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 respect_n and_o consideration_n only_o that_o it_o be_v long_o broad_a and_o thick_a it_o bear_v the_o name_n of_o a_o solid_a or_o body_n â—nd_n have_v the_o nature_n and_o property_n thereof_o there_o be_v add_v to_o the_o endâ—_n of_o the_o definition_n of_o a_o solid_a that_o the_o term_n and_o limit_v of_o a_o solid_a â—s_n a_o superficies_n of_o thing_n infinity_n there_o iâ—_z no_o art_n or_o science_n all_o quantity_n therefore_o in_o this_o art_n entreat_v of_o be_v imagine_v to_o be_v finite_a infinite_a and_o to_o have_v their_o end_n and_o border_n as_o have_v be_v show_v in_o the_o first_o book_n that_o the_o limit_n and_o end_n of_o a_o line_n be_v point_n and_o the_o limit_n or_o border_n of_o a_o superficies_n be_v line_n so_o now_o he_o say_v thaâ—_z the_o end_n limit_n or_o border_n of_o a_o solideâ—_n be_v superficiece_n as_o the_o side_n of_o any_o square_a piece_n of_o timber_n or_o of_o a_o table_n or_o die_v or_o any_o other_o like_a be_v the_o term_n and_o 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
if_o the_o angle_n lmk_n be_v a_o acute_a angle_n then_o be_v that_o angle_n the_o inclination_n of_o the_o superficies_n abcd_o unto_o the_o superficies_n efgh_o by_o this_o definition_n because_o it_o be_v contain_v of_o perpendicular_a line_n draw_v in_o either_o of_o the_o superficiece_n to_o one_o and_o the_o self_n same_o point_n be_v the_o common_a section_n of_o they_o both_o 5_o plain_a superficiece_n be_v in_o like_a sort_n incline_v the_o onâ—_n 〈…〉_o she_o definition_n when_o the_o say_v angle_n of_o inclination_n be_v equal_a the_o one_o to_o the_o o_o 〈…〉_o this_o definition_n need_v no_o declaration_n at_o all_o but_o be_v most_o manifest_a by_o the_o definition_n last_o go_v before_o for_o in_o consider_v the_o inclination_n of_o diverse_a superficiece_n to_o other_o if_o the_o acute_a angle_n contain_v under_o the_o perpendicular_a line_n draw_v in_o they_o from_o one_o point_n assign_v in_o each_o of_o their_o common_a section_n be_v equal_a as_o if_o to_o the_o angle_n lmk_n in_o the_o former_a example_n be_v give_v a_o other_o angle_n in_o the_o inclination_n of_o two_o other_o superficiece_n equal_a then_o be_v the_o inclination_n of_o these_o superficiece_n like_o and_o be_v by_o this_o definition_n say_v in_o like_a sort_n to_o incline_v the_o one_o to_o the_o other_o now_o also_o let_v there_o be_v a_o other_o ground_n plain_a superficies_n namely_o the_o superficies_n mnop_n unto_o who_o also_o let_v lean_a and_o incline_v the_o superficies_n qâ—â—t_n and_o let_v the_o common_a section_n or_o segment_n of_o they_o be_v the_o line_n qr_o and_o draw_v in_o the_o superficies_n mnop_n to_o some_o one_o point_n of_o the_o common_a section_n as_o to_o the_o point_n x_o the_o line_n vx_n make_v with_o the_o common_a section_n right_a angle_n namely_o the_o angle_n vxr_n or_o the_o angle_n vxq_n also_o in_o the_o superficies_n stqr_n draw_v the_o right_a line_n yx_n to_o the_o same_o point_n x_o in_o the_o common_a section_n make_v therewith_o right_a angle_n as_o the_o angle_n yxâ—_n or_o the_o angle_n yxq._n now_o as_o say_v the_o definition_n if_o the_o angle_n contain_v under_o the_o right_a line_n draw_v in_o these_o superficiece_n &_o make_v right_a angle_n with_o the_o common_a section_n be_v in_o the_o point_n that_o be_v in_o the_o point_n of_o their_o meet_v in_o the_o common_a section_n equal_a then_o be_v the_o inclination_n of_o the_o superficiece_n equal_a as_o in_o this_o example_n if_o the_o angle_n lgh_n contain_v under_o the_o line_n lg_n be_v in_o the_o incline_a superficies_n â—kef_a and_o under_o the_o line_n hg_o be_v in_o the_o ground_n superficies_n abcd_o be_v equal_a to_o the_o angle_n yxv_n contain_v under_o the_o line_n vx_n be_v in_o the_o ground_n superficies_n mnop_n and_o under_o the_o line_n yx_n be_v in_o the_o incline_a superficies_n stqr_n then_o be_v the_o inclination_n of_o the_o superficies_n ikef_n unto_o the_o superficies_n abcd_o like_a unto_o the_o inclination_n of_o the_o superficies_n stqr_n unto_o the_o superficies_n mnop_n and_o so_o by_o this_o definition_n these_o two_o superficiece_n be_v say_v to_o be_v in_o like_a sort_n incline_v definition_n 6_o parallel_n plain_a superficiece_n be_v those_o which_o be_v produce_v or_o extend_v any_o way_n never_o touch_v or_o concur_v together_o neither_o need_v this_o definition_n any_o declaration_n but_o be_v very_o easy_a to_o be_v understand_v by_o the_o definition_n of_o parallel_a line_n â—or_a as_o they_o be_v draw_v on_o any_o part_n never_o touch_v or_o come_v together_o so_o parallel_v plain_a superâ—icieces_n be_v such_o which_o admit_v no_o touch_n that_o be_v be_v produce_v any_o way_n infinite_o never_o meet_v or_o come_v together_o defâ—inition_n 7_o like_o solid_a or_o bodily_a figure_n be_v such_o which_o be_v contain_v under_o like_o plain_a superficiece_n and_o equal_a in_o multitude_n what_o plain_a superâ—icieces_n be_v call_v like_o have_v in_o the_o begin_n of_o the_o six_o book_n be_v sufficient_o declare_v now_o when_o solid_a figure_n or_o body_n be_v contain_v under_o such_o like_a plain_a superficiece_n as_o there_o be_v define_v and_o equal_a in_o number_n that_o be_v that_o the_o one_o solid_a have_v as_o many_o in_o number_n as_o the_o other_o in_o their_o side_n and_o limit_n they_o be_v call_v like_o solid_a figure_n or_o like_a body_n definition_n 8_o equal_a and_o like_a solid_a or_o bodily_a figure_n be_v those_o which_o be_v contain_v under_o like_a superficiece_n and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n in_o like_o solid_a figure_n it_o be_v sufficient_a that_o the_o superficiece_n which_o contain_v they_o be_v like_a and_o equal_a in_o number_n only_o but_o in_o like_a solid_a figure_n and_o equal_a it_o be_v necessary_a that_o the_o like_a superficiece_n contain_v they_o be_v also_o equal_a in_o magnitude_n so_o that_o beside_o the_o likeness_n between_o they_o they_o be_v each_o be_v compare_v to_o his_o correspondent_a superficies_n oâ—_n one_o greatness_n and_o that_o their_o area_n or_o