the_o square_n of_o agnostus_n be_v double_a to_o the_o square_n of_o ac_fw-la cd_o the_o same_o agnostus_n by_o the_o 47_o of_o the_o 1_o be_v equal_a to_o the_o square_n of_o ad_fw-la bd_o or_o gd_v therefore_o the_o square_n of_o ad_fw-la bd_o be_v double_a the_o square_n of_o ac_fw-la cd_o arithmetical_o let_v ab_fw-la be_v 6_o part_n ac_fw-la 3_o cb_n 3_o bd_o 4_o the_o square_a of_o ad_fw-la 10_o be_v 100_o the_o square_a of_o bd_o 4_o be_v 16_o which_o be_v 116._o the_o square_a of_o ac_fw-la 3_o be_v 9_o the_o square_a of_o cd_o 7_o be_v 49._o now_o 49._o and_o 9_o be_v 58_o the_o half_a of_o 116._o proposition_n xi_o problem_n to_o divide_v a_o line_n so_o that_o the_o rectangle_n comprehend_v under_o the_o whole_a line_n and_o under_o one_o of_o its_o part_n shall_v be_v equal_a to_o the_o square_n of_o the_o other_o part_n it_o be_v propose_v to_o divide_v the_o line_n ab_fw-la so_o that_o the_o rectangle_n comprehend_v under_o the_o whole_a line_n ab_fw-la and_o under_o hb_n be_v equal_a to_o the_o square_n of_o ah_o make_v a_o square_a of_o ab_fw-la by_o the_o 46_o of_o the_o 1_o divide_v ad_fw-la in_o the_o middle_n in_o e_z then_o draw_v ebb_v and_o make_v of_o equal_a to_o ebb_v make_v the_o square_n of_o that_o be_v to_o say_v that_o of_o ah_o be_v equal_a i_o say_v that_o the_o square_a of_o ah_o shall_v be_v equal_a to_o the_o rectangle_n hc_n comprehend_v under_o hb_n and_o the_o line_n bc_n equal_a to_o ab_fw-la demonstration_n the_o line_n ad_fw-la be_v equal_o divide_v in_o e_o and_o there_o be_v add_v thereto_o the_o line_n favorina_n thence_o by_o the_o 6_o the_o rectangle_n dg_n comprehend_v under_o df_n and_o fg_n equal_a to_o of_o with_o the_o square_n of_o ae_n be_v equal_a to_o the_o square_n of_o of_o equal_a to_o ebb_v now_o the_o square_n of_o ebb_n be_v equal_a to_o the_o square_n of_o ab_fw-la ae_n by_o the_o 47_o of_o the_o 1_o therefore_o the_o square_n of_o ab_fw-la ae_n be_v equal_a to_o the_o rectangle_n dg_n and_o to_o the_o square_n of_o ae_n and_o take_v away_o from_o both_o the_o square_a of_o ae_n the_o square_a of_o ab_fw-la which_o be_v ac_fw-la shall_v be_v equal_a to_o the_o rectangle_n dg_n take_v also_o away_o the_o rectangle_n dh_n which_o be_v in_o both_o the_o rectangle_n hc_n shall_v be_v equal_a to_o the_o square_n of_o ag._n use_v this_o proposition_n serve_v to_o cut_v a_o line_n in_o extreme_a and_o mean_a proportion_n as_o shall_v be_v show_v in_o the_o six_o book_n it_o be_v use_v often_o in_o the_o 14_o of_o euclid_n element_n to_o find_v the_o side_n of_o regular_a body_n it_o serve_v for_o the_o 10_o of_o the_o four_o book_n to_o inscribe_v a_o pentagone_fw-mi in_o a_o circle_n as_o also_o a_o pentadecagone_fw-mi you_o shall_v see_v other_o use_n of_o a_o line_n thus_o divide_v in_o the_o 30_o of_o the_o 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triangle_n abc_n the_o line_n de_fw-fr be_v parallel_n to_o the_o base_a bc_n the_o side_n ab_fw-la ac_fw-la shall_v be_v divide_v proportional_o that_o be_v to_o say_v that_o there_o shall_v be_v the_o same_o reason_n of_o ad_fw-la to_o db_n as_o of_o ae_n to_o aec_fw-la draw_v the_o line_n de_fw-fr be._n the_o triangle_n dbe_v dec_n which_o have_v the_o same_o base_a de_fw-fr and_o be_v between_o the_o same_o parallel_n de_fw-fr bc_n be_v equal_a by_o the_o 37th_o of_o the_o one_a demonstration_n the_o triangle_n ade_n dbe_n have_v the_o same_o point_n e_o for_o their_o vertical_a if_o we_o take_v ad_fw-la db_fw-la for_o their_o base_n and_o if_o one_o shall_v draw_v through_o the_o point_n e_o a_o parallel_n to_o ab_fw-la they_o will_v be_v both_o between_o the_o same_o parallel_n they_o shall_v have_v thence_o the_o same_o reason_n as_o their_o base_n by_o the_o one_a that_o be_v to_o say_v that_o there_o be_v the_o same_o reason_n of_o ad_fw-la to_o db_n as_o of_o the_o triangle_n 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bc_n ce_fw-fr be_v on_o the_o same_o line_n and_o continue_v the_o side_n ed_z ba_z see_v the_o angle_n acb_n dec_n be_v equal_a the_o line_n ac_fw-la of_o be_v parallel_n and_o so_o cd_o bf_n by_o the_o 29_o of_o the_o one_a and_o of_o dc_o shall_v be_v a_o parallelogram_n demonstration_n in_o the_o triangle_n bfe_n ac_fw-la be_v parallel_n to_o the_o base_a fe_o thence_o by_o the_o 2d_o there_o shall_v be_v the_o same_o reason_n of_o basilius_n to_o of_o or_o cd_o as_o of_o bc_n to_o ce_fw-fr and_o by_o exchange_n there_o shall_v be_v the_o same_o reason_n of_o ab_fw-la