superficies_n abcd._n who_o length_n be_v take_v by_o the_o line_n ab_fw-la or_o cd_o and_o breadth_n by_o the_o line_n ac_fw-la or_o bd_o and_o by_o reason_n of_o those_o two_o dimension_n a_o superficies_n may_v be_v divide_v two_o way_n namely_o by_o his_o length_n and_o by_o his_o breadth_n but_o not_o by_o thickness_n for_o it_o have_v none_o for_o that_o be_v attribute_v only_o to_o a_o body_n which_o be_v the_o three_o kind_n of_o quantity_n and_o have_v all_o three_o dimension_n length_n breadth_n and_o thickness_n and_o may_v be_v divide_v according_a to_o any_o of_o they_o other_o define_v a_o superficies_n thus_o superficies_n a_o superficies_n be_v the_o term_n or_o end_n of_o a_o body_n as_o a_o line_n be_v the_o end_n and_o term_v of_o a_o superficies_n 6_o extreme_n of_o a_o superficies_n be_v line_n superficies_n as_o the_o end_n limit_n or_o border_n of_o a_o line_n be_v point_n enclose_v the_o line_n so_o be_v line_n the_o limit_n border_n and_o end_n enclose_v a_o 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great_a perfection_n than_o be_v a_o line_n but_o here_o in_o the_o definition_n of_o a_o solid_a or_o body_n euclid_n attribute_v unto_o it_o all_o the_o three_o dimension_n length_n breadth_n and_o thickness_n wherefore_o a_o solid_a be_v the_o most_o perfect_a quantity_n quantity_n which_o want_v no_o dimension_n at_o all_o pass_v a_o line_n by_o two_o dimension_n and_o pass_v a_o superficies_n by_o one_o this_o definition_n of_o a_o solid_a be_v without_o any_o designation_n of_o â—orme_n or_o figure_n easy_o understand_v only_o conceive_v in_o mind_n or_o behold_v with_o the_o eye_n a_o piece_n of_o timber_n or_o stone_n or_o what_o matter_n so_o ever_o else_o who_o dimension_n let_v be_v equal_a or_o unequal_a for_o example_n let_v the_o length_n thereof_o be_v 5._o inch_n the_o breadth_n 4._o and_o the_o thickness_n 2._o if_o the_o dimension_n be_v equal_a the_o reason_n be_v like_a and_o all_o one_o as_o it_o be_v in_o a_o sphere_n and_o in_o â—_o cube_fw-la for_o in_o that_o 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limit_n of_o they_o 2_o a_o right_a line_n be_v then_o erect_v perpendicular_o to_o a_o pl_z 〈…〉_o erficies_n when_o the_o right_a line_n make_v right_a angle_n with_o all_o the_o line_n 〈…〉_o it_o definition_n and_o be_v draw_v upon_o the_o ground_n plain_a superficies_n suppose_v that_o upon_o the_o ground_n plain_a superficies_n cdef_n from_o the_o point_n b_o be_v erect_v a_o right_a line_n namely_o â—a_o so_o that_o let_v the_o point_v a_o be_v a_o loâ—e_n in_o the_o air_n draw_v also_o from_o the_o point_n â—_o in_o the_o plain_a superficies_n cdbf_n as_o many_o right_a line_n as_o you_o listen_v as_o the_o line_n bc_o bd_o â—â—_o bf_n bg_o hk_o bh_o and_o bl_n if_o the_o erect_a line_n basilius_n with_o all_o these_o line_n draw_v in_o the_o superficies_n cdef_n make_v a_o right_a angle_n so_o that_o all_o those_o â—ngles_v aâ—â—_n aâ—d_n aâ—e_n abfâ—_n aâ—g_v aâ—k_n abh_n abl_n and_o so_o of_o other_o be_v right_a angle_n then_o by_o this_o definition_n the_o line_n ab_fw-la iâ—_z a_o line_n â—â—â—cted_v upon_o the_o superficies_n cdef_n it_o be_v also_o call_v common_o a_o perpendicular_a line_n or_o a_o plumb_n line_n unto_o or_o upon_o a_o superficies_n definition_n 3_o a_o plain_a superficies_n be_v then_o upright_o or_o erect_v perpendicular_o to_o a_o plain_a superficies_n when_o all_o the_o right_a line_n draw_v in_o one_o of_o the_o plain_a superficiece_n unto_o the_o common_a section_n of_o those_o two_o plain_a superficiece_n make_v therewith_o right_a angle_n do_v also_o make_v right_a angle_n to_o the_o other_o plain_a superficies_n inclination_n or_o lean_v of_o a_o right_a line_n to_o a_o plain_a superficies_n be_v a_o acute_a angle_n contain_v under_o a_o right_a line_n fall_v from_o a_o point_n above_o to_o the_o plain_a superficies_n and_o under_o a_o other_o right_a line_n from_o the_o low_a end_n of_o the_o say_a line_n let_v down_o draw_v in_o the_o same_o plain_a superficies_n by_o a_o certain_a point_n assign_v where_o a_o right_a line_n from_o the_o first_o point_n above_o to_o the_o same_o plain_a superficies_n fall_v perpendicular_o touch_v in_o this_o three_o definition_n be_v include_v two_o definition_n the_o first_o be_v of_o a_o plain_a superficies_n erect_v perpendicular_o upon_o a_o plain_a superficies_n definition_n the_o second_o be_v of_o the_o inclination_n or_o lean_v of_o a_o right_a line_n unto_o a_o superficies_n of_o the_o first_o take_v this_o example_n suppose_v you_o have_v two_o superâ—icieces_n abcd_o and_o cdef_a of_o which_o let_v the_o superficies_n cdef_n be_v a_o ground_n plain_a superficies_n and_o let_v the_o superficies_n abcd_o be_v erect_v unto_o it_o and_o let_v the_o line_n cd_o be_v a_o common_a term_n or_o intersection_n to_o they_o both_o that_o be_v let_v it_o be_v the_o end_n or_o bind_v of_o either_o of_o they_o part_n &_o be_v draw_v in_o either_o of_o they_o in_o which_o line_n note_n at_o pleasure_n certain_a point_n as_o the_o point_n g_o h._n from_o which_o point_n unto_o the_o line_n cd_o draw_v perpendicular_a line_n in_o the_o superficies_n abcd_o which_o let_v be_v gl_n and_o hk_o which_o fall_v upon_o the_o superficies_n cdef_n if_o they_o cause_v right_a angle_n with_o it_o that_o be_v with_o line_n draw_v in_o it_o from_o the_o same_o point_n g_z and_o h_o as_o if_o the_o angle_n lgm_n or_o the_o angle_n lgn_v contain_v under_o the_o line_n â—g_z draw_v in_o the_o superficies_n erect_v and_o under_o the_o gm_n or_o gn_n draw_v in_o the_o ground_n superficies_n cdef_n lie_v flat_a be_v a_o right_a angle_n then_o by_o this_o definition_n the_o superficies_n abcd_o be_v upright_o or_o erect_v upon_o the_o superficies_n cdef_n it_o be_v also_o common_o call_v a_o superficies_n perpendicular_a upon_o or_o unto_o a_o superficies_n part_n for_o the_o second_o part_n of_o this_o definition_n which_o be_v of_o the_o inclination_n of_o a_o right_a line_n unto_o a_o plain_a superficies_n take_v this_o example_n let_v abcd_o be_v a_o ground_n plain_a superficies_n upon_o which_o from_o a_o point_n be_v a_o loft_n namely_o the_o point_n e_o suppose_v a_o right_a line_n to_o fall_v which_o let_v be_v the_o line_n eglantine_n touch_v the_o plain_a superficies_n abcd_o at_o the_o point_n g._n again_o from_o the_o point_n e_o be_v the_o top_n or_o high_a limit_n and_o end_n of_o the_o incline_a line_n eglantine_n let_v a_o perpendicular_a line_n fall_v unto_o the_o plain_a superficies_n abcd_o which_o let_v be_v the_o line_n of_o and_o let_v fletcher_n be_v the_o point_n where_o of_o touch_v the_o plain_a superficies_n abcd._n then_o from_o the_o point_n of_o the_o fall_n of_o the_o line_n incline_v upon_o the_o superficies_n unto_o the_o point_n of_o the_o fall_n of_o the_o perpendicular_a line_n upon_o the_o same_o superficies_n that_o be_v from_o the_o point_n g_o to_o the_o point_n fletcher_n draw_v a_o right_a line_n gf_o now_o by_o this_o definition_n the_o acute_a angle_n egf_n be_v the_o inclination_n of_o the_o line_n eglantine_n unto_o the_o superficies_n abcd._n because_o it_o be_v contain_v of_o the_o incline_a line_n and_o of_o the_o right_a line_n draw_v in_o the_o superficies_n from_o the_o point_n of_o the_o fall_n of_o the_o line_n incline_v to_o the_o point_n of_o the_o fall_n of_o the_o perpendicular_a line_n which_o angle_n must_v of_o necessity_n be_v a_o acute_a angle_n for_o the_o angle_n efg_o be_v by_o construction_n a_o right_a angle_n and_o three_o angle_n in_o a_o triangle_n be_v equâ—_fw-la 〈…〉_o â—ight_a angle_n wherefore_o the_o other_o two_o angle_n namely_o the_o angle_n egf_n and_o gef_n be_v equâ—_fw-la 〈…〉_o right_n angle_n wherefore_o either_o of_o they_o be_v less_o than_o a_o right_a angle_n wherefore_o the_o angle_n egf_n be_v a_o 〈…〉_o gle_a definition_n 4_o inclination_n of_o a_o plain_a superficies_n to_o a_o plain_a superficies_n be_v a_o acute_a angle_n contain_v under_o the_o right_a line_n which_o be_v draw_v in_o either_o of_o the_o plain_a superficiece_n to_o one_o &_o the_o self_n same_o point_n of_o the_o common_a section_n make_v with_o the_o section_n right_a angle_n suppose_v that_o there_o be_v two_o superficiece_n abcd_o &_o efgh_a and_o let_v the_o superficies_n abcd_o be_v suppose_v to_o be_v erect_v not_o perpendicular_o but_o somewhat_o lean_v and_o incline_v unto_o the_o plain_a superficies_n efgh_o as_o much_o or_o as_o little_a as_o you_o will_v the_o common_a term_n or_o section_n of_o which_o two_o superficiece_n let_v be_v the_o line_n cd_o from_o some_o one_o point_n aâ—_z from_o the_o point_n m_o assign_v in_o the_o common_a section_n of_o the_o two_o superficiece_n namely_o in_o the_o line_n cd_o draw_v a_o perpendicular_a line_n in_o either_o superficies_n in_o the_o ground_n superficies_n efgh_o draw_v the_o line_n mk_o and_o in_o the_o superficies_n abcd_o draw_v the_o line_n ml_o now_o
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wherefore_o by_o this_o definition_n of_o theodosius_n it_o be_v the_o axe_n of_o the_o sphere_n 14_o the_o centre_n of_o a_o sphere_n be_v that_o point_n which_o be_v also_o the_o centre_n of_o the_o semicircle_n definition_n this_o definition_n of_o the_o centre_n of_o a_o sphere_n be_v give_v as_o be_v the_o other_o definition_n of_o the_o axe_n namely_o have_v a_o relation_n to_o the_o definition_n of_o a_o sphere_n here_o give_v of_o euclid_n where_o it_o be_v say_v that_o a_o sphere_n be_v make_v by_o the_o revolution_n of_o a_o semicircle_n who_o diameter_n abide_v fix_v the_o diameter_n of_o a_o circle_n and_o of_o a_o semicrcle_n be_v all_o one_o and_o in_o the_o diameter_n either_o of_o a_o circle_n or_o of_o a_o semicircle_n be_v contain_v the_o centre_n of_o either_o of_o they_o for_o that_o they_o diameter_n of_o each_o ever_o pass_v by_o the_o centre_n now_o say_v euclid_n the_o point_n which_o be_v the_o centre_n of_o the_o semicircle_n by_o who_o motion_n the_o sphere_n be_v describe_v be_v also_o the_o centre_n of_o the_o sphere_n as_o in_o the_o example_n there_o give_v the_o point_n d_o be_v the_o centre_n both_o of_o the_o semicircle_n &_o also_o of_o the_o sphere_n theodosius_n give_v as_o other_o definition_n of_o the_o centre_n of_o a_o sphere_n which_o be_v thus_o sphere_n the_o centre_n of_o a_o sphere_n be_v a_o point_n with_o in_o the_o sphere_n from_o which_o all_o line_n draw_v to_o the_o superficies_n of_o the_o sphere_n be_v equal_a as_o in_o a_o circle_n be_v a_o plain_a figure_n there_o be_v a_o point_n in_o the_o midst_n from_o which_o all_o line_n draw_v to_o the_o circumfrence_n be_v equal_a which_o be_v the_o centre_n of_o the_o circle_n so_o in_o like_a manner_n with_o in_o a_o sphere_n which_o be_v a_o solid_a and_o bodily_a figure_n there_o must_v be_v conceive_v a_o point_n in_o the_o midst_n thereof_o from_o which_o all_o line_n draw_v to_o the_o superficies_n thereof_o be_v equal_a and_o this_o point_n be_v the_o centre_n 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compare_v they_o all_o with_o triangle_n &_o also_o together_o the_o one_o with_o the_o other_o in_o it_o also_o be_v teach_v how_o a_o figure_n of_o any_o