field_n be_v equal_a when_o such_o superâ—icieces_n contain_v body_n or_o solid_n then_o be_v such_o body_n equal_a and_o like_a solid_n or_o body_n dissilition_n 9_o a_o solid_a or_o bodily_a angle_n be_v a_o inclination_n of_o more_o than_o two_o line_n to_o all_o the_o line_n which_o touch_v themselves_o mutual_o and_o be_v not_o in_o one_o and_o the_o self_n same_o superficies_n or_o else_o thus_o a_o solid_a or_o bodily_a angle_n be_v that_o which_o be_v contain_v under_o more_o than_o two_o plain_a angle_n not_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n but_o consist_v all_o at_o one_o point_n of_o a_o solid_a angle_n do_v euclid_n here_o geve_v two_o several_a definition_n the_o first_o be_v give_v by_o the_o concourse_n and_o touch_n of_o many_o line_n the_o second_o by_o the_o touch_n &_o concourse_n of_o many_o superficial_a angle_n and_o both_o these_o definition_n tend_v to_o one_o and_o be_v not_o much_o different_a for_o that_o line_n be_v the_o limit_n and_o term_n of_o superficiece_n but_o the_o second_o give_v by_o superâ—iciall_a angle_n be_v the_o more_o natural_a definition_n because_o that_o supeâ—ficieces_n aâ—e_v the_o next_o and_o immediate_a limit_n of_o body_n and_o so_o be_v not_o line_n a_o example_n of_o a_o solid_a angle_n can_v well_o and_o at_o â—ully_o be_v give_v or_o describe_v in_o a_o plaâ—â—e_a superficies_n but_o touch_v this_o first_o definition_n lay_v before_o you_o a_o cube_fw-la or_o a_o die_n and_o consider_v any_o of_o the_o corner_n or_o angle_n thereof_o so_o shall_v you_o see_v that_o at_o every_o angle_n there_o concur_v three_o line_n for_o two_o line_n concur_v can_v make_v a_o solid_a angle_n namely_o the_o line_n or_o edge_n of_o his_o breadth_n of_o his_o length_n and_o of_o his_o thickness_n which_o their_o so_o incline_v &_o concur_v tovether_o make_v a_o solid_a angle_n and_o so_o of_o other_o and_o now_o concern_v the_o second_o definition_n what_o superficial_a or_o plain_a angle_n be_v have_v be_v teach_v before_o in_o the_o first_o book_n namely_o that_o it_o be_v the_o touch_n of_o two_o right_a line_n and_o as_o a_o superâ—iciall_a or_o plain_a angle_n be_v cause_v &_o contain_v of_o right_a line_n so_o si_fw-mi a_o solid_a angle_n cause_v &_o contain_v of_o plain_a superficial_a angle_n two_o right_a line_n touch_v together_o make_v a_o plain_a angle_n but_o two_o plain_a angle_n join_v together_o can_v not_o make_v a_o solid_a angle_n but_o according_a to_o the_o definition_n they_o must_v be_v more_o they_o two_o as_z three_o â—oure_n â—ive_a or_o moâ—_n which_o also_o must_v not_o be_v in_o one_o &_o the_o self_n same_o superficies_n but_o must_v be_v in_o diverse_a superficiece_n â—eeting_v at_o one_o point_n this_o definition_n be_v not_o hard_o but_o may_v easy_o be_v conceive_v in_o a_o cube_fw-la or_o a_o die_n where_o you_o see_v three_o angle_n of_o any_o three_o superficiece_n or_o side_n of_o the_o die_v concur_v and_o meet_v together_o in_o one_o point_n which_o three_o plain_a angle_n so_o join_v together_o make_v a_o solid_a angle_n likewise_o in_o a_o pyramiâ—_n or_o a_o spiâ—e_n of_o a_o steeple_n or_o any_o other_o such_o thing_n all_o the_o side_n thereof_o tend_v upward_o narrow_n and_o narrow_a at_o length_n end_v their_o angle_n at_o the_o height_n or_o top_n thereof_o in_o one_o point_n so_o all_o their_o angle_n there_o join_v together_o make_v a_o solid_a angle_n and_o for_o the_o better_a â—igâ—t_n thereof_o i_o have_v set_v here_o a_o figure_n whereby_o you_o shall_v more_o easy_o conceive_v â—â—_o the_o base_a of_o the_o figure_n be_v a_o triangle_n namely_o abc_n if_o on_o every_o side_n of_o the_o triangle_n abc_n you_o raise_v up_o a_o triangle_n as_o upon_o the_o side_n ab_fw-la you_o raise_v up_o the_o triangle_n afb_o and_o upon_o the_o side_n ac_fw-la the_o
signify_v last_o of_o all_o a_o dodecahedron_n heaven_n for_o that_o it_o be_v make_v of_o pâ—ntagoâ—_n who_o angle_n be_v more_o ample_a and_o large_a than_o the_o angle_n of_o the_o other_o body_n and_o by_o that_o â—eaâ—â—â—_n draw_v more_o â—â—_o roundness_n 〈◊〉_d &_o to_o the_o form_n and_o nature_n of_o a_o sphere_n they_o assign_v to_o a_o sphere_n namely_o 〈…〉_o who_o so_o will_v 〈…〉_o in_o his_o tineus_n shall_v â—ead_a of_o these_o figure_n and_o of_o their_o mutual_a proportionâ—â—â—raunge_a maâ—terâ—_n which_o hâ—re_o be_v not_o to_o be_v entreat_v of_o this_o which_o be_v say_v shall_v be_v sufficient_a for_o the_o 〈◊〉_d of_o they_o and_o for_o thâ—_z declaration_n of_o their_o definition_n after_o all_o these_o definition_n here_o set_v of_o euclid_n flussas_n have_v add_v a_o other_o definition_n which_o 〈◊〉_d of_o a_o parallelipipedon_n which_o because_o it_o have_v not_o hitherto_o of_o euclid_n in_o any_o place_n be_v define_v and_o because_o it_o be_v very_o good_a and_o necessary_a to_o be_v have_v i_o think_v good_a not_o to_o omit_v it_o thus_o it_o be_v a_o parallelipipedon_n be_v a_o solid_a figure_n comprehend_v under_o four_o plain_a quadrangle_n figure_n parallelipipedon_n of_o which_o those_o which_o be_v opposite_a be_v parallel_n because_o these_o five_o regular_a body_n here_o define_v be_v not_o by_o these_o figure_n here_o set_v so_o full_o and_o lively_o express_v that_o the_o studious_a beholder_n can_v thorough_o according_a to_o their_o definition_n conceive_v they_o i_o have_v here_o give_v of_o they_o other_o description_n draw_v in_o a_o plain_n by_o which_o you_o may_v easy_o attain_v to_o the_o knowledge_n of_o they_o for_o if_o you_o draw_v the_o like_a form_n in_o matter_n that_o will_v bow_v and_o geve_v place_n as_z most_o apt_o you_o may_v do_v in_o fine_a past_v paper_n such_o as_o pastwive_n make_v woman_n paste_n of_o &_o they_o with_o a_o knife_n cut_v every_o line_n fine_o not_o through_o but_o half_a way_n only_o if_o they_o you_o bow_v and_o bend_v they_o according_o you_o shall_v most_o plain_o and_o manifest_o see_v the_o form_n and_o shape_n of_o these_o body_n even_o as_o their_o definition_n show_v and_o it_o shall_v be_v very_o necessary_a for_o you_o to_o hadâ—â—tore_v of_o that_o past_v paper_n by_o you_o for_o