to_o bc_n as_o of_o dc_o to_o ce._n in_o like_a manner_n in_o the_o same_o triangle_n cd_o be_v parallel_n to_o the_o base_a bf_n there_o shall_v be_v the_o same_o reason_n of_o fd_n or_o ac_fw-la to_o de_fw-fr as_o of_o bc_n to_o ge_z by_o the_o 2d_o and_o by_o exchange_n there_o shall_v be_v the_o same_o reason_n of_o ac_fw-la to_o bc_n as_o of_o de_fw-fr to_z ce._n use_v this_o proposition_n be_v of_o a_o great_a extent_n and_o may_v pass_v for_o a_o universal_a principle_n in_o all_o sort_n of_o measure_v for_o in_o the_o first_o place_n the_o ordinary_a practice_n in_o measure_v inaccessible_a line_n by_o make_v a_o little_a triangle_n like_o unto_o that_o which_o be_v make_v or_o imagine_v to_o be_v make_v on_o the_o ground_n be_v found_v on_o this_o proposition_n as_o also_o the_o great_a part_n of_o those_o instrument_n on_o which_o be_v make_v triangle_n like_v unto_o those_o that_o we_o will_v measure_v as_o the_o geometrical_a square_n sinical_a quadrant_n jacob_n staff_n and_o other_o moreover_o we_o can_v not_o take_v the_o plane_n of_o a_o place_n but_o by_o this_o proposition_n wherefore_o to_o explain_v its_o use_n we_o shall_v be_v force_v to_o bring_v in_o the_o first_o book_n of_o practical_a geometry_n proposition_n v._o theorem_fw-la triangle_n who_o side_n be_v proportional_a be_v equiangule_a if_o the_o triangle_n abc_n def_n have_v their_o side_n proportional_a that_o be_v to_o say_v if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o bc_n as_o of_o de_fw-fr to_o of_o as_o also_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o ac_fw-la as_o of_o de_fw-fr to_o df_n the_o angel_n abc_n def_n a_o and_o d_o c_z and_z f_o shall_v be_v equal_a make_v the_o angle_n feg_v equal_a to_o the_o angle_n b_o and_o efg_v equal_a to_o the_o angle_n c._n demonstration_n the_o triangle_n abc_n efg_v have_v two_o angle_n equal_a they_o be_v thence_o equiangle_v by_o the_o cor._n of_o the_o 32d_o of_o the_o one_a and_o by_o the_o four_o there_o be_v the_o same_o reason_n of_o de_fw-fr to_o of_o as_o of_o eglantine_n to_o ef._n now_o it_o be_v suppose_v that_o there_o be_v the_o same_o reason_n of_o de_fw-fr to_o of_o as_o of_o eglantine_n to_o ef._n thence_o by_o the_o seven_o of_o the_o 5_o de_fw-fr eglantine_n be_v equal_a in_o like_a manner_n df_n fg_n be_v also_o equal_a and_o by_o the_o 8_o of_o the_o one_a the_o triangle_n def_n gef_n be_v equiangular_a now_o the_o angle_n gef_n be_v make_v equal_a to_o the_o angle_n b_o thence_o def_n be_v equal_a to_o the_o angle_n b_o and_o the_o angle_n dfe_n to_o the_o angle_n c._n so_o that_o the_o triangle_n abc_n def_n be_v equiangular_a proposition_n vi_o theorem_fw-la triangle_n which_o have_v their_o side_n proportional_a which_o include_v a_o equal_a angle_n be_v equiangular_a if_o the_o angle_n b_o and_o e_o of_o the_o triangle_n abc_n def_n be_v equal_a there_o be_v the_o same_o reason_n of_o ab_fw-la to_o bc_n as_o of_o de_fw-fr to_o of_o the_o triangle_n abc_n def_n shall_v be_v equiangular_a make_v the_o angle_n feg_n equal_a to_o the_o angle_n b_o and_o
the_o line_n be_v in_o sub-duplicate_a ratio_fw-la shall_v be_v also_o proportional_a use_v a_o b_o c_o d._n 3._o 2._o 6._o 4._o 9_o 4._o 36._o 16_o e_z f_o g_o h._n this_o proposition_n may_v be_v easy_o apply_v to_o number_n if_o the_o number_n a_o b_o c_o d_o be_v proportional_a their_o square_n of_o gh_o shall_v be_v so_o likewise_o which_o we_o make_v use_v of_o in_o arithmetic_n and_o yet_o more_o use_n thereof_o in_o algebra_n proposition_n xxiii_o theorem_fw-la equiangular_a parallellogram_n have_v their_o ratio_fw-la compound_v of_o the_o ratio_fw-la of_o their_o side_n if_o the_o parallellogram_n l_o and_o m_o be_v equiangular_a the_o ratio_fw-la of_o l_o to_o m_n shall_v be_v compound_v of_o that_o of_o ab_fw-la to_o de_fw-fr and_o that_o of_o db_n to_o df._n join_v the_o parallellogram_n in_o such_o a_o manner_n that_o their_o side_n bd_o df_n be_v on_o a_o straight_a line_n as_o also_o cd_o de_fw-fr which_o may_v be_v if_o they_o be_v equiangular_a complete_a the_o parallelogram_n bdeh_n demonstration_n the_o parallelogram_n l_o have_v the_o same_o ratio_fw-la to_o the_o parallelogram_n bdeh_n as_o the_o base_a ab_fw-la have_v to_o the_o base_a bh_n or_o de_fw-fr by_o the_o first_o the_o parallelogram_n bdeh_n have_v the_o same_o ratio_fw-la to_o the_o parallelogram_n dfge_n that_o be_v to_o say_v m_n as_o the_o base_a bd_o have_v to_o the_o base_a df._n now_o the_o ratio_fw-la of_o the_o