form_n may_v be_v change_v into_o a_o figure_n of_o a_o other_o form_n and_o for_o that_o it_o entreat_v of_o these_o most_o common_a and_o general_a thing_n this_o book_n be_v more_o universal_a than_o be_v the_o second_o three_o or_o any_o other_o and_o therefore_o just_o occupy_v the_o first_o place_n in_o order_n as_o that_o without_o which_o the_o other_o book_n of_o eâ—clide_n which_o follow_v and_o also_o the_o work_n of_o other_o which_o have_v write_v in_o geometry_n can_v be_v perceive_v nor_o understand_v and_o forasmuch_o â—s_v all_o the_o demonstration_n and_o proof_n of_o all_o the_o proposition_n in_o this_o whole_a book_n depend_v of_o these_o ground_n and_o principle_n follow_v which_o by_o reason_n of_o their_o plainness_n need_v no_o great_a declaration_n yet_o to_o remove_v all_o be_v it_o never_o so_o little_a 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be_v understand_v by_o the_o definition_n of_o a_o right_a line_n for_o every_o contrary_n be_v well_o manifest_v &_o set_v forth_o by_o his_o contrary_n one_o crooked_a line_n may_v be_v more_o crooked_a than_o a_o other_o and_o from_o one_o point_n to_o a_o other_o may_v be_v draw_v infinite_a crooked_a line_n but_o one_o right_a line_n can_v be_v right_a than_o a_o other_o and_o therefore_o from_o one_o point_n to_o a_o other_o there_o may_v be_v draw_v but_o one_o tight_a line_n as_o by_o figure_n above_o set_n you_o may_v see_v 5_o a_o superficies_n be_v that_o which_o have_v only_a length_n and_o breadth_n superficies_n a_o superficies_n be_v the_o second_o kind_n of_o quantity_n and_o to_o it_o be_v attribute_v two_o dimension_n namely_o length_n and_o breadth_n way_n as_o in_o the_o
be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n wherefore_o from_o the_o one_o to_o the_o other_o there_o be_v some_o short_a draught_n which_o be_v a_o right_a line_n likewise_o any_o two_o right_a line_n howsoever_o they_o be_v set_v be_v imagine_v to_o be_v in_o one_o superficies_n and_o therefore_o from_o any_o one_o line_n to_o any_o one_o line_n may_v be_v draw_v a_o superficies_n 2_o to_o produce_v a_o right_a line_n finite_a straight_o forth_o continual_o as_o to_o draw_v in_o length_n continual_o the_o right_a line_n ab_fw-la who_o will_v not_o grant_v for_o there_o be_v no_o magnitude_n so_o great_a but_o that_o there_o may_v be_v a_o great_a nor_o any_o so_o little_a but_o that_o there_o may_v be_v a_o less_o and_o a_o line_n be_v a_o draught_n from_o one_o point_n to_o a_o other_o therefore_o from_o the_o point_n b_o which_o be_v the_o end_n of_o the_o line_n ab_fw-la may_v be_v draw_v a_o line_n to_o some_o other_o point_n as_o to_o the_o point_n c_o and_o from_o that_o to_o a_o otherâ—_n and_o so_o infinite_o 3_o upon_o any_o centre_n and_o at_o any_o distance_n â—o_z describe_v a_o circle_n a_o plain_a superficies_n may_v in_o compass_n be_v extend_v infinite_o as_o from_o any_o point_n to_o any_o point_n may_v be_v draw_v a_o right_a line_n by_o reason_n whereof_o it_o come_v to_o pass_v that_o a_o circle_n may_v be_v describe_v upon_o any_o centre_n and_o at_o any_o space_n or_o distance_n as_o upon_o the_o centre_n a_o and_o upon_o the_o space_n ab_fw-la you_o may_v describe_v the_o circle_n bc_o &_o upon_o the_o same_o centre_n upon_o the_o distance_n ad_fw-la you_o may_v describe_v the_o circle_n de_fw-fr or_o upon_o the_o same_o centre_n a_o according_a to_o the_o distance_n of_o you_o may_v describe_v the_o circle_n fg_o and_o so_o infinite_o extend_v your_o space_n 4_o all_o right_a angle_n be_v equal_a the_o one_o to_o the_o other_o this_o petition_n be_v most_o plain_a and_o offer_v itself_o even_o to_o the_o sense_n for_o as_o much_o as_o a_o right_a angle_n be_v 〈…〉_o riâ—ht_v line_n fall_v perpendicular_o upon_o a_o other_o and_o no_o one_o line_n can_v fall_v more_o perpendicular_o upon_o a_o line_n then_o a_o otherâ—_n therefore_o no_o one_o right_a angle_n can_v be_v great_a 〈◊〉_d a_o otherâ—_n neither_o dâ—_z the_o length_n or_o shortness_n of_o the_o line_n alter_v the_o greatness_n of_o the_o angle_n for_o in_o the_o example_n the_o right_a angle_n abc_n though_o it_o be_v make_v of_o much_o long_a line_n than_o the_o right_a angle_n def_n who_o line_n be_v much_o short_a yet_o be_v that_o angle_n no_o great_a than_o the_o other_o for_o if_o you_o set_v the_o point_n eâ—ust_v â—ust_z upon_o the_o point_n bâ—_n â—_z then_o shall_v the_o line_n ed_z even_o and_o just_o fall_v upon_o the_o line_n abâ—_n â—_z and_o the_o line_n of_o shall_v also_o fall_v equal_o upon_o the_o line_n bc_o and_o so_o shall_v the_o angle_n def_n be_v equal_a to_o the_o angle_n abc_n for_o that_o the_o line_n which_o cause_v they_o be_v of_o like_a inclination_n it_o may_v evident_o also_o be_v see_v at_o the_o centre_n of_o a_o circle_n for_o if_o you_o draw_v in_o a_o circle_n two_o diameter_n the_o one_o cut_v the_o other_o in_o the_o centre_n by_o right_a angle_n you_o shall_v divide_v the_o circle_n into_o four_o equal_a part_n of_o which_o each_o contain_v one_o right_a angle_n so_o be_v all_o the_o four_o right_a angle_n about_o the_o centre_n of_o the_o circle_n equal_a 5_o when_o a_o right_a line_n fall_v upon_o too_o right_a line_n do_v make_v â—n_o one_o &_o the_o self_n same_o side_n the_o two_o inward_a angle_n less_o than_o two_o right_a angle_n then_o shall_v these_o two_o right_a line_n be_v produce_v 〈◊〉_d length_n concur_v on_o that_o part_n in_o which_o be_v the_o two_o angle_n less_o than_o two_o right_a angle_n as_o if_o the_o right_a line_n ab_fw-la fall_v upon_o two_o right_a line_n namely_o cd_o and_o of_o so_o that_o it_o make_v the_o two_o inward_a angle_n on_o the_o one_o side_n as_o the_o angle_n dhi_fw-mi &_o fih_o less_o than_o two_o right_a angle_n as_o in_o the_o example_n they_o do_v the_o say_v two_o line_n cd_o and_o of_o be_v draw_v forth_o in_o length_n on_o that_o part_n whereon_o the_o two_o angle_n be_v lessâ—_n 〈◊〉_d too_o right_a angleâ—_n consist_v shall_v 〈◊〉_d length_n concur_v and_o meet_v together_o as_o in_o the_o point_n d_o as_o it_o be_v easy_a to_o see_v for_o the_o part_n of_o the_o line_n towards_o df_o be_v more_o incline_v the_o one_o to_o the_o other_o than_o the_o part_n of_o the_o line_n towards_o ce_fw-fr be_v wherefore_o the_o more_o these_o part_n be_v produce_v the_o more_o they_o shall_v approach_v near_o and_o near_o till_o at_o length_n they_o shall_v mete_v in_o one_o point_n contrariwise_o the_o same_o line_n draw_v in_o length_n on_o the_o other_o side_n for_o that_o the_o angle_n on_o that_o side_n namely_o the_o angle_n chb_n and_o the_o angle_n eia_n be_v great_a than_o two_o right_a angle_n so_o much_o as_o the_o other_o two_o angle_n be_v less_o thâ—n_n two_o right_a angle_n shall_v never_o mete_v but_o the_o further_o they_o be_v draw_v the_o further_o they_o shall_v be_v distant_a the_o one_o from_o the_o other_o 6_o that_o two_o right_a line_n include_v not_o a_o superficies_n if_o the_o line_n ab_fw-la and_o ac_fw-la be_v right_a line_n shall_v enclose_v a_o superficies_n they_o must_v of_o necessity_n be_v join_v together_o at_o both_o the_o end_n and_o the_o superficies_n must_v be_v between_o they_o join_v they_o on_o the_o one_o side_n together_o in_o the_o point_n a_o and_o imagine_v the_o point_n b_o to_o be_v draw_v to_o the_o point_n c_o so_fw-mi shall_v the_o line_n ab_fw-la fall_v on_o the_o line_n ac_fw-la and_o cover_v it_o and_o so_o be_v all_o one_o with_o it_o and_o never_o enclose_v a_o space_n or_o superficies_n common_a sentence_n 1_o thing_n equal_a to_o one_o and_o the_o self_n same_o thing_n be_v equal_a also_o the_o one_o to_o the_o other_o be_v after_o definition_n and_o petition_n now_o be_v set_v common_a sentence_n which_o be_v the_o three_o and_o last_o kind_n of_o principle_n which_o be_v certain_a general_a proposition_n common_o know_v of_o all_o man_n of_o themselves_o most_o manifest_a &_o clear_a &_o therefore_o be_v call_v also_o dignity_n not_o able_a to_o be_v deny_v of_o any_o petition_n also_o be_v very_o manifest_a but_o not_o so_o full_o as_o be_v the_o common_a sentence_n sentence_n and_o therefore_o be_v require_v or_o desire_v to_o be_v grant_v petition_n also_o be_v more_o peculiar_a to_o the_o art_n whereof_o they_o be_v as_o those_o before_o put_v be_v proper_a to_o geometry_n but_o common_a sentence_n be_v general_a to_o all_o thing_n whereunto_o they_o can_v be_v apply_v again_o petition_n consist_v in_o action_n or_o do_v of_o somewhat_o most_o easy_a to_o be_v do_v but_o common_a sentence_n consist_v in_o consideration_n of_o mind_n but_o yet_o of_o such_o thing_n which_o be_v most_o easy_a to_o be_v understand_v as_o be_v that_o before_o set_v as_o if_o the_o line_n a_o be_v equal_a to_o the_o line_n b_o and_o if_o the_o line_n c_o be_n also_o equal_a to_o the_o line_n b_o then_o of_o necessity_n the_o line_n a_o and_o c_o shall_v be_v equal_a the_o one_o to_o the_o other_o so_o be_v it_o in_o all_o superâ—iciesses_n angle_n &_o number_n &_o in_o all_o other_o thing_n of_o one_o kind_n that_o may_v be_v compare_v together_o 2_o and_o if_o you_o add_v equal_a thing_n to_o equal_a thing_n the_o whole_a shall_v be_v equal_v as_o if_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o &_o to_o the_o line_n ab_fw-la be_v add_v the_o line_n be_v &_o to_o the_o line_n cd_o be_v add_v also_o a_o other_o line_n df_o be_v equal_a to_o the_o line_n be_v so_o that_o two_o equal_a line_n namely_o be_v and_o df_o be_v add_v to_o two_o equal_a line_n ab_fw-la &_o cd_o then_o shall_v the_o whole_a line_n ae_n be_v equal_a to_o the_o whole_a line_n cf_o and_o so_o of_o all_o quantity_n general_o 3_o and_o if_o from_o equal_a thing_n you_o take_v away_o equal_a thing_n the_o thing_n remain_v shall_v be_v equal_a as_o if_o from_o the_o two_o line_n ab_fw-la and_o cd_o be_v equal_a you_o take_v away_o two_o equal_a line_n namely_o ebb_n and_o fd_o then_o may_v you_o conclude_v by_o this_o common_a sentence_n that_o the_o part_n remain_v namely_o ae_n and_o cf_o be_v equal_a the_o one_o to_o the_o other_o and_o so_o of_o all_o other_o quantity_n 4_o and_o if_o from_o unequal_a thing_n you_o take_v away_o equal_a thing_n the_o thing_n which_o remain_v shall_v be_v unequal_a as_o if_o the_o line_n ab_fw-la and_o cd_o be_v unequal_a