so_o shall_v yoâ—_n upon_o it_o 〈…〉_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n 〈…〉_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a 〈◊〉_d of_o thâ—se_a body_n there_o mention_v which_o will_v make_v these_o book_n concern_v body_n as_o easy_a unto_o you_o as_o be_v the_o other_o book_n who_o figure_n you_o may_v plain_o see_v upon_o a_o plain_a superficies_n describe_v thiâ—_z figurâ—_n which_o consi_v of_o twâ—luâ—â—quilâ—â—â—â—_n and_o equiangle_n pâ—ntâ—â—â—â—â—_n upoâ—_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v â—â—ught_v tâ—â—â—lâ—uâ—n_v line_n contain_v within_o thâ—_z figurâ—_n and_o bow_n and_o fold_n the_o penâ—â—gonâ—_n according_o and_o they_o will_v so_o close_o together_o thaâ—_z thâ—y_n will_v â—â—kâ—_n thâ—_z very_a form_n of_o a_o dodecahedron_n dâ—dâ—â—â—â—edron_n icosaâ—edron_n if_o you_o describe_v this_o figure_n which_o consist_v of_o twenty_o equilater_n and_o equiangle_n triangle_n upon_o the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v show_v the_o ninâ—tâ—ne_a line_n which_o be_v contain_v within_o the_o figure_n and_o then_o bow_n and_o fold_n they_o according_o they_o will_v in_o such_o sort_n close_o together_o that_o therâ—_n will_v be_v make_v a_o perfect_v form_n of_o a_o icosahedron_n because_o in_o these_o five_o book_n there_o be_v sometime_o require_v other_o body_n beside_o the_o foresay_a five_o regular_a body_n as_o pyramid_n of_o diverse_a form_n prism_n and_o other_o i_o have_v here_o set_v forth_o three_o figure_n of_o three_o sundry_a pyramid_n one_o have_v to_o his_o base_a a_o triangle_n a_o other_o a_o quadrangle_n figure_n the_o other_o â—_o pentagonâ—_n which_o if_o you_o describe_v upon_o the_o foresay_a matter_n &_o fine_o cut_v as_o it_o be_v before_o teach_v the_o line_n contain_v within_o each_o figure_n namely_o in_o the_o first_o three_o line_n in_o the_o second_o four_o line_n and_o in_o the_o three_o five_o line_n and_o so_o bend_v and_o fold_v they_o according_o they_o will_v so_o close_o together_o at_o the_o top_n that_o they_o will_v â—ake_v pyramid_n of_o that_o form_n that_o their_o base_n be_v of_o and_o if_o you_o conceive_v well_o the_o describe_v of_o these_o you_o may_v most_o easy_o describe_v the_o body_n of_o a_o pyramid_n of_o what_o form_n so_o ever_o you_o will._n because_o these_o five_o book_n follow_v be_v somewhat_o hard_o for_o young_a beginner_n by_o reason_n they_o must_v in_o the_o figure_n describe_v in_o a_o plain_a imagine_v line_n and_o superficiece_n to_o be_v elevate_v and_o erect_v the_o one_o to_o the_o other_o and_o also_o conceive_v solid_n or_o body_n which_o for_o that_o they_o have_v not_o hitherto_o be_v acquaint_v with_o will_v at_o the_o first_o sight_n be_v somewhat_o strange_a unto_o they_o i_o have_v for_o their_o more_o â—ase_a in_o this_o eleven_o book_n at_o the_o end_n of_o the_o demonstration_n of_o every_o proposition_n either_o set_v new_a figure_n if_o they_o concern_v the_o elevate_n or_o erect_v of_o line_n or_o superficiece_n or_o else_o if_o they_o concern_v body_n i_o have_v show_v how_o they_o shall_v describe_v body_n to_o be_v compare_v with_o the_o construction_n and_o demonstration_n of_o the_o proposition_n to_o they_o belong_v and_o if_o they_o diligent_o weigh_v the_o manner_n observe_v in_o this_o eleven_o book_n touch_v the_o description_n of_o new_a figure_n agree_v with_o the_o figure_n describe_v in_o the_o plain_a it_o shall_v not_o be_v hard_o for_o they_o of_o themselves_o to_o do_v the_o like_a in_o the_o other_o book_n follow_v when_o they_o come_v to_o a_o proposition_n which_o concern_v either_o the_o elevate_n or_o erect_v of_o line_n and_o superficiece_n or_o any_o kind_n of_o body_n to_o be_v imagine_v ¶_o the_o 1._o theorem_a the_o 1._o proposition_n that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n &_o part_v elevate_v upward_o be_v impossible_a for_o if_o it_o be_v possible_a let_v part_n of_o the_o right_a line_n abc_n namely_o the_o part_n ab_fw-la be_v in_o a_o ground_n plain_a superficies_n and_o the_o other_o part_n thereof_o namely_o bc_o be_v elevate_v upward_o and_o produce_v direct_o upon_o the_o ground_n plain_a superficies_n the_o right_a line_n ab_fw-la beyond_o the_o point_n b_o unto_o the_o point_n d._n wherefore_o unto_o two_o right_a line_n give_v abc_n and_o abdella_n the_o line_n ab_fw-la be_v a_o common_a section_n or_o part_n which_o be_v impossible_a impossibility_n for_o a_o right_a line_n can_v not_o touch_v a_o right_a line_n in_o 〈◊〉_d point_n then_o one_o uâ—lesse_o those_o right_n be_v exact_o agree_v and_o lay_v the_o one_o upon_o the_o other_o wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a which_o be_v require_v to_o be_v prove_v this_o figure_n more_o plain_o set_v forth_o the_o foresay_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o the_o line_n bc._n an_o other_o demonstration_n after_o flâ—sâ—â—s_n if_o it_o be_v possible_a let_v there_o be_v a_o right_a line_n abg_o who_o part_n ab_fw-la let_v be_v in_o the_o ground_n plain_a superficies_n aed_n flussas_n and_o let_v the_o rest_n thereof_o bg_o be_v elevate_v on_o high_a that_o be_v without_o the_o plain_n aed_n then_o i_o say_v that_o abg_o be_v not_o one_o right_a line_n for_o forasmuch_o as_o aed_n be_v a_o plain_a superficies_n produce_v direct_o &_o equal_o upon_o the_o say_a plain_n aed_n the_o right_a line_n ab_fw-la towards_o d_z which_o by_o the_o 4._o definition_n of_o the_o first_o shall_v be_v a_o right_a line_n and_o from_o some_o one_o point_n of_o the_o right_a line_n abdella_n namely_o from_o c_o draâ—_n unto_o the_o point_n g_o a_o right_a line_n cg_o wherefore_o in_o the_o triangle_n 〈…〉_o the_o outward_a angâ—â—_n abâ—_n be_v equal_a to_o the_o two_o inward_a and_o opposite_a angle_n by_o the_o 32._o of_o the_o first_o and_o therefore_o it_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o same_o wherefore_o the_o line_n abg_o forasmuch_o as_o it_o make_v a_o angle_n be_v not_o â—_o right_n line_n wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a if_o you_o mark_v well_o the_o figure_n before_o add_v for_o the_o plain_a declaration_n of_o euclides_n demonstration_n iâ—_z will_v not_o be_v hard_o for_o you_o to_o coâ—â—â—â—e_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â—_o wherein_o