parallelogram_n l_o to_o the_o parallelogram_n m_o be_v compound_v of_o that_o of_o l_o to_o the_o parallelogram_n bdeh_n and_o that_o of_o bdeh_n to_o the_o parallelogram_n m._n thence_o the_o ratio_fw-la of_o l_o to_o m_n be_v compound_v of_o that_o of_o ab_fw-la to_o de_fw-fr and_o of_o that_o of_o bd_o to_o eglantine_n for_o example_n if_o ab_fw-la be_v eight_o and_o bh_n five_o bd_o four_o df_n seven_a say_v as_o four_o be_v to_o seven_o so_o be_v five_o to_o eight_o ¾_n you_o shall_v have_v these_o three_o number_n eight_o five_o eight_o ¾_n eight_o to_o five_o shall_v be_v the_o ratio_fw-la of_o the_o parallelogram_n l_o to_o bdeh_n the_o same_o as_o that_o of_o ab_fw-la to_o de_fw-fr five_o to_o eight_o ¾_n shall_v be_v that_o of_o the_o parallelogram_n bdeh_n to_o m._n so_o then_o take_v away_o the_o middle_a term_n which_o be_v five_o you_o shall_v have_v the_o ratio_fw-la of_o eight_o to_o eight_o â—_o for_o the_o ratio_fw-la compound_v of_o both_o or_o as_o 4_o time_n 8_o or_o 32_o to_o 5_o time_n 7_o or_o 35._o proposition_n xxiv_o theorem_fw-la in_o all_o sort_n of_o parallellogram_n those_o through_o which_o the_o diameter_n pass_v be_v like_a to_o the_o great_a let_v the_o diameter_n of_o the_o parallelogram_n ac_fw-la pass_v through_o the_o parallellogram_n of_o gh_o i_o say_v they_o be_v like_a unto_o the_o parallelogram_n ac_fw-la demonstration_n the_o parallelograms_n ac_fw-la of_o have_v the_o same_o angle_n b_o and_o because_o in_o the_o triangle_n bcd_v if_o be_v parallel_n to_o the_o base_a dc_o the_o triangle_n bfi_n bcd_n be_v equiangular_a there_o be_v therefore_o by_o the_o four_o the_o same_o ratio_fw-la of_o bc_n to_o cd_o as_o of_o bf_n to_o fi_n and_o consequent_o the_o side_n be_v in_o the_o same_o ratio_fw-la in_o like_a manner_n ih_o be_v parallel_n to_o bc_n there_o shall_v then_o be_v the_o same_o ratio_fw-la of_o dh_n to_o he_o as_o of_o dc_o to_o bc_n and_o the_o angle_n be_v also_o equal_a all_o the_o side_n be_v parallel_n thence_o by_o the_o 8_o def._n the_o parallellogram_n of_o gh_o be_v like_a unto_o the_o parallelogram_n ac_fw-la use_v i_o make_v use_v of_o this_o proposition_n in_o the_o ten_o proposition_n of_o the_o last_o book_n of_o perspective_n to_o show_v that_o a_o image_n be_v draw_v like_o unto_o the_o original_a with_o the_o help_n of_o a_o parallelogram_n compose_v of_o four_o line_n proposition_n xxv_o problem_n to_o describe_v a_o polygon_n like_v unto_o a_o give_v polygon_n and_o equal_a to_o any_o other_o right_o line_v figure_n if_o you_o will_v describe_v a_o polygon_n equal_a to_o the_o right_o line_a figure_n a_o and_o like_v unto_o the_o polygon_n b_o make_v a_o parallelogram_n ce_fw-fr equal_a to_o the_o polygon_n b_o by_o the_o 34th_o of_o the_o first_o and_o on_o de_fw-fr make_v a_o parallelogram_n of_o equal_a to_o the_o right_o line_a figure_n a_o by_o the_o 45th_o of_o the_o first_o then_o seek_v a_o mean_a proportional_a gh_o between_o cd_o and_o df_n by_o the_o 13_o last_o make_v on_o gh_a a_o polygon_n o_o like_v unto_o b_o by_o the_o 18_o it_o shall_v be_v equal_a to_o the_o right_o line_a figure_n a._n demonstration_n see_v that_o cd_o gh_n df_n be_v continual_o proportional_a the_o right_o line_a figure_n b_o describe_v on_o the_o first_o shall_v be_v to_o the_o right_o line_a figure_n o_o describe_v on_o the_o second_o as_o cd_o be_v to_o df_n by_o the_o coral_n of_o the_o 20_o now_o as_o cd_o be_v to_o df_n so_o be_v the_o parallelogram_n ce_fw-fr to_o the_o parallelogram_n of_o or_o as_o b_o to_o a_o see_v they_o be_v equal_a there_o be_v thence_o the_o same_o ratio_fw-la of_o b_o to_z o_o as_o of_o b_o to_o a._n so_o then_o by_o the_o seven_o of_o the_o 5_o a_o and_o o_o be_v equal_a use_v this_o proposition_n contain_v a_o change_n of_o figure_n keep_v always_o the_o same_o area_n which_o be_v very_o useful_a principal_o in_o practical_a geometry_n to_o reduce_v a_o irregular_a figure_n into_o a_o square_n proposition_n xxvi_o theorem_fw-la if_o in_o one_o of_o the_o angle_n of_o a_o parallelogram_n there_o be_v describe_v a_o lesser_a parallelogram_n like_o unto_o the_o great_a the_o diameter_n of_o the_o great_a shall_v meet_v the_o angle_n of_o the_o lesser_a if_o in_o the_o angle_n d_o of_o the_o parallelogram_n ac_fw-la be_v describe_v another_o lesser_a dg_n like_v thereto_o the_o diameter_n db_n shall_v pass_v through_o the_o point_n g._n for_o if_o it_o do_v not_o but_o it_o pass_v through_o ay_o as_o do_v the_o line_n bid_v