draw_v by_o the_o 11._o of_o the_o firsâ—_o a_o perpendicular_a line_n ab_fw-la and_o let_v ab_fw-la be_v a_o rational_a line_n and_o make_v perfectâ—_n the_o parallelogram_n bc._n wherefore_o bg_o be_v irrational_a by_o that_o which_o be_v declare_v and_o prove_v in_o manner_n of_o a_o assumpt_n in_o the_o end_n of_o the_o demonstration_n of_o the_o 38_o and_o the_o line_n that_o contain_v it_o iâ—_z power_n be_v also_o irrational_a let_v the_o line_n cd_o contain_v in_o power_n the_o superficies_n bc._n wherefore_o cd_o be_v irrational_a &_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o for_o the_o square_n of_o the_o line_n cd_o apply_v to_o a_o rational_a line_n namely_o ab_fw-la make_v the_o breadth_n a_o medial_a line_n namely_o ac_fw-la but_o the_o square_n of_o none_o of_o the_o foresay_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o medial_a line_n again_o make_v perfect_v the_o parallelogram_n ed._n wherefore_o the_o parallelogram_n ed_z be_v also_o irrational_a by_o the_o say_v assumpt_v in_o the_o end_n of_o the_o 98._o his_o demonstration_n brieâ—ly_o prove_v and_o the_o line_n which_o contain_v it_o in_o power_n be_v irrational_o let_v the_o line_n which_o contain_v it_o in_o power_n be_v df._n wherefore_o df_o be_v irrational_a and_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o the_o foresay_a irrational_a line_n for_o the_o square_n of_o none_o of_o the_o foresay_a irrational_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n the_o line_n cd_o wherefore_o of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 92._o theorem_a the_o 116._o proposition_n now_o let_v we_o prove_v that_o in_o square_a figure_n the_o diameter_n be_v incommensurable_a in_o length_n to_o the_o side_n svppose_v that_o abcd_o be_v a_o square_a and_o let_v the_o diameter_n thereof_o be_v ac_fw-la then_o i_o say_v that_o the_o diameter_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o side_n ab_fw-la for_o if_o it_o be_v possible_a impossibility_n let_v it_o be_v commensurable_a in_o length_n i_o say_v that_o they_o this_o will_v follow_v that_o one_o and_o the_o self_n same_o number_n shall_v be_v both_o a_o even_a number_n &_o a_o odd_a number_n it_o be_v manifest_a by_o the_o 47._o of_o the_o first_o that_o the_o square_a of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o for_o that_o the_o line_n ac_fw-la be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o supposition_n therefore_o the_o line_n ac_fw-la have_v unto_o the_o line_n ab_fw-la that_o proportion_n that_o a_o number_n have_v to_o a_o number_n by_o the_o 5._o of_o the_o ten_o let_v the_o line_n ac_fw-la have_v unto_o the_o line_n ab_fw-la that_o proportion_n that_o the_o number_n of_o have_v to_o the_o number_n g._n and_o let_v of_o and_o g_z be_v the_o least_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o they_o wherefore_o of_o be_v not_o unity_n for_o if_o of_o be_v unity_n and_o it_o have_v to_o the_o number_n g_o that_o proportion_n that_o the_o line_n ac_fw-la have_v to_o the_o line_n ab_fw-la and_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n ab_fw-la wherefore_o unity_n of_o be_v great_a than_o the_o number_n g_o which_o be_v impossible_a wherefore_o fe_o be_v not_o unity_n wherefore_o it_o be_v a_o number_n and_o for_o that_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v to_o the_o square_n of_o the_o line_n ab_fw-la so_o be_v the_o square_a number_n of_o the_o number_n of_o to_o the_o square_a number_n of_o the_o number_n g_o for_o in_o each_o be_v the_o proportion_n of_o their_o side_n double_v by_o the_o corollary_n of_o the_o 20_o of_o the_o six_o and_o 11._o of_o the_o eight_o and_o the_o proportion_n of_o the_o line_n ac_fw-la to_o the_o line_n ab_fw-la double_v be_v equal_a to_o the_o proportion_n of_o the_o number_n of_o to_o the_o number_n g_o double_a for_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n ab_fw-la so_o be_v the_o number_n of_o to_o the_o number_n g._n but_o the_o square_n of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o square_a number_n produce_v of_o the_o number_n of_o be_v double_a to_o the_o square_a number_n produce_v of_o the_o number_n g._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v a_o even_a number_n wherefore_o of_o be_v also_o a_o even_a number_n for_o if_o of_o be_v a_o odd_a number_n the_o square_a number_n also_o produce_v of_o it_o shall_v by_o the_o 23._o and_o 29._o of_o the_o nine_o be_v a_o odd_a number_n for_o if_o odd_a number_n how_o many_o soever_o be_v add_v together_o and_o if_o the_o multitude_n of_o they_o be_v odd_a the_o whole_a also_o shall_v be_v odd_a wherefore_o of_o be_v a_o even_a number_n divide_v the_o number_n of_o into_o two_o equal_a part_n in_o h._n and_o forasmuch_o as_o the_o number_n of_o and_o g_z be_v the_o least_a number_n in_o that_o proportion_n therefore_o by_o the_o 24._o of_o the_o seven_o they_o be_v prime_a number_n the_o one_o to_o the_o other_o and_o of_o be_v a_o even_a number_n wherefore_o g_z be_v a_o odd_a number_n for_o if_o g_z be_v a_o even_a number_n the_o number_n two_z shall_v measure_v both_o the_o number_n of_o and_o the_o number_n g_z for_o every_o even_a number_n have_v a_o half_a part_n by_o the_o definition_n but_o these_o number_n of_o &_o g_o be_v prime_a the_o one_o to_o the_o other_o wherefore_o it_o be_v impossible_a that_o they_o shall_v be_v measure_v by_o two_o or_o by_o any_o other_o number_n beside_o unity_n wherefore_o g_z be_v a_o odd_a number_n and_o forasmuch_o as_o the_o number_n of_o be_v double_a to_o the_o number_n eh_o therefore_o the_o square_a number_n produce_v of_o of_o be_v quadruple_a to_o the_o square_a number_n produce_v of_o eh_o and_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o g_o be_v double_a to_o the_o square_a number_n produce_v of_o eh_o wherefore_o the_o square_a number_n produce_v of_o g_o be_v a_o even_a number_n wherefore_o also_o by_o those_o thing_n which_o have_v be_v before_o speak_v the_o number_n g_z be_v a_o even_a number_n but_o it_o be_v prove_v that_o it_o be_v a_o odd_a number_n which_o be_v impossible_a wherefore_o the_o line_n ac_fw-la be_v not_o commensurable_a in_o length_n to_o the_o line_n ab_fw-la wherefore_o it_o be_v incommensurable_a an_o other_o demonstration_n we_o may_v by_o a_o other_o demonstration_n prove_v that_o the_o diameter_n of_o a_o square_n be_v incommensurable_a to_o the_o side_n thereof_o impossibility_n suppose_v that_o there_o be_v a_o square_a who_o diameter_n let_v be_v a_o and_o let_v the_o side_n thereof_o be_v b._n then_o i_o say_v that_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n b._n for_o if_o it_o be_v possible_a let_v it_o be_v commensurable_a in_o length_n and_o again_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o let_v the_o number_n of_o be_v to_o the_o number_n g_o and_o let_v they_o be_v the_o least_o that_o have_v one_o and_o the_o same_o proportion_n with_o they_o wherefore_o the_o number_n of_o and_o g_z be_v prime_a the_o one_o to_o the_o other_o first_o i_o say_v that_o g_z be_v not_o unity_n for_o if_o it_o be_v possible_a let_v it_o be_v unity_n and_o for_o that_o the_o square_a of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o as_o the_o square_a number_n produce_v of_o of_o be_v to_o the_o square_a number_n produce_v of_o g_z as_o it_o be_v prove_v in_o the_o â—ormer_a demonstration_n but_o the_o square_n of_o the_o line_n a_o be_v double_a to_o the_o square_n of_o the_o line_n b._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n and_o by_o your_o supposition_n g_z be_v unity_n wherefore_o the_o square_a number_n produce_v of_o of_o be_v the_o number_n two_o which_o be_v impossible_a wherefore_o g_z be_v not_o unity_n wherefore_o it_o be_v a_o number_n and_o for_o that_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a number_n produce_v of_o of_o to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o g._n
the_o word_n of_o psellus_n in_o his_o epitome_n of_o geometry_n where_o he_o entreat_v of_o the_o production_n and_o constitution_n of_o these_o body_n his_o word_n be_v these_o psellus_n all_o rectiline_v figure_n be_v erect_v upon_o their_o plain_n or_o base_n by_o right_a angle_n make_v prism_n who_o perceive_v not_o but_o that_o a_o pentagon_n erect_v upon_o his_o base_a of_o â—ive_a side_n make_v by_o his_o motion_n a_o side_v column_n of_o five_o side_n likewise_o a_o hexagon_n erect_v at_o right_a angle_n produce_v a_o column_n have_v six_o side_n and_o so_o of_o all_o other_o rectillne_v figure_n all_o which_o solid_n or_o body_n so_o produce_v whether_o they_o be_v side_v column_n or_o parallelipipedon_n be_v here_o in_o most_o plain_a word_n of_o this_o excellent_a and_o ancient_a greek_a author_n psellus_n call_v prism_n wherefore_o if_o the_o definition_n of_o a_o prism_n give_v of_o euclid_n shall_v extend_v itself_o so_o large_o as_o flussas_n imagine_v and_o shall_v include_v such_o figure_n or_o body_n as_o he_o note_v he_o ought_v not_o yet_o for_o all_o that_o so_o much_o to_o be_v offend_v and_o so_o narrow_o to_o have_v seek_v fault_n for_o euclid_n in_o so_o define_v may_v have_v that_o meaning_n &_o sense_n of_o a_o prism_n which_o psellus_n have_v so_o you_o see_v that_o euclid_n may_v be_v defend_v either_o of_o these_o two_o way_n either_o by_o that_o that_o the_o definition_n extend_v not_o to_o these_o figure_n and_o so_o not_o to_o be_v over_o general_n nor_o stretch_v far_o than_o it_o ought_v or_o else_o by_o that_o that_o if_o it_o shall_v stretch_v so_o far_o it_o be_v not_o so_o heinous_a for_o that_o as_o you_o see_v many_o have_v takâ—_n it_o in_o that_o sense_n in_o deed_n common_o a_o prism_n be_v take_v in_o that_o signification_n and_o meaning_n in_o which_o campaâ—â—â—_n flussas_n and_o other_o take_v it_o in_o which_o sense_n it_o seem_v also_o that_o in_o diverse_a proposition_n in_o these_o book_n follow_v it_o ought_v of_o necessity_n to_o be_v take_v 12_o a_o sphere_n be_v a_o figure_n which_o be_v make_v definition_n when_o the_o diameter_n of_o a_o semicircle_n abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v unto_o the_o self_n same_o place_n from_o whence_o it_o begin_v to_o be_v move_v to_o the_o end_n we_o may_v full_o and_o perfect_o understand_v this_o definition_n how_o a_o sphere_n be_v produce_v of_o the_o motion_n of_o a_o semicircle_n it_o shall_v be_v expedient_a to_o consider_v how_o quantity_n mathematical_o be_v by_o imagination_n conceive_v to_o be_v produce_v by_o flow_v and_o motion_n as_o be_v somewhat_o touch_v in_o the_o begin_n of_o the_o first_o book_n ever_o the_o less_o quantity_n by_o his_o motion_n bring_v forâ—h_v the_o quantity_n next_o above_o it_o as_o a_o point_n move_v flow_a or_o glide_v bring_v forth_o a_o line_n which_o be_v the_o first_o quantity_n and_o next_o to_o a_o point_n a_o line_n move_v produce_v a_o superficies_n which_o be_v the_o second_o quantity_n and_o next_o unto_o a_o line_n and_o last_o of_o all_o a_o superficies_n move_v bring_v forth_o a_o solid_a or_o body_n which_o be_v the_o three_o &_o last_o quantity_n these_o thing_n well_o mark_v it_o shall_v not_o be_v very_o hard_a to_o attain_v to_o the_o right_a understanding_n of_o this_o definition_n upon_o the_o line_n ab_fw-la be_v the_o diameter_n describe_v a_o semicircle_n acb_o who_o centre_n let_v be_v d_o the_o diameter_n ab_fw-la be_v six_v on_o his_o end_n orâ—_n point_n imagine_v the_o whole_a superficies_n of_o the_o semicircle_n to_o move_v round_o from_o some_o one_o point_n assign_v till_o it_o return_v to_o the_o same_o point_n again_o so_o shall_v it_o produce_v a_o perfect_a sphere_n or_o globe_n the_o form_n whereof_o you_o see_v in_o a_o ball_n or_o bowl_n and_o it_o be_v full_o round_a and_o solid_a for_o that_o it_o be_v describe_v of_o a_o semicircle_n which_o be_v perfect_o round_a as_o our_o country_n man_n johannes_n de_fw-fr sacro_fw-la busco_n in_o his_o book_n of_o the_o sphere_n busco_n of_o this_o definition_n which_o he_o take_v out_o of_o euclid_n do_v well_o collect_n but_o it_o be_v to_o be_v note_v