be_v no_o difference_n but_o only_o the_o draught_n of_o the_o line_n gc_o ¶_o the_o 2._o theorem_a the_o 2._o proposition_n if_o two_o right_a line_n cut_v the_o ouâ—_n to_o the_o other_o they_o be_v â—â—â—ne_v and_o the_o self_n same_o plain_a superficies_n &_o every_o triangle_n be_v in_o one_o &_o the_o self_n same_o superficies_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o do_v cut_v the_o one_o the_o other_o in_o the_o point_n e._n then_o i_o say_v that_o these_o line_n ab_fw-la and_o cd_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o that_o every_o triangle_n be_v in_o one_o &_o self_n same_o plain_a superficies_n construction_n take_v in_o the_o line_n ec_o and_o ebb_n point_v at_o all_o aventure_n and_o let_v the_o same_o be_v fletcher_n and_o g_o and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c_o and_o a_o other_o from_o the_o point_n f_o to_o the_o point_n g._n and_o draw_v the_o line_n fh_o and_o gk_o first_o i_o say_v that_o the_o triangle_n ebc_n be_v in_o one_o and_o the_o same_o ground_n superficies_n impossibility_n for_o if_o part_n of_o the_o triangle_n ebc_n namely_o the_o triangle_n fch_o or_o the_o triangle_n gbk_n be_v in_o the_o ground_n superficies_n and_o the_o residue_n be_v in_o a_o other_o then_o also_o part_n of_o one_o of_o the_o right_a line_n ec_o or_o ebb_n shall_v be_v in_o the_o ground_n superficies_n and_o part_n in_o a_o other_o so_o also_o if_o part_n of_o the_o triangle_n ebc_n namely_o the_o part_n efg_o be_v in_o the_o ground_n superficies_n and_o the_o residue_n be_v in_o a_o other_o then_o also_o one_o part_n of_o each_o of_o the_o right_a line_n ec_o and_o ebb_n shall_v be_v in_o the_o ground_n superficies_n &_o a_o other_o part_n in_o a_o other_o superficies_n which_o by_o the_o first_o of_o the_o eleven_o be_v prove_v to_o be_v impossible_a wherefore_o the_o triangle_n ebc_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n for_o in_o what_o superficies_n the_o triangle_n be_v be_v in_o the_o same_o also_o be_v either_o of_o the_o line_n ec_o and_o ebb_n and_o in_o what_o superficies_n either_o of_o the_o line_n ec_o and_o ebb_n be_v in_o the_o self_n same_o also_o be_v the_o line_n ab_fw-la and_o cd_o wherefore_o the_o right_a line_n line_n ab_fw-la and_o cd_o be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n and_o every_o triangle_n be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v in_o this_o figure_n here_o set_v may_v you_o more_o plain_o conceive_v the_o demonstration_n of_o the_o former_a proposition_n where_o 〈◊〉_d may_v eleâ—â—â—â—_n what_o part_n of_o the_o triangle_n ecb_n you_o will_v namely_o the_o part_n fch_n or_o the_o part_n gbk_n or_o final_o the_o part_n fcgb_n as_o be_v require_v in_o the_o demonstration_n ¶_o the_o 3._o theorem_a the_o 3._o proposition_n if_o two_o plain_a superficiece_n cut_v the_o one_o the_o other_o their_o common_a section_n be_v a_o right_a line_n svppose_v that_o these_o two_o superficiece_n ab_fw-la &_o bc_o do_v cut_v the_o one_o the_o other_o and_o let_v their_o common_a sectiâ—â—â—e_n the_o line_n db._n then_o i_o say_v that_o db_o be_v a_o right_a line_n for_o if_o not_o draw_v from_o the_o point_n d_o to_o the_o point_n b_o a_o right_a line_n dfb_n in_o the_o plain_a superficies_n ab_fw-la impossibility_n and_o likewise_o from_o the_o same_o point_n draw_v a_o other_o right_a line_n deb_n in_o the_o plain_a superficies_n bc._n now_o therefore_o two_o right_a line_n deb_n and_o dfb_n shall_v â—ave_n the_o self_n saâ—â—_n eâ—deâ—_n and_o therefore_o do_v include_v a_o superficies_n which_o by_o the_o last_o common_a sentence_n be_v impossibleâ—_n wherefore_o the_o line_n deb_n and_o dfb_n be_v not_o right_a line_n in_o like_a sort_n also_o may_v we_o prove_v that_o no_o other_o right_a line_n can_v be_v draw_v from_o the_o point_n d_o to_o the_o point_n b_o beside_o the_o line_n db_o which_o be_v the_o common_a section_n of_o the_o two_o superficiece_n ab_fw-la and_o bc._n if_o therefore_o two_o plain_a superficiece_n cut_v the_o one_o the_o other_o their_o common_a section_n be_v a_o right_a line_n which_o be_v require_v to_o be_v demonstrate_v this_o figure_n here_o set_v show_v most_o plain_o not_o only_a this_o three_o proposition_n but_o also_o the_o demonstration_n thereof_o if_o you_o elevate_v the_o superficies_n ab_fw-la and_o so_o compare_v it_o with_o the_o demonstration_n ¶_o the_o 4._o theorem_a the_o 4._o proposition_n if_o from_o two_o right_a line_n cut_v the_o one_o the_o other_o at_o their_o common_a section_n a_o right_a line_n be_v perpendicular_o erect_v the_o same_o shall_v also_o be_v perpendicular_o erect_v from_o the_o plain_a superficies_n by_o the_o say_v two_o line_n pass_v svppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o cd_o cut_v the_o one_o the_o other_o in_o the_o point_n e._n and_o from_o the_o point_n e_o let_v there_o be_v erect_v a_o right_a line_n of_o perpendicular_o to_o the_o say_v two_o right_a line_n ab_fw-la and_o cd_o then_o i_o say_v that_o the_o right_a line_n of_o construction_n be_v also_o erect_v perpendicular_a to_o the_o plain_a superficies_n which_o pass_v by_o the_o line_n a_o b_o and_o cd_o let_v these_o line_n ae_n ebb_n ec_o and_o ed_z be_v put_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o point_n e_o extend_v a_o right_a line_n at_o all_o aventure_n and_o let_v the_o same_o be_v geh_a and_o draw_v these_o right_a line_n ad_fw-la cb_o favorina_n fg_o fd_o fc_o fh_o and_o fb_o demonstration_n and_o forasmuch_o as_o these_o two_o right_a line_n ae_n &_o ed_z be_v equal_a to_o these_o two_o line_n ce_fw-fr and_o ebb_n and_o they_o comprehend_v equal_a angle_n by_o the_o 15._o of_o the_o first_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ad_fw-la be_v equal_a to_o the_o base_a cb_o and_o the_o triangle_n aed_n be_v equal_a to_o the_o triangle_n ceb_fw-mi wherefore_o also_o the_o angle_n dae_n be_v equal_a to_o the_o angle_n ebc_n but_o the_o angle_n