draw_v the_o line_n je_n parallel_n to_o hd_n demonstration_n the_o parallelogram_n diego_n be_v like_a unto_o the_o parallelogram_n ac_fw-la by_o the_o 24_o now_o it_o be_v suppose_v that_o the_o parallelogram_n dg_n be_v also_o like_a thereto_o thence_o the_o parallellogram_n dii_fw-la dg_n will_v be_v like_a which_o be_v impossible_a otherwise_o there_o will_v be_v the_o same_o ratio_fw-la of_o he_o to_o je_n or_o gf_n as_o of_o hg_v to_o gf_n and_o by_o the_o seven_o of_o the_o 5_o the_o line_n he_o hg_n will_v be_v equal_a the_o twenty_o seven_o twenty_o eight_o and_o twenty_o nine_o proposition_n be_v unnecessary_a proposition_n xxx_o problem_n to_o cut_v a_o line_n give_v into_o extreme_a and_o mean_a proportion_n it_o be_v propose_v to_o cut_v the_o line_n ab_fw-la in_o extreme_a and_o mean_a proportion_n that_o be_v to_o say_v in_o such_o a_o manner_n that_o there_o may_v be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o ac_fw-la as_o of_o ac_fw-la to_o cb._n divide_v the_o line_n ab_fw-la by_o the_o 11_o of_o the_o second_o in_o such_o a_o manner_n that_o the_o rectangle_n comprehend_v under_o ab_fw-la cb_n be_v equal_a to_o the_o square_n of_o ac_fw-la demonstration_n see_v the_o rectangle_n of_o ab_fw-la bc_n be_v equal_a to_o the_o square_n of_o ac_fw-la there_o will_v be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o ac_fw-la as_o of_o ac_fw-la to_o bc_n by_o the_o 17_o use_v this_o proposition_n be_v necessary_a in_o the_o thirteen_o book_n of_o euclid_n to_o find_v the_o length_n of_o the_o side_n of_o some_o of_o the_o five_o regular_a body_n father_n lucas_n of_o st._n sepulcher_n have_v compose_v a_o book_n of_o the_o propriety_n of_o a_o line_n which_o be_v cut_v into_o extreme_a and_o mean_a proportion_n proposition_n xxxi_o theorem_fw-la a_o polygon_n which_o be_v describe_v on_o the_o base_a of_o a_o right_a angle_a triangle_n be_v equal_a to_o the_o other_o two_o like_v polygon_n describe_v on_o the_o other_o side_n of_o the_o same_o triangle_n if_o the_o triangle_n abc_n have_v a_o right_a angle_n bac_n the_o polygon_n d_o describe_v on_o the_o base_a bc_n be_v equal_a to_o the_o like_a polygon_n f_o and_o e_o describe_v on_o the_o side_n ab_fw-la ac_fw-la demonstration_n the_o polygon_n d_o e_o f_o be_v among_o themselves_o in_o duplicate_v ratio_fw-la of_o their_o homologous_n side_n bc_n ac_fw-la ab_fw-la by_o the_o 20_o if_o there_o be_v describe_v a_o square_a on_o those_o side_n they_o will_v be_v also_o in_o duplicate_v ratio_fw-la to_o their_o side_n now_o by_o the_o 47th_o of_o the_o first_o the_o square_a of_o bc_n will_v be_v equal_a to_o the_o square_n of_o ac_fw-la ab_fw-la thence_o the_o polyligon_n d_o describe_v on_o bc_n be_v equal_a to_o the_o like_a polygon_n e_o and_o f_o describe_v on_o ab_fw-la ac_fw-la use_v this_o proposition_n be_v make_v use_n
of_o to_o make_v all_o sort_n of_o figure_n great_a or_o lesser_a for_o it_o be_v more_o universal_a then_o the_o 47th_o of_o the_o one_a which_o notwithstanding_o be_v so_o useful_a that_o it_o seem_v that_o almost_o all_o geometry_n be_v establish_v on_o this_o principle_n the_o 32d_o proposition_n be_v unnecessary_a proposition_n xxxiii_o theorem_fw-la in_o equal_a circle_n the_o angle_n as_o well_o those_o at_o the_o centre_n as_o those_o at_o the_o circumference_n as_o also_o the_o sector_n be_v in_o the_o same_o ratio_fw-la as_o the_o ark_n which_o serve_v to_o they_o as_o base_a if_o the_o circle_n anc_n dof_n be_v equal_a there_o shall_v be_v the_o same_o ratio_fw-la of_o the_o angle_n abc_n to_o the_o angle_n def_n as_o of_o the_o ark_n ac_fw-la to_o the_o ark_n df._n let_v the_o ark_n agnostus_n gh_n hc_n he_o equal_a ark_n and_o consequent_o aliquot_fw-la part_n of_o the_o ark_n ac_fw-la and_o let_v be_v divide_v the_o ark_n df_n into_o so_o many_o equal_a to_o agnostus_n as_o may_v be_v find_v therein_o and_o let_v the_o line_n ei_o eke_o and_o the_o rest_n be_v draw_v demonstration_n all_o the_o angles_n abg_n gbh_n hbc_n dei_fw-la jek_n and_o the_o rest_n be_v equal_a by_o the_o 37th_o of_o the_o three_o so_o then_o agnostus_n a_o aliquot_fw-la part_n of_o ac_fw-la be_v find_v in_o the_o ark_n df_n as_o many_o time_n as_o the_o angle_n abg_n a_o aliquot_fw-la part_n of_o the_o angle_n abc_n be_v find_v in_o def_n there_o be_v therefore_o the_o same_o ratio_fw-la of_o the_o ark_n ac_fw-la to_o the_o ark_n df_n as_o of_o the_o angle_n abc_n to_o the_o angle_n def_n and_o because_o n_n and_o o_o be_v the_o