and_o take_v heed_n of_o that_o none_o be_v deceive_v by_o the_o definition_n of_o a_o sphere_n give_v by_o johannes_n de_fw-fr sacro_fw-la busco_n a_o sphere_n say_v he_o be_v the_o passage_n or_o move_v of_o the_o circumference_n of_o a_o semicircle_n till_o it_o return_v unto_o the_o place_n where_o it_o begin_v which_o agree_v not_o with_o euclid_n euclid_n plain_o say_v that_o a_o sphere_n be_v the_o passage_n or_o motion_n of_o a_o semicircle_n and_o not_o the_o passage_n or_o motion_n of_o the_o circumference_n of_o a_o semicircle_n neither_o can_v it_o be_v true_a that_o the_o circumference_n of_o a_o semicircle_n which_o be_v a_o line_n shall_v describe_v a_o body_n it_o be_v before_o note_v that_o every_o quantity_n move_v describe_v and_o produce_v the_o quantity_n next_o unto_o it_o wherefore_o a_o line_n move_v can_v not_o bring_v forth_o a_o body_n but_o a_o superficies_n only_o as_o if_o you_o imagine_v a_o right_a line_n fasten_v at_o one_o of_o his_o end_n to_o move_v about_o from_o some_o one_o point_n till_o it_o return_v to_o the_o same_o again_o it_o shall_v describe_v a_o plain_a superficies_n namely_o a_o circle_n so_o also_o if_o you_o likewise_o conceive_v of_o a_o crooked_a line_n such_o as_o be_v the_o circumference_n of_o a_o semicircle_n that_o his_o diameter_n fasten_v on_o both_o the_o end_n it_o shall_v move_v from_o a_o point_n assign_v till_o it_o return_v to_o the_o same_o again_o it_o shall_v describe_v &_o produce_v a_o â—ound_a superficies_n only_o which_o be_v the_o superficies_n and_o limit_n of_o the_o sphere_n and_o shall_v not_o produce_v the_o body_n and_o solidity_n of_o the_o sphere_n but_o the_o whole_a semicircle_n which_o be_v a_o superficies_n by_o his_o motion_n as_o be_v before_o say_v produce_v a_o body_n that_o be_v a_o perfect_a sphere_n so_o see_v you_o the_o error_n of_o this_o definition_n of_o the_o author_n of_o the_o sphere_n which_o whether_o it_o happen_v by_o the_o author_n himself_o which_o i_o think_v not_o or_o that_o that_o particle_n be_v thrust_v in_o by_o some_o one_o after_o he_o which_o be_v more_o likely_a it_o it_o not_o certain_a but_o it_o be_v certain_a that_o it_o be_v unapt_o put_v in_o and_o make_v a_o untrue_a definition_n which_o thing_n be_v not_o here_o speak_v any_o thing_n to_o derogate_v the_o author_n of_o the_o book_n which_o assure_o be_v a_o man_n of_o excellent_a knowledge_v neither_o to_o the_o hindrance_n or_o diminish_n of_o the_o worthiness_n of_o the_o book_n which_o undoubted_o be_v a_o very_a necessary_a book_n than_o which_o i_o know_v none_o more_o mere_a to_o be_v teach_v and_o read_v in_o school_n touch_v the_o ground_n and_o principle_n of_o astronomy_n and_o geography_n but_o only_o to_o admonish_v the_o young_a and_o unskilful_a reader_n of_o not_o fall_v into_o error_n sphere_n theodosius_n in_o his_o book_n de_fw-fr sphericis_n a_o book_n very_o necessary_a for_o all_o those_o which_o will_v see_v the_o ground_n and_o principle_n of_o geometry_n and_o astronomy_n which_o also_o i_o have_v translate_v into_o our_o vulgar_a tongue_n ready_a to_o the_o press_n define_v a_o sphere_n after_o this_o manner_n a_o sphere_n be_v a_o solid_a or_o body_n contain_v under_o one_o superficies_n in_o the_o middle_n whereof_o there_o be_v a_o point_n from_o which_o all_o line_n draw_v to_o the_o circumference_n be_v equal_a this_o definition_n of_o theodosius_n be_v more_o essential_a and_o natural_a then_o be_v the_o other_o give_v by_o euclid_n the_o other_o do_v not_o so_o much_o declare_v the_o inward_a nature_n and_o substance_n of_o a_o sphere_n as_o it_o show_v the_o industry_n and_o knowledge_n of_o the_o produce_v of_o a_o sphere_n and_o therefore_o be_v a_o causal_n definition_n give_v by_o the_o cause_n efficient_a or_o rather_o a_o description_n then_o a_o definition_n but_o this_o definition_n be_v very_o essential_a declare_v the_o nature_n and_o substance_n of_o a_o sphere_n as_o if_o a_o circle_n shall_v be_v thus_o define_v as_o it_o well_o may_v a_o circle_n be_v the_o passage_n or_o move_v of_o a_o line_n from_o a_o point_n till_o it_o return_v to_o the_o same_o point_n against_o it_o be_v a_o causal_n definition_n show_v the_o efficient_a cause_n whereof_o a_o circle_n be_v produce_v namely_o of_o the_o motion_n of_o a_o line_n and_o it_o be_v a_o very_a good_a description_n full_o show_v what_o a_o circle_n be_v such_o like_a description_n be_v the_o definition_n of_o a_o sphere_n give_v oâ—_n euclideâ—_n â—_z by_o the_o motion_n of_o a_o semicircle_n but_o when_o a_o circle_n be_v define_v to_o be_v a_o plain_a superficies_n
same_o superficies_n wherefore_o these_o right_a line_n ab_fw-la bd_o and_o dc_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o either_o of_o these_o angle_n abdella_n and_o bdc_o be_v a_o right_a angle_n by_o supposition_n wherefore_o by_o the_o 28._o of_o the_o first_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v here_o for_o the_o better_a understanding_n of_o this_o 6._o proposition_n i_o have_v describe_v a_o other_o figure_n as_o touch_v which_o if_o you_o erect_v the_o superficies_n abdella_n perpendicular_o to_o the_o superficies_n bde_v and_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n a_o to_o the_o point_n e_o if_o you_o will_v you_o may_v extend_v a_o thread_n from_o the_o say_a point_n a_o to_o the_o point_n e_o and_o so_o compare_v it_o with_o the_o demonstration_n it_o will_v make_v both_o the_o proposition_n and_o also_o the_o demonstration_n most_o clear_a unto_o you_o ¶_o a_o other_o demonstration_n of_o the_o six_o proposition_n by_o m._n dee_n suppose_v that_o the_o two_o right_a line_n ab_fw-la &_o cd_o be_v perpendicular_o erect_v to_o one_o &_o the_o same_o plain_a superficies_n namely_o the_o plain_a superficies_n open_v then_o i_o say_v that_o â—â—_o and_o cd_o be_v parallel_n let_v the_o end_n point_v of_o the_o right_a line_n ab_fw-la and_o cd_o which_o touch_v the_o plain_a superficies_n oâ—_n be_v the_o point_n â—_o and_o d_o fronâ—_n to_o d_o let_v a_o straight_a line_n be_v draw_v by_o the_o first_o petition_n and_o by_o the_o second_o petition_n let_v the_o straight_a line_n â—d_a be_v extend_v as_o to_o the_o point_n m_o &_o n._n now_o forasmuch_o as_o the_o right_a line_n ab_fw-la from_o the_o point_n â—_o produce_v do_v cut_v the_o line_n mn_v by_o construction_n therefore_o by_o the_o second_o proposition_n of_o this_o eleven_o book_n the_o right_a line_n ab_fw-la &_o mn_o be_v in_o one_o plainâ—_n superficies_n which_o let_v be_v qr_o cut_v the_o superficies_n open_v in_o the_o right_a line_n mn_o by_o the_o same_o mean_n may_v we_o conclude_v the_o right_a line_n cd_o to_o be_v in_o one_o plain_a superficies_n with_o the_o right_a line_n mn_o but_o the_o right_a line_n mn_o by_o supposition_n be_v in_o the_o plain_a superficies_n qr_o wherefore_o cd_o be_v in_o the_o plain_a superficies_n qr_o and_o aâ—_z the_o right_a line_n be_v prove_v to_o be_v in_o the_o same_o plain_a superficies_n qr_o therefore_o ab_fw-la and_o cd_o be_v in_o one_o plain_a superficies_n namely_o qr_o and_o forasmuch_o as_o the_o line_n aâ—_z and_o cd_o by_o supposition_n be_v perpendicular_a upon_o the_o plain_a superficies_n open_v therefore_o by_o the_o second_o definition_n of_o this_o book_n with_o all_o the_o right_a line_n draw_v in_o the_o superficies_n open_v and_o touch_v ab_fw-la and_o cd_o the_o same_o perpendiculars_n aâ—_z and_o cd_o do_v make_v right_a angle_n but_o by_o construction_n mn_v be_v draw_v in_o the_o plain_a superficies_n open_a touch_v the_o perpendicular_n ab_fw-la and_o cd_o at_o the_o point_n â—_o and_o d._n therefore_o the_o perpendicular_n aâ—_z and_o cd_o make_v with_o the_o right_a line_n mn_v two_o right_a angle_n namely_o abn_n and_o cdm_o and_o mn_v the_o right_a line_n be_v prove_v to_o be_v in_o the_o one_o and_o the_o same_o plain_a superficies_n with_o the_o right_a line_n ab_fw-la &_o cd_o namely_o in_o the_o plain_a superficies_n qr_o wherefore_o by_o the_o second_o part_n of_o the_o 28._o proposition_n of_o the_o first_o book_n the_o right_a lineâ—_z ab_fw-la and_o cd_o be_v parallel_n if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o m._n dee_n hereby_o it_o be_v evident_a that_o any_o two_o right_a line_n perpendicular_o erect_v to_o one_o and_o the_o self_n same_o plain_a superficies_n be_v also_o themselves_o in_o one_o and_o the_o same_o plain_a superficies_n which_o be_v likewise_o perpendicular_o erect_v to_o the_o same_o plain_a superficies_n unto_o which_o the_o two_o right_a line_n be_v perpendicular_a the_o first_o part_n hereof_o be_v prove_v by_o the_o former_a construction_n and_o demonstration_n that_o the_o right_a line_n ab_fw-la and_o cd_o be_v in_o one_o and_o the_o same_o plain_a superficies_n qâ—_n the_o second_o part_n be_v also_o manifest_a that_o be_v that_o the_o plain_a superficies_n qr_o be_v perpendicular_o erect_v upon_o the_o plain_a superficies_n open_v for_o that_o aâ—_z and_o cd_o be_v in_o the_o plain_a superficies_n qr_o be_v by_o supposition_n perpendicular_a to_o the_o plain_a superficies_n open_v wherefore_o by_o the_o three_o definition_n of_o this_o book_n qr_o be_v perpendicular_o erect_v to_o or_o upon_o open_a which_o be_v require_v to_o be_v prove_v io._n do_v you_o his_o advice_n upon_o the_o assumpt_n of_o the_o 6._o as_o concern_v the_o make_n of_o the_o line_n de_fw-fr equal_a to_o the_o right_a line_n ab_fw-la very_o the_o second_o of_o the_o first_o without_o some_o far_a consideration_n be_v not_o proper_o enough_o allege_v and_o no_o wonder_n it_o be_v for_o that_o in_o the_o former_a bookeâ—_n whatsoeâ—â—â—â—aâ—h_v of_o line_n be_v speak_v the_o same_o have_v always_o be_v imagine_v to_o be_v in_o one_o only_a plain_a superficies_n consider_v or_o execute_v but_o here_o the_o perpendicular_a line_n ab_fw-la be_v not_o in_o the_o same_o plaint_n superficies_n that_o the_o right_a line_n db_o be_v therefore_o some_o other_o help_n must_v be_v put_v into_o the_o hand_n of_o young_a beginner_n how_o to_o bring_v this_o problem_n to_o execution_n which_o be_v this_o most_o plain_a and_o brief_a understand_v that_o bd_o the_o right_n line_n be_v the_o common_a section_n of_o the_o plain_a superficies_n wherein_o the_o perpendicular_n ab_fw-la and_o cd_o be_v &_o of_o the_o other_o plain_a superficies_n to_o which_o they_o be_v perpendicular_n the_o first_o of_o these_o in_o my_o former_a demonstration_n of_o the_o 6_o â—_o i_o note_v by_o the_o plain_a superficies_n qr_o and_o the_o other_o i_o note_v by_o the_o plain_a superficies_n open_v wherefore_o bd_o be_v a_o right_a line_n common_a to_o both_o the_o plain_a supâ—rficieces_n qr_o &_o op_n thereby_o the_o ponit_n b_o and_o d_o be_v common_a to_o the_o plain_n qr_o and_o op_n now_o from_o bd_o sufficient_o extend_v cut_v a_o right_a line_n equal_a to_o ab_fw-la which_o suppose_v to_o be_v bf_o by_o the_o three_o of_o the_o first_o and_o orderly_a to_o bf_o make_v de_fw-fr equal_a by_o the_o 3._o oâ—_n the_o first_o if_o de_fw-fr be_fw-mi great_a than_o bf_o which_o always_o you_o may_v cause_v so_o to_o be_v by_o produce_v of_o de_fw-fr sufficient_o now_o forasmuch_o as_o bf_o by_o