aeg_n be_v equal_a to_o the_o angle_n beh_n by_o the_o 15_o of_o the_o first_o wherefore_o there_o be_v two_o triangle_n age_n and_o beh_n 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 each_o to_o his_o correspondent_a angle_n 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 one_o of_o the_o side_n which_o lie_v between_o the_o equal_a angle_n namely_o the_o side_n ae_n be_v equal_a to_o the_o side_n ebb_n wherefore_o by_o the_o 26._o of_o the_o first_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ge_z be_v equal_a to_o the_o side_n eh_o and_o the_o side_n agnostus_n to_o the_o side_n bh_o and_o forasmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n and_o the_o line_n fe_o be_v common_a to_o they_o both_o and_o make_v with_o they_o right_a angle_n wherefore_o by_o the_o four_o of_o the_o first_o the_o base_a favorina_n it_o equal_a to_o the_o base_a fb_o and_o by_o the_o same_o reason_n the_o base_a fc_n be_v equal_a to_o the_o base_a fd._n and_o forasmuch_o as_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n bc_o and_o the_o line_n favorina_n be_v equal_a to_o the_o line_n fb_o as_o it_o have_v be_v prove_v therefore_o these_o two_o line_n favorina_n and_o ad_fw-la be_v equal_a to_o these_o two_o line_n fb_o &_o bc_o the_o one_o to_o the_o other_o &_o the_o base_a fd_o be_v equal_a to_o the_o base_a fc_n wherefore_o also_o the_o angle_n fad_n be_v equal_a to_o the_o angle_n fbc_n and_o again_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o line_n agnostus_n be_v equal_a to_o the_o line_n bh_o but_o the_o line_n favorina_n be_v equal_a to_o the_o line_n fb_o wherefore_o there_o be_v two_o line_n favorina_n and_o agnostus_n equal_a to_o two_o line_n fb_o and_o bh_o and_o it_o be_v prove_v that_o the_o angle_n fag_n be_v equal_a to_o the_o angle_n fbh_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a fg_o be_v equal_a to_o the_o base_a fh_o again_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o line_n ge_z be_v equal_a to_o the_o line_n eh_o and_o the_o line_n of_o be_v common_a to_o they_o both_o wherefore_o these_o two_o line_n ge_z and_o of_o be_v equal_a to_o these_o two_o line_n he_o and_o of_o and_o the_o base_a fh_o be_v equal_a to_o the_o base_a fg_o wherefore_o the_o angle_n gef_n be_v equal_a to_o the_o angle_n hef_fw-fr wherefore_o either_o of_o the_o angle_n gef_n and_o hef_n be_v a_o right_a angle_n wherefore_o the_o line_n of_o be_v erect_v from_o the_o point_n e_o
perpendicular_o to_o the_o line_n gh_o in_o like_a sort_n may_v we_o prove_v that_o the_o same_o line_n fe_o make_v right_a angle_n with_o all_o the_o right_a line_n which_o be_v draw_v upon_o the_o ground_n plain_a superficies_n and_o touch_v the_o point_n b._n but_o a_o right_a line_n be_v then_o erect_v perpendicular_o to_o a_o plain_a superficies_n when_o it_o make_v right_a angle_n with_o all_o the_o line_n which_o touch_v it_o and_o be_v draw_v upon_o the_o ground_n plain_a superficies_n by_o the_o 2._o definition_n of_o the_o eleven_o wherefore_o the_o right_a line_n fe_o be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n and_o the_o ground_n plain_a superficies_n be_v that_o which_o pass_v by_o these_o right_a line_n ab_fw-la and_o cd_o wherefore_o the_o right_a line_n fe_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o pass_v by_o the_o right_a line_n ab_fw-la and_o cd_o if_o therefore_o from_o two_o right_a line_n cut_v the_o one_o the_o other_o and_o at_o their_o common_a section_n a_o right_a line_n be_v perpendicular_o erect_v it_o shall_v also_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n by_o the_o say_v two_o line_n pass_v which_o be_v require_v to_o be_v prove_v in_o this_o figure_n you_o may_v most_o evident_o conceive_v the_o former_a proposition_n and_o demonstration_n if_o you_o erect_v perpendicular_o unto_o the_o ground_n plain_a superficies_n acbd_v theâ—_z triangle_n afb_o and_o elevate_v the_o triangle_n afd_v &_o cfb_n in_o such_o sort_n that_o the_o line_n of_o of_o the_o triangle_n afb_o may_v join_v &_o make_v one_o line_n with_o the_o line_n of_o of_o the_o triangle_n afd_v and_o likewise_o that_o the_o line_n bf_o of_o the_o triangle_n afb_o may_v join_v &_o make_v one_o right_a line_n with_o the_o line_n bf_o of_o the_o triangle_n bfc_n ¶_o the_o 5._o theorem_a the_o 5._o proposition_n if_o unto_o three_o right_a line_n which_o touch_v the_o one_o the_o other_o be_v erect_v a_o perpendicular_a line_n from_o the_o common_a point_n where_o those_o three_o line_n touch_v those_o three_o right_a line_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n svppose_v that_o unto_o these_o three_o right_a line_n bc_o bd_o and_o be_v touch_v the_o one_o the_o other_o in_o the_o point_n b_o be_v erect_v perpendicular_o from_o the_o point_n b_o the_o line_n ab_fw-la then_o i_o say_v that_o those_o three_o right_a line_n bc_o bd_o and_o be_v be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o line_n bd_o &_o be_v be_v in_o the_o ground_n superficies_n and_o let_v the_o line_n bc_o be_v erect_v upward_o now_o the_o line_n ab_fw-la and_o bc_o be_v in_o one_o and_o the_o same_o plain_a superficies_n by_o the_o 2._o of_o the_o eleven_o for_o they_o touch_v the_o one_o the_o other_o in_o the_o point_n b_o extend_v the_o plain_a superficies_n wherein_o the_o line_n ab_fw-la and_o bc_o be_v impossibility_n and_o it_o shall_v make_v at_o the_o length_n a_o common_a section_n with_o the_o ground_n superficies_n which_o common_a section_n shall_v be_v a_o right_a line_n by_o the_o 3._o of_o the_o eleven_o let_v that_o common_a section_n be_v the_o line_n bf_o wherefore_o the_o three_o right_a line_n ab_fw-la bc_o and_o bf_o be_v in_o one_o and_o the_o self_n same_o superficies_n namely_o in_o the_o superficies_n wherein_o the_o line_n ab_fw-la and_o bc_o be_v and_o forasmuch_o as_o the_o right_a line_n ab_fw-la be_v erect_v perpendicular_o to_o either_o of_o these_o line_n bd_o and_o be_v therefore_o the_o line_n ab_fw-la be_v also_o by_o the_o 4._o of_o the_o eleven_o erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o the_o line_n bd_o and_o be_v be_v but_o the_o superficies_n