half_n of_o the_o angel_n abc_n def_n they_o shall_v be_v in_o the_o same_o ratio_fw-la as_o be_v those_o angles_n there_o be_v therefore_o the_o same_o ratio_fw-la of_o the_o angle_n n_n to_o the_o angle_n o_o as_o of_o the_o ark_n ac_fw-la to_o the_o ark_n df._n it_o be_v the_o same_o with_o sector_n for_o if_o the_o line_n agnostus_n gh_n hc_n 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the_o four_o be_v tripled_a of_o that_o of_o the_o first_o to_o the_o second_o use_v you_o may_v comprehend_v by_o this_o proposition_n that_o the_o celebrate_a proposition_n of_o the_o duplication_n of_o the_o cube_n propose_v by_o the_o oracle_n consist_v in_o this_o to_o find_v two_o mean_n continual_o proportional_a for_o if_o you_o propose_v for_o this_o first_o term_n the_o side_n of_o the_o first_o cube_n and_o that_o the_o four_o term_n be_v the_o double_a of_o this_o first_o if_o you_o find_v two_o mean_a proportional_n the_o cube_n describe_v on_o the_o first_o line_n shall_v have_v the_o same_o ratio_fw-la to_o that_o describe_v on_o the_o second_o as_o the_o first_o line_n have_v to_o the_o four_o which_o shall_v be_v as_o one_o to_o two_n we_o also_o correct_v by_o this_o proposition_n the_o false_a opinion_n of_o those_o which_o imagine_v that_o like_a solid_n be_v in_o the_o same_o ratio_fw-la as_o be_v their_o side_n as_o if_o a_o cube_n of_o one_o foot_n long_o be_v the_o half_a of_o a_o cube_n of_o two_o foot_n long_o although_o it_o be_v but_o the_o eight_o part_n thereof_o this_o be_v the_o principle_n of_o the_o calibre_n which_o may_v be_v apply_v not_o only_o to_o cannon_n shoot_v but_o also_o to_o all_o like_a body_n for_o example_n i_o have_v see_v a_o person_n which_o will_v make_v a_o naval_a architecture_n and_o will_v keep_v the_o same_o proportion_n in_o all_o sort_n of_o vessel_n but_o argue_v thus_o if_o a_o ship_n of_o one_o hundred_o tun_n aught_o to_o be_v fifty_o foot_n long_o a_o ship_n of_o two_o hundred_o ought_v to_o be_v one_o hundred_o foot_n by_o the_o keel_n in_o which_o he_o be_v much_o mistake_v for_o instead_o of_o make_v a_o ship_n of_o twice_o that_o burden_n he_o will_v make_v one_o octuple_a he_o ought_v to_o have_v give_v to_o the_o second_o ship_n to_o be_v double_a in_o burden_n to_o the_o first_o somewhat_o less_o than_o sixty_o three_o foot_n proposition_n xxxiv_o theorem_fw-la equal_a parallelepipedon_n have_v their_o base_n and_o height_n reciprocal_a and_o those_o which_o have_v their_o base_n and_o height_n reciprocal_a be_v equal_a if_o the_o parallepipedons_n ab_fw-la cd_o be_v equal_a they_o shall_v have_v their_o base_n and_o height_n reciprocal_a that_o be_v to_o say_v there_o shall_v be_v the_o same_o ratio_fw-la of_o the_o base_a ae_n to_o the_o base_a cf_n as_o of_o the_o height_n ch_z to_o the_o height_n ag._n have_v make_v ci_o equal_a to_o agnostus_n draw_v the_o plane_n ik_fw-mi parallel_n to_o the_o base_a cf._n demonstration_n the_o parallelepipedon_n ab_fw-la have_v the_o same_o ratio_fw-la to_o ck_n of_o the_o same_o height_n as_o have_v the_o base_a ae_n to_o cf_n by_o the_o 32d_o now_o as_o ab_fw-la to_o ck_n so_o be_v cd_o to_o the_o same_o ck_n see_v that_o ab_fw-la and_o cd_o be_v equal_a and_o as_o cd_o be_v to_o ck_n which_o have_v the_o same_o base_a so_o be_v the_o height_n ch_z to_o the_o height_n ci_o by_o the_o coral_n of_o the_o 32d_o wherefore_o as_o the_o base_a ae_n be_v to_o the_o base_a cf_n so_o be_v the_o height_n ch_z to_o the_o height_n ci_o or_o ag._n i_o further_o add_v that_o if_o there_o be_v the_o same_o ratio_fw-la of_o ae_n to_o cf_n as_o of_o the_o height_n ch_z to_o the_o height_n agnostus_n the_o solid_n ab_fw-la cd_o shall_v be_v equal_a demonstration_n there_o be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o ck_n of_o the_o same_o height_n as_o of_o the_o base_a ae_n to_o the_o base_a cf_n by_o the_o 32d_o there_o be_v also_o the_o same_o ratio_fw-la of_o the_o height_n ch_z to_o the_o height_n ci_o or_o agnostus_n as_o of_o cd_o to_o ck_n we_o suppose_v that_o the_o ratio_fw-la of_o ae_n to_o cf_n be_v the_o same_o of_o that_o of_o ch_z to_o ci_o or_o agnostus_n so_o then_o there_o shall_v be_v the_o same_o ratio_fw-la of_o the_o solid_a ab_fw-la to_o the_o solid_a ck_n as_o of_o the_o solid_a cd_o to_o the_o same_o solid_a ck_n therefore_o by_o the_o 8_o of_o the_o 5_o the_o solid_n ab_fw-la cd_o be_v equal_a use_v this_o reciprocation_n of_o base_n render_v these_o solid_n easy_a to_o be_v measure_v it_o have_v