construction_n be_v cut_v equal_a to_o ab_fw-la and_o de_fw-fr also_o by_o construction_n put_v equal_a to_o bf_o therefore_o by_o the_o 1._o common_a sentence_n de_fw-fr be_v put_v equal_a to_o ab_fw-la which_o be_v require_v to_o be_v do_v in_o like_a sort_n if_o de_fw-fr be_v a_o line_n give_v to_o who_o ab_fw-la be_v to_o be_v cut_v and_o make_v equal_a first_o out_o of_o the_o line_n db_o sufficient_o produce_v cut_v of_o dg_o equal_a to_o de_fw-fr by_o the_o three_o of_o the_o first_o and_o by_o the_o same_o 3._o cut_v from_o basilius_n sufficient_o produce_v basilius_n equal_a to_o dg_v then_o be_v it_o evident_a that_o to_o the_o right_a line_n de_fw-fr the_o perpendicular_a line_n ab_fw-la be_v put_v equal_a and_o though_o this_o be_v easy_a to_o conceive_v yet_o i_o have_v design_v the_o figure_n according_o whereby_o you_o may_v instruct_v your_o imagination_n many_o such_o help_n be_v in_o this_o book_n requisite_a as_o well_o to_o inform_v the_o young_a student_n therewith_o as_o also_o to_o master_n the_o froward_a gaynesayer_n of_o our_o conclusion_n or_o interrupter_n of_o our_o demonstration_n course_n ¶_o the_o 7._o theorem_a the_o 7._o proposition_n if_o there_o be_v two_o parallel_n right_a line_n and_o in_o either_o of_o they_o be_v take_v a_o point_n at_o all_o adventure_n a_o right_a line_n draw_v by_o the_o say_a point_n be_v in_o the_o self_n same_o superficies_n with_o the_o parallel_n right_a line_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v parallel_n and_o in_o either_o of_o they_o take_v a_o point_n at_o all_o adventure_n namely_o e_o and_o f._n then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o point_n fletcher_n be_v in_o the_o self_n same_o plain_a superficies_n that_o the_o
at_o all_o adventure_n namely_o d_o five_o g_o s_o and_o a_o right_a line_n be_v draw_v from_o the_o point_n d_o to_o the_o point_n g_o and_o a_o other_o from_o the_o point_n five_o to_o the_o point_n s._n wherefore_o by_o the_o 7._o of_o the_o eleven_o the_o line_n dg_o and_o we_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bg_o therefore_o by_o the_o 24._o of_o the_o first_o the_o angle_n edt_n be_v equal_a to_o the_o angle_n bgt_fw-mi for_o they_o be_v alternate_a angle_n and_o likewise_o the_o angle_n dtv_n be_v equal_a to_o the_o angle_n gts_fw-fr now_o then_o there_o be_v two_o triangle_n that_o be_v dtv_n and_o gts_fw-fr have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o the_o side_n which_o subtend_v the_o equal_a angle_n that_o be_v the_o side_n dv_o to_o the_o side_n g_n for_o they_o be_v the_o half_n of_o the_o line_n de_fw-fr and_o bg_o wherefore_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o line_n dt_n be_v equal_a to_o the_o line_n tg_n and_o the_o line_n vt_fw-la to_o the_o line_n it_o be_v if_o therefore_o the_o opposite_a side_n of_o a_o parallelipipedon_n be_v divide_v into_o two_o equal_a part_n and_o by_o their_o section_n be_v extend_v plain_a superficiece_n the_o common_a section_n of_o those_o plain_a superficiece_n and_o the_o diameter_n of_o the_o parallelipipedon_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n every_o plain_a superficies_n extend_v by_o the_o centre_n of_o a_o parallelipipedon_n divide_v that_o solid_a into_o two_o equal_a part_n and_o so_o do_v not_o any_o other_o plain_a superficies_n not_o extend_v by_o the_o centre_n for_o every_o plain_n extend_v by_o the_o centre_n cut_v the_o diameter_n of_o the_o parallelipipedon_n in_o the_o centre_n into_o two_o equal_a part_n for_o it_o be_v prove_v that_o plain_a superficiece_n which_o cut_v the_o solid_a into_o two_o equal_a part_n do_v cut_v the_o dimetient_fw-la into_o two_o equal_a part_n in_o the_o centre_n wherefore_o all_o the_o line_n draw_v by_o the_o centre_n in_o that_o plain_a superficies_n shall_v make_v angle_n with_o the_o dimetient_fw-la and_o forasmuch_o as_o the_o diameter_n fall_v upon_o the_o parallel_n right_a line_n of_o the_o solid_a which_o describe_v the_o opposite_a side_n of_o the_o say_v solid_a or_o upon_o the_o parallel_n plain_a superficiece_n of_o the_o solid_a which_o make_v angel_n at_o the_o end_n of_o the_o diameter_n the_o triangle_n contain_v under_o the_o diameter_n and_o the_o right_a line_n extend_v in_o that_o plain_n by_o the_o centre_n and_o the_o right_a line_n which_o be_v draw_v in_o the_o opposite_a superficiece_n of_o the_o solid_a join_v together_o the_o end_n of_o the_o foresay_a right_a line_n namely_o the_o end_n of_o the_o diameter_n and_o the_o end_n of_o the_o line_n draw_v by_o the_o centre_n in_o the_o superficies_n extend_v by_o the_o centre_n shall_v always_o be_v equal_a and_o equiangle_n by_o the_o 26._o of_o the_o first_o for_o the_o opposite_a right_a line_n draw_v by_o the_o opposite_a plain_a superficiece_n of_o the_o solid_a do_v make_v equal_a angle_n with_o the_o diameter_n forasmuch_o as_o they_o be_v parallel_a line_n by_o the_o 16._o of_o this_o book_n but_o the_o angle_n at_o the_o centre_n be_v equal_a by_o the_o 15._o of_o the_o first_o for_o they_o be_v head_n angle_n &_o one_o side_n be_v equal_a to_o one_o side_n namely_o half_o the_o dimetient_fw-la wherefore_o the_o triangle_n contain_v under_o every_o right_a line_n draw_v by_o the_o centre_n of_o the_o parallelipipedon_n in_o the_o superficies_n which_o be_v extend_v also_o by_o the_o say_a centre_n and_o the_o diameter_n thereof_o who_o end_n be_v the_o angle_n of_o the_o solid_a be_v equal_a equilater_n &_o equiangle_n by_o the_o 26._o of_o the_o first_o wherefore_o it_o follow_v that_o the_o plain_a superficies_n which_o cut_v the_o parallelipipedon_n do_v make_v the_o part_n of_o the_o base_n on_o the_o opposite_a side_n equal_a and_o equiangle_n and_o therefore_o like_a and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n wherefore_o the_o two_o solid_a section_n of_o that_o solid_a shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n and_o now_o that_o no_o other_o plain_a superficies_n beside_o that_o which_o be_v extend_v by_o the_o centre_n divide_v the_o parallelipipedon_n into_o two_o equal_a part_n it_o be_v manifest_a if_o unto_o the_o plain_a superficies_n which_o be_v not_o extend_v by_o the_o centre_n we_o extend_v by_o the_o centre_n a_o parallel_n plain_a superficies_n by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o that_o superficies_n which_o be_v extend_v by_o the_o centre_n do_v divide_v the_o parallelipipedom_n into_o two_o equal_a parâ—â—_n it_o be_v manifest_a that_o the_o other_o plain_a superficies_n which_o be_v parallel_n to_o the_o superficies_n which_o divide_v the_o solid_a into_o two_o equal_a part_n be_v in_o one_o of_o the_o equal_a part_n of_o the_o solid_a wherefore_o see_v that_o the_o whole_a be_v ever_o great_a than_o his_o part_n it_o must_v needs_o be_v that_o one_o of_o these_o section_n be_v less_o than_o the_o half_a of_o the_o solid_a and_o therefore_o the_o other_o be_v great_a for_o the_o better_a understanding_n of_o this_o former_a proposition_n &_o also_o of_o this_o corollary_n add_v by_o flussas_n it_o shall_v be_v very_o needful_a for_o you_o to_o describe_v of_o past_v paper_n or_o such_o like_a matter_n a_o parallelipipedom_n or_o a_o cube_n and_o to_o divide_v all_o the_o parallelogram_n thereof_o into_o two_o equal_a part_n by_o draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n which_o centre_n be_v the_o point_n make_v by_o the_o cut_n of_o diagonal_a line_n draw_v from_o thâ—_z opposite_a angle_n of_o the_o say_a parallelogram_n line_n parallel_n to_o the_o side_n of_o the_o parallelogram_n as_o in_o the_o former_a figure_n describe_v in_o a_o plain_a you_o may_v see_v be_v the_o six_o parallelogram_n de_fw-fr eh_o ha_o ad_fw-la dh_o and_o cg_o who_o these_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n namely_o xo_o or_o pr._n and_o px_n do_v divide_v into_o two_o equal_a part_n by_o which_o four_o line_n you_o must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v also_o these_o parallel_a line_n kl_o ln_o nm_o and_o mâ—_n by_o which_o four_o line_n likewise_o yâ—_n must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v you_o may_v if_o you_o will_v put_v within_o your_o body_n make_v thus_o of_o past_v paper_n two_o superficiece_n make_v also_o of_o the_o say_a paper_n have_v to_o their_o limit_n line_n equal_a to_o the_o foresay_a parallel_n line_n which_o superficiece_n must_v also_o be_v divide_v into_o two_o equal_a part_n by_o parallel_a line_n draw_v by_o their_o centre_n and_o must_v cut_v the_o one_o the_o other_o by_o these_o parallel_a line_n and_o for_o the_o diameter_n of_o this_o body_n extend_v a_o thread_n from_o one_o angle_n in_o the_o base_a of_o the_o solid_a to_o his_o opposite_a angle_n which_o shall_v pass_v by_o the_o centre_n of_o the_o parallelipipedon_n as_o do_v the_o line_n dg_o in_o the_o figure_n before_o describe_v in_o the_o plain_n and_o draw_v in_o the_o base_a and_o the_o opposite_a superficies_n unto_o it_o diagonal_a line_n from_o the_o angle_n from_o which_o be_v extend_v the_o diameter_n of_o the_o solid_a as_o in_o the_o former_a description_n be_v the_o line_n bg_o and_o de._n and_o when_o you_o have_v thus_o describe_v this_o body_n compare_v it_o with_o the_o former_a demonstration_n and_o it_o will_v make_v it_o very_o plain_a unto_o you_o so_o your_o letter_n agree_v with_o the_o letter_n of_o the_o figure_n describe_v in_o the_o book_n and_o this_o description_n will_v plain_o set_v forth_o unto_o you_o the_o corollary_n follow_v that_o proposition_n for_o where_o as_o to_o the_o understanding_n of_o the_o demonstration_n of_o the_o proposition_n the_o superficiece_n put_v within_o the_o body_n be_v extend_v by_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o base_n of_o the_o parallelipipedon_n to_o the_o understanding_n of_o the_o say_a corollary_n you_o may_v extend_v a_o superficies_n by_o any_o other_o line_n draw_v in_o the_o say_a base_n so_o that_o yet_o it_o pass_v through_o the_o midst_n of_o the_o thread_n which_o be_v suppose_v to_o be_v the_o centre_n of_o the_o parallelipipedon_n ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o there_o be_v