wherein_o the_o line_n bd_o and_o be_v be_v be_v the_o ground_n superficies_n wherefore_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n wherefore_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n ab_fw-la make_v right_a angle_n with_o all_o the_o line_n which_o be_v draw_v upon_o the_o ground_n superficies_n and_o touch_v it_o but_o the_o line_n bf_o which_o be_v in_o the_o ground_n superficies_n do_v touch_v it_o wherefore_o the_o angle_n abf_n be_v a_o right_a angle_n and_o it_o be_v suppose_v that_o the_o angle_n abc_n be_v a_o right_a angle_n wherefore_o the_o angle_n abf_n be_v equal_a to_o the_o angle_n abc_n and_o they_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n which_o be_v impossible_a wherefore_o the_o right_a line_n bc_o be_v not_o in_o a_o high_a superficies_n wherefore_o the_o right_a line_n bc_o bd_o be_v be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n if_o therefore_o unto_o three_o right_a line_n touch_v the_o one_o the_o one_o the_o other_o be_v erect_v a_o perpendicular_a line_n from_o the_o common_a point_n where_o those_o three_o line_n touch_v those_o three_o right_a line_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n which_o be_v require_v to_o be_v demonstrate_v this_o figure_n here_o set_v more_o plain_o declare_v the_o demonstration_n of_o the_o former_a proposition_n if_o you_o erect_v perpendicular_o unto_o the_o ground_n superficies_n the_o sâ—â—perficies_n wherein_o be_v draw_v the_o line_n 〈◊〉_d and_o so_o compare_v it_o with_o the_o say_a defenestration_n the_o 6._o theorem_a the_o 6._o proposition_n if_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o &_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 svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v erect_v perpendicular_o to_o a_o ground_n plain_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o let_v the_o point_n which_o those_o two_o right_a line_n touch_v in_o the_o plain_a superficies_n be_v b_o and_o d._n construction_n and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n bd_o in_o the_o ground_n superficies_n a_o perpendicular_a line_n de._n and_o by_o the_o 2._o of_o the_o first_o demonstration_n put_v the_o line_n de_fw-fr equal_a to_o the_o line_n ab_fw-la and_o draw_v these_o right_a line_n be_v ae_n and_o ad._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n ab_fw-la make_v right_a angle_n with_o all_o the_o line_n which_o be_v draw_v upon_o the_o ground_n plain_a superficies_n and_o touch_v it_o but_o either_o of_o tâ—ese_n line_n bd_o and_o be_v which_o be_v in_o the_o ground_n superficies_n touch_v the_o line_n ab_fw-la wherefore_o either_o of_o these_o angle_n abdella_n and_o abe_n be_v a_o right_a angleâ—_n and_o by_o the_o same_o reason_n also_o either_o of_o the_o angle_n cdb_n &_o cde_o be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n bd_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n ab_fw-la and_o bd_o be_v equal_a to_o these_o two_o line_n ed_z and_o db_o and_o they_o contain_v right_a angle_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ad_fw-la be_v equal_a to_o the_o base_a be._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n ad_fw-la to_o the_o line_n be_v therefore_o these_o two_o line_n ab_fw-la and_o be_v be_v equal_a to_o these_o two_o line_n ed_z and_o da_z and_o the_o line_n ae_n be_v a_o common_a base_a to_o they_o both_o wherefore_o the_o angle_n abe_n be_v by_o the_o 8._o of_o the_o first_o equal_a to_o the_o angle_n eda_n but_o the_o angle_n abe_n be_v a_o right_a angle_n wherefore_o also_o the_o angle_n eda_n be_v a_o right_a angle_n wherefore_o the_o line_n ed_z be_v erect_v perpendicular_o to_o the_o line_n dam_fw-ge and_o it_o be_v also_o erect_v perpendicular_o to_o either_o of_o these_o line_n bd_o and_o dc_o wherefore_o the_o line_n ed_z be_v unto_o these_o three_o right_a line_n bd_o dam_fw-ge and_o dc_o erect_v perpendicular_o from_o the_o point_n where_o these_o three_o right_a line_n touch_v the_o one_o the_o other_o wherefore_o by_o the_o 5._o of_o the_o eleven_o these_o three_o right_a line_n bd_o dam_fw-ge and_o dc_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o in_o what_o superficies_n the_o line_n bd_o and_o da_z be_v in_o the_o self_n same_o also_o be_v the_o line_n basilius_n for_o every_o triangle_n be_v by_o the_o 2._o of_o the_o eleven_o in_o one_o and_o the_o self_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 parallel_a line_n be_v for_o if_o not_o then_o if_o it_o be_v possible_a let_v it_o be_v in_o a_o high_a superficies_n impossibility_n as_o the_o line_n egf_n be_v and_o draw_v the_o superficies_n wherein_o the_o line_n egf_n be_v &_o extend_v it_o and_o it_o shall_v make_v a_o common_a section_n with_o the_o ground_n superficies_n which_o
section_n shall_v by_o the_o 3._o of_o the_o eleven_o be_v a_o right_a line_n let_v that_o section_n be_v the_o right_a line_n ef._n wherefore_o two_o right_a line_n egf_n and_o of_o include_v a_o superficies_n which_o by_o the_o last_o common_a sentence_n be_v impossible_a wherefore_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 not_o in_o a_o high_a superficies_n wherefore_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 superficies_n wherein_o be_v the_o parallel_n right_a line_n ab_fw-la and_o cd_o if_o therefore_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 thâ—se_a point_n be_v in_o the_o self_n same_o plain_a superficies_n with_o the_o parallel_n right_a line_n which_o be_v require_v to_o be_v demonstrate_v by_o this_o figure_n it_o be_v easy_a to_o see_v the_o former_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o be_v draw_v the_o line_n egf_n the_o 8._o theorem_a the_o 8._o proposition_n if_o there_o be_v two_o parallel_n right_a line_n of_o which_o one_o be_v erect_v perpendicular_o to_o a_o round_a plain_a superficies_n the_o other_o also_o be_v erect_v perpendicular_o to_o the_o self_n same_o ground_n plain_a superficies_n svppose_v that_o there_o be_v two_o parallel_n right_a line_n ab_fw-la and_o cd_o and_o let_v one_o of_o they_o namely_o ab_fw-la be_v erect_v perpendicular_o to_o a_o ground_n