also_o a_o kind_n of_o analogy_n with_o the_o sixteenth_o proposition_n of_o the_o six_o book_n which_o bear_v this_o that_o parallellogram_n equiangle_v and_o equal_a have_v their_o side_n reciprocal_a and_o it_o demonstrate_v also_o as_o well_o the_o practice_n of_o the_o rule_n of_o three_o proposition_n xxxvi_o theorem_fw-la if_o three_o line_n be_v continual_o proportional_a the_o solid_a parallelepipedon_n make_v of_o they_o be_v equal_a to_o a_o equiangular_a parallelepipedon_n which_o have_v all_o it_o side_n equal_a to_o the_o middle_a line_n if_o the_o line_n a_o b_o c_o be_v continual_o proportional_a the_o parallelepipedon_n fe_o make_v of_o they_o that_o be_v to_o say_v which_o have_v the_o side_n fi_n equal_a to_o the_o line_n a_o and_o he_o equal_a to_o b_o he_o equal_a to_o c_o be_v equal_a to_o the_o equiangular_a parallelepipedon_n kl_n which_o have_v its_o side_n lm_o mn_a kn_n equal_a to_o the_o line_n b._n let_v there_o be_v draw_v from_o the_o point_n h_o and_o n_n the_o line_n hp_n nq_n perpendicular_a to_o the_o plane_n of_o the_o base_n they_o shall_v be_v equal_a see_v that_o the_o solid_a angle_n e_z and_o k_o be_v suppose_v equal_a in_o such_o manner_n that_o if_o they_o do_v penetrate_v each_o other_o they_o will_v not_o surpass_v each_o other_o and_o that_o the_o line_n eh_n kn_n be_v suppose_v equal_a therefore_o the_o height_n hp_n nq_n be_v equal_a demonstration_n there_o be_v the_o same_o ratio_fw-la of_o a_o to_z b_o or_o of_o fi_n to_o lm_o as_o of_o b_o to_z c_z or_o of_o lm_o to_o he_o so_o than_o the_o parallelogram_n fh_v comprehend_v under_o fi_n ih_o be_v equal_a to_o the_o parallelogram_n ln_o comprehend_v under_o lm_o mn_a equal_a to_o b_o by_o the_o 15_o of_o the_o 6_o the_o base_n be_v thence_o equal_a now_o the_o height_n hp_n nq_n be_v so_o likewise_o therefore_o by_o the_o 31st_o the_o parallelepipedon_n be_v equal_a proposition_n xxxvii_o theorem_fw-la if_o four_o line_n be_v proportional_a the_o parallepipedon_n describe_v on_o those_o line_n shall_v be_v proportional_a and_o if_o like_a parallelepipedon_n be_v proportional_a their_o homologous_a side_n shall_v be_v so_o likewise_o if_o a_o to_o b_o bein_z the_o same_o ratio_fw-la as_o c_o to_z d_o the_o like_a parallepipedons_n which_o shall_v have_v for_o their_o homologous_n side_n a_o b_o c_o d_o shall_v be_v in_o the_o same_o ratio_fw-la demonstration_n the_o parallelepipedon_n a_o to_o the_o parallelepipedon_n b_o be_v in_o triple_a ratio_fw-la of_o that_o of_o the_o line_n a_o to_o the_o line_n b_o or_o of_o that_o of_o the_o line_n c_o to_o the_o line_n d._n now_o the_o parallelepipedon_n c_o to_o the_o parallelepipedon_n d_o be_v also_o in_o triple_a ratio_fw-la of_o that_o of_o c_o to_o d_o by_o the_o 33d_o there_o be_v therefore_o the_o same_o ratio_fw-la of_o the_o parallepipedon_n a_o to_o the_o parallepipedon_n b_o as_o of_o the_o parallelepipedon_n c_o to_o the_o parallelepipedon_n d._n proposition_n xxxviii_o theorem_fw-la if_o two_o plane_n be_v perpendicular_a to_o each_o other_o the_o perpendicular_a draw_v from_o one_o point_n of_o one_o of_o the_o plane_n to_o the_o other_o shall_v fall_v on_o the_o common_a section_n if_o the_o plane_n ab_fw-la cd_o be_v perpendicular_a to_o each_o other_o there_o be_v draw_v from_o the_o point_n e_o of_o the_o plane_n ab_fw-la a_o line_n perpendicular_a to_o the_o plane_n cd_o it_o shall_v fall_v on_o agnostus_n the_o common_a section_n of_o the_o plane_n draw_v of_o perpendicular_a to_o the_o common_a section_n ag._n demonstration_n the_o line_n of_o perpendicular_a to_o agnostus_n the_o common_a section_n of_o the_o plane_n which_o we_o suppose_v perpendicular_a shall_v be_v perpendicular_a to_o the_o plane_n cd_o by_o the_o 3d._n def._n and_o see_v there_o can_v be_v draw_v from_o the_o point_n e_o two_o perpendicular_o to_o the_o plane_n cd_o by_o the_o 13_o the_o perpendicular_a shall_v fall_v on_o the_o common_a section_n ag._n use_v this_o proposition_n ought_v to_o have_v be_v after_o the_o 17_o see_v it_o have_v respect_n to_o solid_n in_o general_n it_o be_v of_o use_n in_o the_o treatise_n of_o astrolabe_fw-la to_o prove_v that_o in_o the_o analemma_n all_o the_o circle_n which_o be_v perpendicular_a to_o the_o meridian_n be_v strait_a line_n proposition_n thirty-nine_o theorem_fw-la if_o there_o be_v draw_v in_o a_o parallelepipedon_n two_o plane_n divide_v the_o opposite_a side_n equal_o their_o common_a section_n and_o the_o diameter_n do_v also_o cut_v each_o other_o equal_o let_v the_o opposite_a side_n of_o the_o parallelepipedon_n ab_fw-la be_v divide_v into_o two_o equal_o by_o the_o plane_n cd_o of_o their_o common_a section_n gh_o and_o