superficies_n or_o solidity_n in_o the_o hole_n or_o in_o partâ—_n such_o certain_a knowledge_n demonstrative_a may_v arise_v and_o such_o mechanical_a exercise_n thereby_o be_v devise_v that_o sure_a i_o be_o to_o the_o sincere_a &_o true_a student_n great_a light_n aid_n and_o comfortable_a courage_n far_a to_o wade_v will_v enter_v into_o his_o heart_n and_o to_o the_o mechanical_a witty_a and_o industrous_a deviser_n new_a manner_n of_o invention_n &_o execution_n in_o his_o work_n will_v with_o small_a travail_n for_o foot_n application_n come_v to_o his_o perceiveraunce_n and_o understanding_n therefore_o even_o a_o manifold_a speculation_n &_o practice_n may_v be_v have_v with_o the_o circle_n his_o quantity_n be_v not_o know_v in_o any_o kind_n of_o small_a certain_a measure_n so_o likewise_o of_o the_o sphere_n many_o problem_n may_v be_v execute_v and_o his_o precise_a quantity_n in_o certain_a measure_n not_o determine_v or_o know_v yet_o because_o both_o one_o of_o the_o first_o humane_a occasion_n of_o invent_v and_o stablish_v this_o art_n be_v measure_v of_o the_o earth_n and_o therefore_o call_v geometria_n that_o be_v earthmeasure_v and_o also_o the_o chief_a and_o general_a end_n in_o deed_n be_v measure_n and_o measure_n require_v a_o determination_n of_o quantity_n in_o a_o certain_a measure_n by_o number_n express_v it_o be_v needful_a for_o mechanical_a earthmeasure_n not_o to_o be_v ignorant_a of_o the_o measure_n and_o content_n of_o the_o circle_n neither_o of_o the_o sphere_n his_o measure_n and_o quantity_n as_o near_o as_o sense_n can_v imagine_v or_o wish_v and_o in_o very_a deed_n the_o quantity_n and_o measure_n of_o the_o circle_n be_v know_v make_v not_o only_o the_o cone_n and_o cylinder_n but_o also_o the_o sphere_n his_o quantity_n to_o be_v as_o precise_o know_v and_o certain_a therefore_o see_v in_o respect_n of_o the_o circle_n quantity_n by_o archimedes_n specify_v this_o theorem_a be_v note_v unto_o you_o i_o will_v by_o order_n upon_o that_o as_o a_o supposition_n infer_v the_o conclusion_n of_o this_o our_o theorem_n note_n 1._o wherefore_o if_o you_o divide_v the_o one_o side_n as_o tq_n of_o the_o cube_fw-la tx_n into_o 21._o equal_a part_n and_o where_o 11._o part_n do_v end_n reckon_v from_o t_o suppose_v the_o point_n p_o and_o by_o that_o point_n p_o imagine_v a_o plain_a pass_v parallel_n to_o the_o opposite_a base_n to_o cut_v the_o cube_fw-la tx_n and_o thereby_o the_o cube_fw-la tx_n to_o be_v divide_v into_o two_o rectangle_n parallelipipedon_n namely_o tn_n and_o px_n it_o be_v manifest_a give_v tn_n to_o be_v equal_a to_o the_o sphere_n a_o by_o construction_n and_o the_o 7._o of_o the_o five_o note_n 2._o second_o the_o whole_a quantity_n of_o the_o sphere_n a_o assign_v be_v contain_v in_o the_o rectangle_n parallelipipedon_n tn_n you_o may_v easy_o transform_v the_o same_o quantity_n into_o other_o parallelipipedon_n rectangle_v of_o what_o height_n and_o of_o what_o parallelogram_n base_a you_o listen_v by_o my_o first_o and_o second_o problem_n upon_o the_o 34._o of_o this_o book_n and_o the_o like_a may_v you_o do_v to_o any_o assign_a part_n of_o the_o sphere_n a_o by_o the_o like_a mean_n devide_v the_o parallelipipedon_n tn_n as_o the_o part_n assign_v do_v require_v as_o if_o a_o three_o four_o five_o or_o six_o part_n of_o the_o sphere_n a_o be_v to_o be_v have_v in_o a_o parallelipipedon_n of_o any_o parallelograâ—â—e_n base_a assign_v or_o of_o any_o heith_n assign_v then_o devide_v tp_n into_o so_o many_o part_n as_o into_o 4._o if_o a_o four_o part_n be_v to_o be_v transform_v or_o into_o five_o if_o a_o five_o part_n be_v to_o be_v transform_v etc_n etc_n and_o then_o proceed_v â—s_v you_o do_v with_o cut_v of_o tn_n from_o tx_n and_o that_o i_o say_v of_o parallelipipedon_n may_v in_o like_a sort_n by_o my_o â—â—yd_n two_o problem_n add_v to_o the_o 34._o of_o this_o book_n be_v do_v in_o any_o side_v column_n pyramid_n and_o prismeâ—_n so_o thâ—â—_n in_o pyramid_n and_o some_o prism_n you_o use_v the_o caution_n necessary_a in_o respect_n of_o their_o quan_fw-mi 〈…〉_o odyes_n have_v parallel_n equal_a and_o opposite_a base_n who_o part_n 〈…〉_o re_fw-mi in_o their_o proposition_n be_v by_o euclid_n demonstrate_v and_o final_o 〈…〉_o addition_n you_o have_v the_o way_n and_o order_n how_o to_o geve_v to_o a_o sphere_n or_o any_o segment_n oâ—_n the_o same_o cones_n or_o cylinder_n equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n give_v with_o many_o other_o most_o necessary_a speculation_n and_o practice_n about_o the_o sphere_n i_o trust_v that_o i_o have_v sufficient_o â—raughted_v your_o imagination_n for_o your_o honest_a and_o profitable_a study_n herein_o and_o also_o give_v you_o reaâ—â—_n â—â—tter_n wheâ—â—_n with_o to_o sâ—â—p_v the_o mouth_n of_o the_o malicious_a ignorant_a and_o arrogant_a despiser_n of_o the_o most_o excellent_a discourse_n travail_n and_o invention_n mathematical_a sting_v aswell_o the_o heavenly_a sphere_n &_o star_n their_o spherical_a solidity_n geometry_n with_o their_o convene_n spherical_a superficies_n to_o the_o earth_n at_o all_o time_n respect_v and_o their_o distance_n from_o the_o earth_n as_o also_o the_o whole_a earthly_a sphere_n and_o globe_n itself_o and_o infinite_a other_o case_n concern_v sphere_n or_o globe_n may_v hereby_o with_o as_o much_o ease_n and_o certainty_n be_v determine_v of_o as_o of_o the_o quantity_n of_o any_o bowl_n ball_n or_o bullet_n which_o we_o may_v gripe_v in_o our_o hand_n reason_n and_o experience_n be_v our_o witness_n and_o without_o these_o aid_n such_o thing_n of_o importance_n never_o able_a of_o we_o certain_o to_o be_v know_v or_o attain_a unto_o here_o end_n m._n john_n do_v you_o his_o addition_n upon_o the_o last_o proposition_n of_o the_o twelve_o book_n a_o proposition_n add_v by_o flussas_n if_o a_o sphere_n touch_v a_o plain_a superficiesâ—_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n suppose_v that_o there_o be_v a_o sphere_n bcdl_n who_o centre_n let_v be_v the_o point_n a._n and_o let_v the_o plain_a superficies_n gci_n touch_v the_o spear_n in_o the_o point_n c_o and_o extend_v a_o right_a line_n from_o the_o centre_n a_o to_o the_o point_n c._n then_o i_o say_v that_o the_o line_n ac_fw-la be_v erect_v perpendicular_o to_o tâ—e_z plain_a gic._n let_v the_o sphere_n be_v cut_v by_o plain_a superficiece_n pass_v by_o the_o right_a line_n lac_n which_o plain_n let_v be_v abcdl_n and_o acel_n which_o let_v cut_v the_o plain_n gci_n by_o the_o right_a line_n gch_o and_o kci_fw-la now_o it_o be_v manifest_a by_o the_o assumpt_n put_v before_o the_o 17._o of_o this_o book_n that_o the_o two_o section_n of_o the_o sphere_n shall_v be_v circle_n have_v to_o their_o diameter_n the_o line_n lac_n which_o be_v also_o the_o diameter_n of_o the_o sphere_n wherefore_o the_o right_a line_n gch_o and_o kci_fw-la which_o be_v draw_v in_o the_o plain_n gci_n do_v at_o the_o point_n c_o fall_n without_o the_o circle_n bcdl_n and_o ecl._n wherefore_o they_o touch_v the_o circle_n in_o the_o point_n c_o by_o the_o second_o definition_n of_o the_o three_o wherefore_o the_o right_a line_n lac_n make_v right_a angle_n with_o the_o line_n gch_v and_o kci_fw-la by_o the_o 16._o of_o the_o three_o wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o right_a line_n ac_fw-la be_v erect_v perpendicular_o to_o to_o the_o plain_a superficies_n gci_n wherein_o be_v draw_v the_o line_n gch_v and_o kci_fw-la if_o therefore_o a_o sphere_n touch_v a_o plain_a superficies_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o twelve_o book_n of_o euclides_n element_n ¶_o the_o thirteen_o book_n of_o euclides_n element_n in_o this_o thirteen_o book_n be_v set_v forth_o certain_a most_o wonderful_a and_o excellent_a passion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n book_n a_o matter_n undoubted_o of_o great_a and_o infinite_a use_n in_o geometry_n as_o you_o shall_v both_o in_o this_o book_n and_o in_o the_o other_o book_n follow_v most_o evident_o perceive_v it_o teach_v moreover_o the_o composition_n of_o the_o five_o regular_a solid_n and_o how_o to_o inscribe_v they_o in_o a_o sphere_n give_v and_o also_o set_v forth_o certain_a comparison_n of_o the_o say_a body_n both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o sphere_n wherein_o they_o be_v describe_v the_o 1._o theorem_a the_o 1._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o
eglantine_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a wherefore_o the_o parallelogram_n abcd_o be_v by_o the_o first_o definition_n of_o the_o six_o like_v unto_o the_o parallelogram_n eglantine_n and_o by_o the_o same_o reason_n also_o the_o parallelogram_n abcd_o be_v like_a to_o the_o parallelogram_n kh_o abcd_o wherefore_o either_o of_o these_o parallelogram_n eglantine_n and_o kh_o be_v like_a unto_o the_o parallelogram_n abcd._n but_o rectiline_a figure_n which_o be_v like_a to_o one_o and_o the_o same_o rectiline_a figure_n be_v also_o by_o the_o 21._o of_o the_o six_o like_o the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n eglantine_n be_v like_a to_o the_o parallelogram_n hk_o other_o wherefore_o in_o every_o parallelogram_n the_o parallelogram_n about_o the_o dimecient_a be_v like_a unto_o the_o whole_a and_o also_o like_o the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o more_o brief_a demonstration_n after_o flussates_n suppose_v that_o there_o be_v a_o parallelogram_n abcd_o who_o dimetient_a let_v bâ—_z aâ—_z flussates_n about_o which_o let_v consist_v these_o parallelogram_n eke_o and_o ti_o have_v the_o angle_n at_o the_o point_n â—_o and_o 〈…〉_o with_o the_o whole_a parallelogram_n abcd._n then_o i_o say_v that_o those_o parallelogram_n eke_o and_o ti_fw-mi be_v like_a to_o the_o whole_a parallelogram_n db_o and_o also_o alâ—_z like_o the_o one_o to_o the_o other_o for_o forasmuch_o as_o bd_o eke_o and_o ti_fw-mi be_v parallelogram_n therefore_o the_o right_a line_n azg_v fall_v upon_o these_o parallel_a line_n aeb_fw-mi kzt_n and_o di_z g_z or_o upon_o these_o parallel_a line_n akd_v ezi_n and_o btg_n make_v these_o angle_n equal_a the_o one_o to_o the_o other_o namely_o the_o angle_n eaz_n to_o the_o angle_n kza_n &_o the_o angle_n eza_n to_o the_o angle_n kaz_n and_o the_o angle_n tzg_v to_o the_o angle_n zgi_n and_o the_o angle_n tgz_n to_o the_o angle_n izg_v and_o the_o angle_n bag_n to_o the_o angle_n agd_v and_o final_o the_o angle_n bga_n to_o the_o angle_n dag_n wherefore_o by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o and_o by_o the_o 34._o of_o the_o first_o the_o angle_n remain_v be_v equal_v the_o one_o to_o the_o other_o namely_o the_o angle_n b_o to_o the_o angle_n d_o and_o the_o angle_n e_o to_o the_o angle_n king_n and_o the_o angle_n t_o to_o the_o angle_n i_o wherefore_o these_o triangle_n be_v equiangle_n and_o therefore_o like_o the_o one_o to_o the_o other_o namely_o the_o triangle_n abg_o to_o the_o triangle_n gda_n and_o the_o triangle_n aez_n to_o the_o triangle_n zka_n &_o the_o triangle_n ztg_n to_o the_o triangle_n giz._n wherefore_o as_o the_o side_n ab_fw-la be_v to_o the_o side_n bg_o so_o be_v the_o side_n ae_n to_o the_o side_n ez_fw-fr and_o the_o side_n zt_n to_o the_o side_n tg_n wherefore_o the_o parallelogram_n contain_v under_o those_o right_a line_n namely_o the_o parallelogram_n abgd_v eke_o &_o ti_fw-mi be_v like_o the_o one_o to_o the_o other_o by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o every_o parallelogram_n the_o parallelogram_n etc_n etc_n as_o before_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o problem_n add_v by_o pelitarius_n two_o equiangle_n parallelogrammes_n be_v give_v so_o that_o they_o be_v not_o like_a to_o cut_v of_o from_o one_o of_o they_o a_o parallelogram_n like_a unto_o the_o other_o pelitarius_n suppose_v that_o the_o two_o equiangle_n parallelogram_n be_v abcd_o and_o cefg_n which_o let_v not_o be_v like_o the_o one_o to_o the_o other_o it_o be_v require_v from_o the_o parallelogram_n abcd_o to_o cut_v of_o a_o parallelogram_n like_a unto_o the_o parallelogram_n cefg_n let_v the_o angle_n c_o of_o the_o one_o be_v equal_a to_o the_o angle_n c_o of_o the_o other_o and_o