superficies_n construction_n then_o i_o say_v that_o the_o line_n cd_o be_v also_o erect_v perpendicular_o to_o the_o self_n same_o ground_n superficies_n let_v the_o line_n ab_fw-la and_o cd_o fall_n upon_o the_o ground_n superficies_n in_o tâ—e_z point_n b_o and_o d_o and_o by_o the_o first_o petition_n draw_v a_o righâ—_n line_n from_o the_o point_n b_o to_o the_o point_n d._n and_o draw_v by_o the_o 11._o of_o the_o first_o in_o the_o ground_n superficies_n from_o the_o point_n d_o unto_o the_o line_n bd_o a_o perpendicular_a line_n de_fw-fr and_o by_o the_o 2._o of_o the_o first_o put_v the_o line_n de_fw-fr equal_a to_o the_o line_n ab_fw-la and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n a_o to_o the_o point_n e_o and_o a_o other_o from_o the_o â—oint_n a_o to_o the_o point_n d._n demonstration_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v ere_v perpendicular_o to_o the_o ground_n superficiece_n therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o all_o the_o right_a line_n that_o be_v in_o the_o ground_n superficies_n and_o touch_v it_o wherefore_o either_o of_o these_o angle_n abdella_n &_o abe_n be_v a_o right_a angle_n and_o forasmuch_o as_o upon_o these_o parallel_a line_n ab_fw-la and_o cd_o fall_v a_o certain_a right_a line_n bd_o therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n abdella_n and_o cdb_n be_v equal_a to_o two_o right_a angle_n but_o the_o angle_n abdella_n be_v a_o right_a angle_n wherefore_o also_o the_o angle_n cdb_n be_v a_o right_a angle_n wherefore_o the_o line_n cd_o be_v erect_v perpendicular_o to_o the_o line_n bd._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n â—d_a be_v common_a to_o they_o both_o therefore_o these_o two_o line_n ab_fw-la and_o bd_o be_v equal_a to_o these_o two_o line_n ed_z and_o db_o and_o the_o angle_n abdella_n be_v equal_a to_o the_o angle_n edb_n for_o either_o of_o they_o be_v a_o right_a angle_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ad_fw-la be_v equal_a to_o the_o base_a be._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n be_v to_o the_o linâ—_n ad_fw-la therefore_o these_o two_o line_n ab_fw-la and_o be_v be_v equal_a to_o these_o two_o line_n ad_fw-la &_o de_fw-fr the_o onâ—_n to_o the_o other_o and_o the_o line_n ae_n be_v a_o common_a base_a to_o they_o both_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n abe_n be_v equal_a to_o the_o angle_n ade_n but_o the_o angle_n aâ—e_n be_v a_o right_a angle_n wherefore_o thâ—â—ngle_a eda_n be_v also_o a_o right_a angle_n wherefore_o the_o line_n ed_z be_v erect_v perpendicular_o to_o the_o line_n ad_fw-la and_o it_o be_v also_o erect_v perpendicular_o to_o thâ—_z line_n db._n wherefore_o the_o line_n ed_z be_v erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o thâ—_z lâ—nâ—s_v bd_o and_o ba_o be_v by_o the_o 4._o of_o â—his_n book_n wherefore_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n ed_z be_v erect_v perpendicular_o to_o all_o the_o right_a line_n that_o touch_v it_o and_o be_v in_o the_o superficies_n wherein_o the_o line_n bd_o and_o ad_fw-la be_v but_o in_o what_o superficies_n the_o line_n bd_o and_o da_z be_v in_o the_o self_n same_o superficies_n be_v the_o line_n dc_o for_o the_o line_n ad_fw-la be_v draw_v from_o two_o point_n take_v in_o the_o parallel_a line_n ab_fw-la and_o cd_o be_v by_o the_o former_a proposition_n in_o the_o self_n same_o superficies_n with_o they_o now_o forasmuch_o as_o the_o line_n ab_fw-la and_o bd_o arâ—_n in_o the_o superficies_n wherein_o the_o line_n bd_o and_o da_z be_v but_o in_o what_o superficies_n the_o line_n ab_fw-la &_o bd_o be_v in_o the_o same_o be_v the_o line_n dc_o wherefore_o the_o line_n ed_z be_v erect_v perpendicular_o to_o the_o line_n dc_o wherefore_o also_o the_o line_n cd_o be_v erect_v perpendicular_o to_o the_o line_n de._n and_o the_o line_n cd_o be_v erect_v perpendicular_o to_o the_o line_n db._n for_o by_o the_o 29._o of_o the_o first_o the_o angle_n cdb_n be_v equal_a to_o the_o angle_n abdella_n be_v a_o right_a angle_n wherefore_o the_o line_n cd_o be_v from_o the_o point_n d_o erect_v perpendicular_o to_o two_o right_a line_n de_fw-fr and_o db_o cut_v the_o one_o the_o other_o in_o the_o point_n d._n wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o line_n cd_o be_v erect_v perpendiculaaly_a to_o the_o plain_a superficies_n wherein_o be_v the_o line_n de_fw-fr and_o db._n but_o the_o ground_n plain_a superficies_n be_v that_o wherein_o be_v the_o line_n de_fw-fr and_o db_o to_o which_o superficies_n also_o the_o line_n ab_fw-la be_v suppose_v to_o be_v erect_v perpendicular_o wherefore_o the_o line_n cd_o be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n whereunto_o the_o line_n ab_fw-la be_v erect_v perpendicular_o if_o therefore_o there_o be_v two_o parallel_n right_a line_n of_o which_o one_o be_v erect_v perpendicular_o to_o a_o ground_n plain_a superficies_n the_o other_o also_o be_v erect_v perpendicular_o to_o the_o self_n same_o ground_n plain_a superficies_n which_o be_v require_v to_o be_v demonstrate_v this_o figure_n will_v more_o clear_o set_v forth_o the_o former_a demonstration_n if_o you_o erect_v perpendicular_o the_o superficies_n abdella_n to_o the_o superficies_n bde_v and_o imagine_v a_o line_n to_o be_v draw_v from_o the_o point_n a_o to_o the_o point_n d_o in_o stead_n whereof_o as_o in_o the_o 6._o proposition_n you_o may_v extend_v a_o thread_n ¶_o the_o 9_o theorem_a the_o 9_o pro_fw-la 〈◊〉_d right_a line_n which_o be_v parallel_n to_o one_o and_o the_o self_n same_o right_a line_n and_o be_v not_o in_o the_o self_n same_o superficies_n that_o it_o be_v in_o be_v also_o parallel_v the_o one_o to_o the_o other_o svppose_v that_o either_o of_o these_o right_a line_n ab_fw-la and_o cd_o be_v a_o parallel_n to_o the_o line_n of_o not_o be_v in_o the_o self_n same_o superficies_n with_o it_o then_o i_o say_v that_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o take_v in_o the_o line_n of_o a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v g._n construction_n and_o from_o the_o point_n g_o raise_v up_o in_o the_o superficies_n wherein_o be_v the_o line_n of_o and_o ab_fw-la unto_o the_o line_n of_o a_o perpendicular_a line_n gh_o and_o again_o in_o the_o superficies_n wherein_o be_v the_o line_n of_o and_o cd_o raise_v up_o from_o the_o same_o point_n g_o to_o the_o line_n of_o a_o perpendicular_a line_n gk_o demonstration_n and_o forasmuch_o as_o the_o line_n of_o be_v erect_v perpendicular_o to_o either_o of_o the_o line_n gh_o and_o gk_o therefore_o by_o the_o 4._o of_o the_o eleven_o