the_o diameter_n basilius_n shall_v divide_v each_o other_o equal_o in_o o._n draw_v the_o line_n bg_n gk_n ah_o lh_n i_o prove_v in_o the_o first_o place_n that_o they_o make_v but_o one_o strait_a line_n for_o the_o triangle_n dbg_n kmg_n have_v their_o side_n bd_o km_n equal_a see_v they_o be_v the_o half_n of_o equal_a side_n as_o also_o gd_v gm_n moreover_o db_n km_fw-la be_v parallel_n the_o alternate_a angel_n bdg_n gmk_n shall_v be_v equal_a by_o the_o 29_o of_o the_o first_o so_o then_o by_o the_o four_o of_o the_o first_o the_o triangle_n dbg_n kgm_n shall_v be_v equal_a in_o every_o respect_n and_o consequent_o the_o angle_n bgd_v kgm_n now_o by_o the_o 16_o of_o the_o one_a bg_n gk_n be_v but_o one_o line_n as_o also_o lh_n ha_o thence_o all_o bk_n be_v but_o one_o plane_n in_o which_o be_v find_v the_o diameter_n ab_fw-la and_o gh_n the_o common_a section_n of_o the_o plane_n the_o plane_n all_o bk_n cut_v the_o parallel_n planes_n a_o cd_o shall_v have_v its_o common_a section_n gh_o ak_v parallel_n and_o by_o the_o four_o of_o the_o 6_o there_o shall_v be_v the_o same_o ratio_fw-la of_o bk_n to_o gk_n as_o of_o basilius_n to_o boy_n and_o of_o ak_n to_o og_n by_o the_o four_o of_o the_o 6_o now_o bk_n be_v double_a to_o bg_n thence_o basilius_n be_v double_a to_o boy_n as_o ak_v equal_a to_o hg_v be_v double_a to_o go_v so_o then_o the_o line_n gh_o ab_fw-la cut_v each_o other_o equal_o in_o the_o point_n o._n coral_n 1._o all_o the_o diameter_n be_v divide_v in_o the_o point_n o._n coral_n 2._o here_o may_v be_v put_v corollary_n which_o depend_v on_o several_a proposition_n for_o example_n that_o triangular_a prism_n of_o equal_a height_n be_v in_o the_o same_o ratio_fw-la as_o be_v their_o base_n for_o the_o parallelepipedon_n of_o which_o they_o be_v the_o half_n by_o the_o 32d_o be_v in_o the_o same_o ratio_fw-la as_o their_o base_n so_o that_o the_o half_a base_n and_o the_o half_a parallelepipedon_n that_o be_v to_o say_v the_o prism_n shall_v be_v in_o the_o same_o ratio_fw-la coral_n 3._o polygon_n prism_n of_o the_o same_o height_n be_v in_o the_o same_o ratio_fw-la as_o be_v their_o base_n see_v they_o may_v be_v reduce_v to_o triangular_a prism_n which_o shall_v each_o of_o they_o be_v in_o the_o same_o ratio_fw-la as_o their_o base_n coral_n 4._o there_o may_v be_v apply_v to_o the_o prism_n the_o other_o proposition_n of_o the_o parallelepipedon_n for_o example_n that_o equal_a prism_n have_v their_o height_n and_o base_n reciprocal_a that_o like_a prism_n be_v in_o triple_a ratio_fw-la of_o their_o homologous_n side_n use_v this_o proposition_n may_v serve_v to_o find_v the_o centre_n of_o gravity_n of_o parallelepipedon_n and_o to_o demonstrate_v some_o proposition_n of_o the_o thirteen_o and_o fourteen_o book_n of_o euclid_n proposition_n xl._n theorem_fw-la that_o prism_n which_o have_v for_o its_o base_a a_o parallelogram_n double_a to_o the_o triangular_a base_a of_o another_o and_o of_o the_o same_o height_n be_v equal_a thereto_o let_v there_o be_v propose_v two_o triangular_a prism_n abe_n cdg_n of_o the_o same_o height_n and_o let_v be_v take_v for_o the_o base_a of_o the_o first_o the_o parallelogram_n ae_n double_a to_o the_o triangle_n fgc_n the_o base_a of_o the_o second_o prism_n i_o say_v that_o those_o prism_n be_v equal_a imagine_v that_o the_o parallelepipedons_n ah_o give_v be_v draw_v demonstration_n it_o be_v suppose_v that_o the_o base_a ae_n be_v double_a to_o the_o triangle_n fgc_n now_o the_o parallelogram_n gk_n be_v also_o double_a to_o the_o same_o triangle_n by_o the_o 34th_o of_o the_o one_a the_o parallellogram_n ae_n gk_n be_v equal_a and_o consequent_o the_o parallelepipedons_n ah_o give_v which_o have_v their_o base_n and_o height_n equal_a be_v equal_a and_o prism_n which_o be_v the_o half_n by_o the_o 28_o shall_v be_v also_o equal_a the_o end_n of_o the_o eleven_o book_n the_o twelve_o book_n of_o euclid_n element_n euclid_n after_o have_v give_v 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circle_n be_v like_a they_o shall_v be_v in_o the_o same_o ratio_fw-la as_o be_v the_o square_n of_o the_o diameter_n be_o fn_n draw_v the_o line_n bm_n gn_n fh_o ac_fw-la demonstration_n it_o be_v suppose_v that_o the_o polygon_n be_v like_a that_o be_v to_o say_v