let_v the_o two_o parallelogram_n be_v so_o 〈◊〉_d that_o the_o line_n bc_o &_o cg_o may_v make_v both_o one_o right_a line_n namely_o bg_o wherefore_o also_o the_o right_a line_n dc_o and_o ce_fw-fr shall_v both_o make_v one_o right_a line_n namely_o de._n and_o draw_v a_o line_n from_o the_o point_n f_o to_o the_o point_n c_o and_o produce_v the_o line_n fc_o till_o it_o concur_v with_o the_o line_n ad_fw-la in_o the_o point_n h._n and_o draw_v the_o line_n hk_o parallel_v to_o the_o line_n cd_o by_o the_o 31._o of_o the_o first_o then_o i_o say_v that_o from_o the_o parallelogram_n ac_fw-la be_v cut_v of_o the_o parallelogram_n cdhk_n like_v unto_o the_o parallelogram_n eglantine_n which_o thing_n be_v manifest_a by_o this_o 24._o proposition_n for_o that_o both_o the_o say_v parallelogram_n be_v describe_v about_o one_o &_o the_o self_n same_o dimetient_fw-la and_o to_o the_o end_n it_o may_v the_o more_o plain_o be_v see_v i_o have_v make_v complete_a the_o parallelogram_n abgl_n ¶_o a_o other_o problem_n add_v by_o pelitarius_n between_o two_o rectiline_a superficiece_n to_o find_v out_o a_o mean_a superficies_n proportional_a pelitarius_n suppose_v that_o the_o two_o superficiece_n be_v a_o and_o b_o between_o which_o it_o be_v require_v to_o place_v a_o mean_a superficies_n proportional_a reduce_v the_o say_v two_o rectiline_a figure_n a_o and_o b_o unto_o two_o like_a parallelogram_n by_o the_o 18._o of_o this_o book_n or_o if_o you_o think_v good_a reduce_v either_o of_o they_o to_o a_o square_a by_o the_o last_o of_o the_o second_o and_o let_v the_o say_v two_o parallelogram_n like_o the_o one_o to_o the_o other_o and_o equal_a to_o the_o superficiece_n a_o and_o b_o be_v cdef_n and_o fghk_n and_o let_v the_o angle_n fletcher_n in_o either_o of_o they_o be_v equal_a which_o two_o angle_n let_v be_v place_v in_o such_o sort_n that_o the_o two_o parallelogram_n ed_z and_o hg_o may_v be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la ck_o which_o be_v do_v by_o put_v the_o right_a line_n of_o and_o fg_o in_o such_o sort_n that_o they_o both_o make_v one_o right_a line_n namely_o eglantine_n and_o make_v complete_a the_o parallelogram_n clk_n m._n then_o i_o say_v that_o either_o of_o the_o supplement_n fl_fw-mi &_o fm_o be_v a_o mean_a proportional_a between_o the_o superficiece_n cf_o &_o fk_o that_o be_v between_o the_o superficiece_n a_o and_o b_o namely_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n for_o by_o this_o 24._o proposition_n the_o line_n hf_o be_v to_o the_o line_n fd_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o but_o by_o the_o first_o of_o this_o book_n as_o the_o line_n hf_o be_v to_o the_o line_n fd_o so_o be_v the_o superficies_n hg_o to_o the_o superficies_n fl_fw-mi and_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o so_o also_o by_o the_o same_o be_v the_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed_z which_o be_v require_v to_o be_v do_v the_o 7._o problem_n the_o 25._o proposition_n unto_o a_o rectiline_a figure_n give_v to_o describe_v a_o other_o figure_n like_o which_o shall_v also_o be_v equal_a unto_o a_o other_o rectiline_a figure_n give_v svppose_v that_o the_o rectiline_a figure_n give_v whereunto_o be_v require_v a_o other_o to_o be_v make_v like_o be_v abc_n and_o let_v the_o other_o rectiline_a figure_n whereunto_o the_o same_o be_v require_v to_o be_v make_v equal_a be_v d._n now_o it_o be_v require_v to_o describe_v a_o rectiline_a figure_n like_o unto_o the_o figure_n abc_n and_o equal_a unto_o the_o figure_n d._n construction_n upon_o the_o line_n bc_o describe_v by_o the_o 44._o of_o the_o first_o a_o parallelogram_n be_v equal_a unto_o the_o triangle_n abc_n and_o by_o the_o same_o upon_o the_o line_n ce_fw-fr describe_v the_o parallelogram_n c_o m_o equal_a unto_o the_o rectiline_a figure_n d_o and_o in_o the_o say_a parallelogram_n let_v the_o angle_n fce_n be_v equal_a unto_o the_o angle_n cbl_n and_o forasmuch_o as_o the_o angle_n fce_n be_v by_o construction_n equal_a to_o the_o angle_n cbl_n demonstration_n add_v the_o angle_n be_v common_a to_o they_o both_o wherefore_o the_o angle_n lbc_n and_o be_v be_v equal_a unto_o the_o angle_n be_v and_o ecf_n but_o the_o angle_n lbc_n and_o be_v be_v equal_a to_o two_o right_a angle_n by_o the_o 29._o of_o the_o first_o wherefore_o also_o the_o angle_n be_v and_o ecf_n be_v equal_a to_o two_o right_a angle_n wherefore_o the_o line_n bc_o and_o cf_o by_o the_o 14._o of_o the_o first_o make_v both_o one_o right_a line_n namely_o bf_o and_o in_o like_a sort_n do_v the_o line_n le_fw-fr and_o em_n make_v both_o one_o right_a line_n namely_o lm_o then_o by_o the_o
two_o other_o proposition_n go_v next_o before_o it_o so_o far_o misplace_v that_o where_o they_o be_v word_n for_o word_n before_o duâ—ly_o place_v be_v the_o 105._o and_o 106._o yet_o here_o after_o the_o book_n end_v they_o be_v repeat_v with_o the_o number_n of_o 116._o and_o 117._o proposition_n zambert_n therein_o be_v more_o faithful_a to_o follow_v as_o he_o find_v in_o his_o greek_n example_n than_o he_o be_v skilful_a or_o careful_a to_o do_v what_o be_v necessary_a yea_o and_o some_o greek_n write_v ancient_a copy_n have_v they_o not_o so_o though_o in_o deed_n they_o be_v well_o demonstrate_v yet_o truth_n disord_v be_v half_a disgracedâ—_n especial_o where_o the_o pattern_n of_o good_a order_n by_o profession_n be_v avouch_v to_o be_v but_o through_o ignorance_n arrogancy_n and_o â—emerltie_n of_o unskilful_a method_n master_n many_o thing_n remain_v yet_o in_o these_o geometrical_a element_n undue_o tumble_v in_o though_o true_a yet_o with_o disgrace_n which_o by_o help_n of_o so_o many_o wit_n and_o hability_n of_o such_o as_o now_o may_v have_v good_a cause_n to_o be_v skilful_a herein_o will_v i_o hope_v ere_o long_o be_v take_v away_o and_o thing_n of_o importance_n want_v supply_v the_o end_n of_o the_o ten_o book_n of_o euclides_n element_n ¶_o the_o eleven_o book_n of_o euclides_n element_n book_n hitherto_o have_v â—vclidâ—_n in_o thâ—sâ—_n former_a book_n with_o a_o wonderful_a method_n and_o order_n entreat_v of_o such_o kind_n of_o figure_n superficial_a which_o be_v or_o may_v be_v describe_v in_o a_o superficies_n or_o plain_a and_o have_v teach_v and_o set_v forth_o their_o property_n nature_n generation_n and_o production_n even_o from_o the_o first_o root_n ground_n and_o beginning_n of_o they_o namely_o from_o a_o point_n which_o although_o it_o be_v indivisible_a yet_o be_v it_o the_o begin_n of_o all_o quantity_n continual_a and_o of_o it_o and_o of_o the_o motion_n and_o slow_v thereof_o be_v produce_v a_o line_n and_o consequent_o all_o quantity_n continual_a as_o all_o figure_n plain_a and_o solid_a what_o so_o ever_o euclid_n therefore_o in_o his_o first_o book_n begin_v with_o it_o boot_n and_o from_o thence_o go_v he_o to_o a_o line_n as_o to_o a_o thing_n most_o simple_a next_o unto_o a_o point_n then_o to_o a_o superficies_n and_o to_o angle_n and_o so_o through_o the_o whole_a first_o book_n boâ—â—e_n he_o entreat_v of_o these_o most_o simple_a and_o plain_a ground_n in_o the_o second_o book_n he_o entreat_v further_o â—â—oâ—e_n and_o go_v unto_o more_o hard_a matter_n and_o teach_v of_o division_n of_o line_n and_o of_o the_o multiplication_n of_o line_n and_o of_o their_o part_n and_o of_o their_o passion_n and_o property_n and_o for_o that_o rightline_v â—_z be_v far_o distant_a in_o nature_n and_o property_n from_o round_a and_o circular_a figure_n in_o the_o three_o book_n he_o instruct_v the_o reader_n of_o the_o nature_n and_o condition_n of_o circle_n booâ—e_n in_o the_o four_o book_n he_o compare_v figure_n of_o right_a line_n and_o circle_n together_o bâ—oâ—e_n and_o teach_v how_o to_o describe_v a_o figure_n of_o right_a line_n with_o in_o or_o about_o a_o circle_n and_o contrariwise_o a_o circle_n with_o in_o or_o about_o a_o rectiline_a figure_n in_o the_o five_o book_n he_o search_v out_o the_o nature_n of_o proportion_n a_o matter_n of_o wonderful_a use_n and_o deep_a consideration_n boâ—â—e_n for_o that_o otherwise_o he_o can_v not_o compare_v â—igure_n with_o figure_n or_o the_o side_n of_o figure_n together_o for_o whatsoever_o be_v compare_v to_o any_o other_o thing_n be_v compare_v unto_o it_o undoubted_o under_o some_o kind_n of_o proportion_n wherefore_o in_o the_o six_o book_n he_o compare_v figure_n together_o booâ—e_n one_o to_o a_o other_o likewise_o their_o side_n and_o for_o that_o the_o nature_n of_o proportion_n can_v not_o be_v full_o and_o clear_o see_v without_o the_o knowledge_n of_o number_n wherein_o it_o be_v first_o and_o chief_o find_v in_o the_o seven_o bookâ—_n eight_o booâ—â—_n and_o nine_o book_n book_n he_o entreatâ—th_v of_o number_n &_o of_o the_o kind_n and_o property_n thereof_o and_o because_o that_o the_o side_n of_o solid_a body_n for_o the_o most_o part_n be_v of_o such_o sort_n that_o compare_v together_o they_o have_v such_o proportion_n the_o one_o to_o the_o other_o booâ—e_n which_o can_v not_o be_v express_v by_o any_o number_n certain_a and_o therefore_o be_v call_v irrational_a line_n he_o in_o the_o ten_o book_n have_v write_v &_o teach_v which_o lineâ—_z be_v commensurable_a or_o incommensurable_a the_o one_o to_o the_o other_o and_o of_o the_o diversity_n of_o kind_n of_o irrational_a line_n with_o all_o the_o condition_n &_o propriety_n of_o they_o and_o thus_o have_v euclid_n in_o these_o ten_o foresay_a book_n full_o &_o most_o plenteous_o in_o a_o marvelous_a order_n teach_v whatsoever_o seem_v necessary_a and_o requisite_a to_o the_o knowledge_n of_o all_o superficial_a figure_n of_o what_o sort_n &_o form_n so_o ever_o they_o be_v now_o in_o these_o book_n follow_v he_o entreat_v of_o figure_n of_o a_o other_o kind_n namely_o of_o bodily_a figure_n follow_v as_o of_o cube_n pyramid_n cones_n column_n cilinder_n parallelipipedon_n sphere_n and_o such_o othersâ—_n and_o show_v the_o diversity_n of_o they_o the_o generation_n and_o production_n of_o they_o and_o demonstrate_v with_o great_a and_o wonderful_a art_n their_o propriety_n and_o passion_n with_o all_o their_o nature_n and_o condition_n he_o also_o compare_v one_o oâ—_n they_o to_o a_o other_o whereby_o to_o know_v the_o reason_n and_o proportion_n of_o the_o one_o to_o the_o other_o chief_o of_o the_o five_o body_n which_o be_v call_v regular_a body_n element_n and_o these_o be_v the_o thing_n of_o all_o other_o entreat_v of_o in_o geometry_n most_o worthy_a and_o of_o great_a dignity_n and_o as_o it_o be_v the_o end_n and_o final_a intent_n of_o the_o whole_a be_v of_o geometry_n and_o for_o who_o cause_n have_v be_v write_v and_o speak_v whatsoever_o have_v hitherto_o in_o the_o former_a book_n be_v say_v or_o write_v as_o the_o first_o book_n be_v a_o ground_n and_o a_o necessary_a entry_n to_o all_o the_o râ—st_o â—ollowing_v so_o be_v this_o eleven_o book_n a_o necessary_a entry_n and_o ground_n to_o the_o rest_n which_o follow_v 〈◊〉_d and_o as_o that_o contain_v the_o declaration_n of_o word_n and_o definition_n of_o thinger_n requisite_a to_o the_o knowledge_n of_o superficial_a figure_n and_o entreat_v of_o line_n and_o of_o their_o division_n and_o section_n which_o be_v the_o term_n and_o limit_n of_o superficial_a figure_n so_o in_o this_o book_n be_v set_v forth_o the_o declaration_n of_o word_n and_o definition_n of_o thing_n pertain_v to_o solid_a and_o corporal_a figure_n and_o also_o of_o superficiece_n which_o be_v the_o term_n &_o limit_n of_o solid_n moreover_o of_o the_o division_n and_o intersection_n of_o they_o and_o diverse_a other_o thing_n without_o which_o the_o knowledge_n of_o bodily_a and_o solid_a form_n can_v not_o be_v attain_a unto_o and_o first_o be_v set_v the_o definition_n as_o follow_v definition_n a_o solid_a or_o body_n be_v that_o which_o have_v length_n breadth_n and_o thickness_n definition_n and_o the_o term_n or_o limit_v of_o a_o solid_a be_v a_o superficies_n there_o be_v three_o kind_n of_o continual_a quantity_n a_o line_n a_o superficies_n and_o a_o solid_a or_o body_n the_o beginning_n of_o all_o which_o as_o before_o have_v be_v say_v be_v a_o point_n which_o be_v indivisible_a two_o of_o these_o quantity_n namely_o a_o line_n and_o a_o superficies_n be_v define_v of_o euclid_n before_o in_o his_o first_o book_n but_o the_o three_o kind_n namely_o a_o solid_a or_o body_n he_o there_o define_v not_o as_o a_o thing_n which_o pertain_v not_o then_o to_o his_o purpose_n but_o here_o in_o this_o place_n he_o set_v the_o definition_n thereof_o as_o that_o which_o chief_o now_o pertain_v to_o his_o purpose_n and_o without_o which_o nothing_o in_o these_o thing_n can_v profitable_o be_v teach_v a_o solid_a say_v he_o be_v that_o which_o have_v length_n breadth_n and_o thickness_n or_o depth_n there_o be_v as_o before_o have_v be_v teach_v three_o reason_n or_o mean_n of_o measure_v which_o be_v call_v common_o dimension_n namely_o length_n breadth_n and_o thickness_n these_o dimension_n be_v ascribe_v unto_o quantity_n only_o by_o these_o be_v all_o kind_n of_o quantity_n define_v â—â—_o be_v count_v perfect_a or_o imperfect_a according_a as_o they_o be_v partaker_n of_o few_o or_o more_o of_o they_o as_o euclid_n define_v a_o line_n ascribe_v unto_o it_o only_o one_o of_o these_o dimension_n namely_o length_n wherefore_o a_o line_n be_v the_o imperfect_a kind_n of_o quantity_n in_o define_v of_o a_o superficies_n he_o ascribe_v unto_o it_o two_o dimension_n namely_o length_n and_o breadth_n whereby_o a_o superficies_n be_v a_o quantity_n of_o
the_o 7._o of_o the_o five_o our_o conclusion_n be_v infer_v the_o superficies_n spherical_a of_o the_o segment_n caesar_n to_o be_v to_o the_o superficies_n spherical_a of_o the_o segment_n fgh_o as_o ad_fw-la be_v to_o give_v a_o theorem_a 6._o to_o any_o solid_a sector_n of_o a_o sphere_n that_o upright_o câ—â—e_n be_v equal_a who_o base_a be_v equal_a to_o the_o connex_v spherical_a superficies_n of_o that_o sector_n and_o heith_n equal_a to_o the_o semidiameter_n of_o the_o same_o sphere_n hereof_o the_o demonstration_n in_o respect_n of_o the_o premise_n and_o the_o common_a argument_n of_o inscription_n and_o circumscription_n of_o figure_n be_v easy_a and_o nevertheless_o if_o your_o own_o write_n will_v not_o help_v you_o sufficient_o you_o may_v take_v help_n at_o archimedes_n hand_n in_o his_o first_o book_n &_o last_o proposition_n of_o the_o sphere_n and_o cylinder_n whether_o if_o you_o have_v recourse_n you_o shalâ—_n perceive_v how_o your_o theorem_a here_o amend_v the_o common_a translation_n there_o and_o also_o our_o delineation_n give_v more_o sâ—uâ—y_a show_n of_o the_o chief_a circumstance_n necessary_a to_o the_o construction_n than_o there_o you_o shall_v find_v of_o the_o sphere_n here_o imagine_v to_o be_v a_o we_o note_v a_o solid_a sector_n by_o the_o letterâ—_n pqro._n so_o that_o pqr_n do_v signify_v the_o spherical_a superficies_n to_o that_o solid_a sector_n belong_v note_n which_o be_v also_o common_a to_o the_o segment_n of_o the_o same_o sphere_n prq_n and_o therefore_o a_o line_n draw_v from_o the_o top_n of_o that_o segmentâ—_n which_o top_n suppose_v to_o be_v queen_n be_v the_o semidiameter_n of_o the_o circle_n which_o be_v equal_a to_o the_o spherical_a superficies_n of_o the_o say_v solid_a sector_n or_o segmentâ—_n as_o before_o be_v teach_v let_v that_o line_n be_v qp_n by_o q_n draw_v a_o line_n contingent_a which_o let_v be_v sqt._n at_o the_o point_n q_o from_o the_o line_n q_n cut_v a_o line_n equal_a to_o pq_n which_o let_v be_v sq_n and_o unto_o sq_n make_v qt_o equal_a then_o draw_v the_o right_a line_n osoâ—_n and_o oq_fw-la about_o which_o oq_fw-la as_o a_o axe_n faâ—â—ened_v if_o you_o imagine_v the_o triangle_n ost_n to_o make_v a_o etc_n half_a circular_a revolution_n you_o shall_v have_v the_o upright_a cone_fw-mi ost_n who_o heith_n be_v oq_fw-la the_o semidiameter_n of_o the_o sphere_n and_o base_a the_o circle_n who_o diameter_n be_v n-ab_n equal_a to_o the_o solid_a sector_n pqro._n a_o theorem_a 7._o to_o any_o segment_n or_o portion_n of_o a_o sphere_n that_o cone_fw-mi iâ—_z equal_a which_o have_v that_o circle_n to_o his_o base_a which_o be_v the_o bâ—se_a of_o the_o segment_n and_o heith_n a_o right_a line_n which_o unto_o the_o heith_n of_o the_o segment_n have_v that_o proportion_n which_o the_o semidiameter_n of_o the_o sphere_n together_o with_o the_o heith_n of_o the_o other_o segment_n remayâ—â—â—g_o have_v to_o the_o heith_n of_o the_o same_o other_o segment_n remaynâ—ng_v this_o be_v well_o demonstrate_v by_o archiâ—â—des_n &_o therefore_o need_v no_o invention_n of_o i_o cylindrâ—_n to_o confirm_v the_o same_o and_o for_o that_o the_o say_a demonstration_n be_v over_o long_o here_o to_o be_v add_v i_o will_v refer_v you_o thither_o for_o the_o demonstration_n and_o here_o supply_v that_o which_o to_o archimedes_n demonstration_n shall_v geve_v light_a and_o to_o your_o far_a speculation_n and_o practice_n shall_v be_v a_o great_a aid_n and_o direction_n suppose_v king_n to_o be_v a_o sphere_n &_o the_o great_a circle_n king_n in_o contain_v let_v be_v abce_n and_o his_o diameter_n be_v &_o centre_n d._n let_v the_o sphere_n king_n be_v cut_v by_o a_o plain_a superficies_n perpendicular_o erect_v upon_o the_o say_v great_a circle_n abce_n let_v the_o section_n be_v the_o circle_n about_o ac_fw-la and_o let_v the_o segment_n of_o the_o sphere_n be_v the_o one_o that_o wherein_o be_v abc_n who_o â—oppe_n be_v bâ—_n and_o the_o other_o let_v be_v that_o wherein_o be_v aec_o and_o his_o top_n let_v be_v e_o i_o say_v that_o a_o cone_fw-mi which_o have_v his_o base_a the_o circle_n about_o ac_fw-la &_o hold_v a_o line_n which_o to_o bf_o the_o heith_n of_o the_o segment_n who_o top_n be_v b_o have_v that_o proportion_n that_o a_o line_n compoâ—ed_v of_o de_fw-fr the_o semidiameter_n of_o the_o sphere_n and_o of_o the_o heith_n of_o the_o other_o remain_v segment_n who_o top_n be_v e_o have_v to_o of_o the_o heith_n of_o that_o other_o segment_n remain_v be_v equal_a to_o the_o segment_n of_o the_o sphere_n king_n who_o top_n be_v b._n to_o make_v this_o cone_fw-mi take_v my_o easy_a order_n thus_o frame_v your_o work_n for_o the_o find_n of_o the_o four_o proportional_a lineâ—_z by_o make_v of_o the_o first_o and_o a_o line_n compose_v of_o de_fw-fr and_o of_o the_o secondâ—_n and_o the_o three_o let_v be_v bf_o then_o by_o the_o 12._o of_o the_o sixâ—h_o let_v the_o four_o proportional_a line_n be_v find_v which_o let_v be_v fgâ—_n upon_o fletcher_n the_o centre_n of_o the_o base_a of_o the_o segment_n who_o top_n be_v b_o erect_v a_o line_n perpendicular_a equal_a to_o fg_o find_v and_o draw_v the_o line_n ga_o and_o gc_o and_o so_o make_v perfect_a the_o cone_n gac_n i_o say_v that_o the_o cone_fw-mi gac_n be_v equal_a to_o the_o segment_n of_o the_o sphere_n king_n who_o top_n be_v b._n in_o like_a manner_n for_o the_o other_o segment_n who_o top_n be_v e_o to_o find_v the_o heith_n due_a for_o a_o cone_fw-mi equal_a to_o it_o by_o the_o order_n of_o the_o theorem_a you_o must_v thus_o frame_v your_o line_n let_v the_o first_o be_v bf_o the_o second_o db_o and_o bf_o compose_v in_o one_o right_a line_n and_o the_o three_o must_v be_v of_o where_o by_o the_o 12._o of_o the_o six_o find_v the_o four_o it_o shall_v be_v the_o heith_n to_o rear_v upon_o the_o base_a the_o circle_n about_o ac_fw-la to_o make_v a_o upright_a cone_fw-mi equal_a to_o the_o segment_n who_o top_n be_v e._n ¶_o logistical_o ¶_o the_o logistical_a find_v hereof_o be_v most_o easy_a the_o diameter_n of_o the_o sphere_n be_v give_v and_o the_o portion_n of_o the_o diameter_n in_o the_o segment_n contain_v or_o axe_n of_o the_o segment_n be_v know_v then_o order_n your_o number_n in_o the_o rule_n of_o proportion_n as_o i_o here_o have_v make_v most_o plain_a in_o order_v of_o the_o line_n for_o the_o â—ought_v heith_n will_v be_v the_o producte_v a_o corollary_n 1._o hereby_o and_o other_o the_o premise_n it_o be_v evident_a that_o to_o any_o segment_n of_o a_o sphere_n note_n who_o whole_a diameter_n be_v know_v and_o the_o axe_n of_o the_o segment_n give_v a_o upright_a cone_n may_v be_v make_v equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n assignedâ—_n and_o therefore_o also_o a_o cylinder_n may_v to_o the_o say_a segment_n of_o the_o sphere_n be_v make_v equal_a â—r_n in_o any_o proportion_n give_v between_o two_o right_a line_n a_o corollary_n 2._o manifest_o also_o of_o the_o former_a theorem_a it_o may_v be_v infer_v that_o a_o sphere_n and_o his_o diameter_n be_v divide_v by_o one_o and_o the_o same_o plain_a superficies_n to_o which_o the_o say_a diameter_n be_v perpendicularâ—_n the_o two_o segment_n of_o the_o sphere_n be_v one_o to_o the_o other_o in_o that_o proportion_n in_o which_o a_o rectangle_n parallelipipedon_n have_v for_o his_o base_a the_o square_n of_o the_o great_a part_n of_o the_o diameter_n and_o his_o heith_n a_o line_n compose_v of_o the_o less_o portion_n of_o the_o diameter_n and_o the_o semidiameter_n to_o the_o rectangle_n parallelipâ—pedon_n have_v for_o his_o base_a the_o square_n of_o the_o less_o portion_n of_o the_o diameter_n &_o his_o heith_n a_o line_n compose_v of_o the_o semidiameter_n &_o the_o great_a part_n of_o the_o diameter_n a_o theorem_a 8._o every_o sphere_n to_o the_o cube_fw-la make_v of_o his_o diameter_n be_v in_o manner_n as_o 11._o to_o 21._o as_o upon_o the_o first_o and_o second_o proposition_n of_o this_o book_n i_o begin_v my_o addition_n with_o the_o circle_n be_v the_o chief_a among_o plain_a figure_n and_o therein_o bring_v manifold_a consideration_n about_o circle_n as_o of_o the_o proportion_n between_o their_o circumference_n and_o their_o diameter_n of_o the_o content_a or_o area_n of_o circle_n of_o the_o proportion_n of_o circle_n to_o the_o square_n describe_v of_o their_o diameter_n &_o of_o circle_n to_o be_v give_v in_o all_o proâ—portions_n to_o other_o circle_n with_o diverse_a other_o most_o necessary_a problem_n who_o use_n be_v partly_o there_o specify_v so_o have_v i_o in_o the_o end_n of_o this_o book_n add_v some_o such_o problem_n &_o theoremesâ—_n about_o the_o sphere_n be_v among_o solid_n the_o chief_a as_o of_o the_o same_o either_o in_o itself_o consider_v or_o to_o cone_n and_o cylinder_n compare_v by_o reason_n of_o