that_o the_o angle_n b_o and_z g_z be_v equal_a and_o that_o there_o be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o bc_n as_o of_o fg_a to_o gh_v whence_o i_o conclude_v by_o the_o 6_o of_o the_o 6_o that_o the_o triangle_n abc_n fgh_a be_v equiangular_a and_o that_o the_o angles_n acb_n fhg_n be_v equal_a so_o then_o by_o the_o 21_o of_o the_o 3d._n the_o angel_n ambiguity_n fng_v be_v equal_a now_o the_o angel_n abm_n fgn_n be_v in_o a_o semicircle_n be_v right_o by_o the_o 31st_o of_o the_o 3d._n and_o consequent_o the_o triangle_n abm_n fgn_n be_v equiangular_a therefore_o by_o the_o 5_o of_o the_o 6_o there_o shall_v be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o fg_n as_o of_o be_o to_o fn_n and_o by_o the_o 22d_o of_o the_o 11_o if_o one_o describe_v two_o similar_v polygon_n on_o ab_fw-la fg_v which_o be_v those_o propose_v and_o also_o two_o other_o like_a polygon_n be_o and_o fn_n which_o shall_v be_v their_o square_n there_o shall_v be_v the_o same_o ratio_fw-la of_o the_o polygon_n abcde_v to_o the_o polygon_n fgnkl_n as_o of_o the_o square_n of_o be_o to_o the_o square_n of_o fn_n this_o proposition_n be_v necessary_a to_o demonstrate_v the_o follow_a proposition_n lemma_n if_o a_o quantity_n be_v less_o than_o a_o circle_n there_o may_v be_v inscribe_v in_o the_o circle_n a_o regular_a polygon_n great_a than_o that_o quantity_n ii_o lemma_n fig._n ii_o let_v the_o figure_n a_o be_v less_o than_o the_o circle_n b_o there_o may_v be_v inscribe_v in_o the_o circle_n a_o regular_a polygon_n great_a than_o the_o figure_n a._n let_v the_o figure_n g_o be_v the_o difference_n of_o the_o figure_n a_o and_o the_o circle_n in_o such_o sort_n that_o the_o figure_n a_o and_o g_o take_v together_o be_v equal_a to_o the_o circle_n b._n inscribe_v in_o the_o circle_n b_o the_o square_n cdef_n by_o the_o 6_o of_o the_o four_o if_o that_o square_n be_v great_a than_o the_o figure_n a_o we_o shall_v have_v what_o we_o pretend_v to_o if_o it_o be_v less_o divide_v the_o quadrant_n of_o the_o circle_n cd_o de_fw-fr of_o fc_n into_o two_o equal_o in_o the_o point_v h_n i_o k_o l_o in_o such_o sort_n as_o you_o may_v have_v a_o octogon_n and_o if_o that_o octogon_n be_v less_o than_o the_o figure_n a_o subdivide_v the_o ark_n and_o you_o shall_v have_v a_o polygon_n of_o sixteen_o side_n than_o thirty_o two_o of_o sixty_o four_o etc._n etc._n i_o say_v in_o fine_a you_o shall_v have_v a_o polygon_n great_a than_o the_o figure_n a_o that_o be_v to_o say_v a_o polygon_n have_v less_o difference_n from_o the_o circle_n than_o the_o figure_n a_o in_o such_o sort_n that_o the_o difference_n shall_v be_v less_o than_o the_o figure_n g._n demonstration_n the_o square_n inscribe_v be_v more_o than_o the_o one_o half_a of_o the_o circle_n it_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o in_o describe_v the_o octagon_n you_o take_v away_o more_o than_o the_o half_a of_o the_o remainder_n that_o be_v to_o say_v the_o four_o segment_n chd_v die_v ekf_n clf_n for_o the_o triangle_n chd_n be_v the_o half_a of_o the_o rectangle_n co_n by_o the_o 34th_o of_o the_o one_a it_o be_v then_o more_o than_o the_o half_a of_o the_o segment_n chd_v it_o be_v the_o same_o of_o the_o other_o arks._n in_o like_a manner_n in_o describe_v a_o polygon_n of_o sixteen_o side_n you_o take_v away_o more_o than_o one_o half_a of_o that_o which_o remain_v in_o the_o circle_n and_o so_o of_o the_o rest_n you_o
perpendicular_a to_o the_o plane_n of_o the_o equinoctial_a in_o like_a manner_n in_o gnomonic_n we_o demonstrate_v by_o this_o proposition_n that_o the_o equinoctial_a be_v perpendicular_a to_o the_o meridian_n in_o horizontal_n dial_n it_o be_v not_o less_o useful_a in_o other_o treatise_n as_o in_o that_o of_o astrolabe_fw-la or_o in_o the_o section_n of_o stone_n proposition_n v._o theorem_fw-la if_o a_o line_n be_v perpendicular_a to_o three_o other_o line_n which_o cut_v one_o the_o other_o in_o the_o same_o point_n these_o three_o line_n shall_v be_v in_o the_o same_o plane_n if_o the_o line_n ab_fw-la be_v perpendicular_a to_o three_o line_n bc_n bd_o be_v which_o cut_v one_o another_o in_o the_o point_n b_o the_o line_n bc_n bd_o be_v be_v in_o the_o same_o plane_n let_v the_o plane_n ae_n be_v the_o plane_n of_o the_o line_n ab_fw-la be_v and_o let_v cf_n be_v the_o plane_n of_o the_o line_n bc_n bd._n if_o be_v be_v the_o common_a section_n of_o the_o two_o plane_n be_v 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