the_o line_n of_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o the_o line_n gh_o and_o gk_o be_v but_o the_o line_n of_o be_v a_o parallel_a line_n to_o the_o line_n ab_fw-la wherefore_o by_o tâ—e_z 8._o of_o the_o eleven_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o be_v the_o line_n gh_o and_o gk_o and_o by_o the_o same_o reason_n also_o the_o line_n cd_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o be_v the_o line_n gh_o &_o gk_o wherefore_o either_o of_o these_o line_n ab_fw-la and_o cd_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o the_o line_n gh_o and_o gk_o be_v but_o if_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 by_o the_o 6._o of_o the_o eleven_o wherefore_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o wherefore_o right_a line_n which_o be_v parallel_n to_o one_o &_o the_o self_n same_o right_a line_n and_o be_v not_o in_o the_o self_n same_o superficies_n with_o it_o be_v also_o parallel_v the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o figure_n more_o clear_o manifest_v the_o former_a proposition_n and_o demonstration_n if_o you_o elevate_v the_o superficiece_n abef_n and_o cdef_n that_o they_o may_v incline_v and_o concur_v in_o the_o line_n ef._n ¶_o the_o 10._o theorem_a the_o 1_o 〈…〉_o if_o two_o right_a line_n touch_v the_o one_o the_o other_o 〈…〉_o she_o right_n line_n touch_v the_o one_o the_o other_o and_o no_o 〈…〉_o lfe_n same_o superficies_n with_o the_o two_o first_o those_o right_a line_n contain_v equal_a angle_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o bc_o touch_v the_o one_o the_o other_o be_v parallel_n to_o these_o two_o line_n de_fw-fr and_o of_o touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o the_o self_n same_o superficies_n that_o the_o line_n ab_fw-la and_o bc_o be_v then_o i_o say_v that_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n construction_n for_o let_v the_o line_n ba_o bc_o ed_z of_o be_v put_v equal_a the_o one_o to_o the_o other_o and_o draw_v these_o right_a line_n ad_fw-la cf_o be_v ac_fw-la and_o df._n demonstration_n and_o forasmuch_o as_o the_o line_n basilius_n be_v equal_a to_o the_o line_n ed_z and_o also_o parallel_v unto_o it_o therefore_o by_o the_o 33._o of_o the_o first_o the_o line_n ad_fw-la be_v equal_a and_o parallel_v to_o the_o line_n be_v and_o by_o the_o same_o reason_n also_o the_o line_n cf_o be_v equal_a &_o parallel_v to_o the_o line_n be._n wherefore_o either_o of_o these_o line_n ad_fw-la and_o cf_o be_v equal_a &_o parallel_v to_o the_o line_n ebb_n but_o right_a line_n which_o be_v parallel_n to_o one_o and_o the_o self_n same_o right_a line_n and_o be_v not_o in_o the_o self_n same_o superficies_n with_o it_o be_v also_o by_o the_o 9_o of_o the_o eleven_o parallel_n the_o one_o to_o the_o other_o wherefore_o the_o line_n ad_fw-la be_v a_o parallel_a line_n to_o the_o line_n cf._n and_o the_o line_n ac_fw-la and_o df_o join_v they_o together_o wherefore_o by_o the_o 33._o of_o the_o first_o the_o line_n ac_fw-la be_v equal_a and_o parallel_v to_o the_o line_n df._n and_o forasmuch_o as_o these_o two_o right_a line_n ab_fw-la &_o bc_o be_v equal_a to_o these_o two_o right_a line_n de_fw-fr and_o of_o and_o the_o base_a ac_fw-la also_o be_v equal_a to_o the_o base_a df_o therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n if_o therefore_o two_o right_a line_n touch_v the_o one_o the_o other_o be_v parallel_n to_o two_o other_o right_a line_n touch_v the_o one_o the_o other_o and_o not_o be_v in_o one_o and_o the_o self_n same_o superficies_n with_o the_o two_o first_o those_o righâ—_n line_n contain_v equal_a angle_n which_o be_v require_v to_o be_v demonstrate_v this_o figure_n here_o set_v more_o plain_o declare_v the_o former_a proposition_n and_o demonstration_n if_o you_o elevate_v the_o superficiece_n dabe_n and_o fcbe_n till_o they_o concur_v in_o the_o line_n fe_o ¶_o the_o 1._o problem_n the_o 11._o proposition_n from_o a_o point_n give_v on_o high_a to_o draw_v unto_o a_o ground_n plain_a superficies_n a_o perpendicular_a right_a line_n case_n svppose_v that_o the_o point_n give_v on_o high_a be_v a_o and_o suppose_v a_o ground_n plain_a superficies_n namely_o bcgh_n it_o be_v require_v from_o the_o point_n a_o to_o draw_v unto_o the_o ground_n superficies_n a_o perpendicular_a line_n draw_v in_o the_o ground_n superficies_n a_o right_a line_n at_o adventure_n and_o let_v the_o same_o be_v bc._n dee_n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n a_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a line_n ad._n practise_v now_o if_o ad_fw-la be_v a_o perpendicular_a line_n to_o the_o ground_n superficies_n then_o be_v that_o do_v which_o be_v seek_v for_o but_o if_o not_o then_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n d_o raise_v up_o in_o the_o ground_n superficies_n unto_o the_o line_n bc_o a_o perpendicular_a line_n de._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n a_o draw_v unto_o the_o line_n de_fw-fr a_o perpendicular_a line_n af._n and_o by_o the_o point_n f_o draw_v by_o the_o 31._o of_o the_o â—irst_n unto_o the_o line_n bc_o a_o parallel_a line_n fh_o and_o extend_v the_o line_n fh_o from_o the_o point_n f_o to_o the_o point_n g._n demonstration_n and_o forasmuch_o as_o the_o line_n bc_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n de_fw-fr and_o da_z therefore_o by_o the_o 4._o of_o the_o eleven_o the_o line_n bc_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o the_o line_n ed_z and_o ad_fw-la be_v and_o to_o the_o line_n bc_o the_o line_n gh_o be_v a_o parallel_n but_o iâ—_z there_o be_v two_o parallel_n right_a line_n of_o which_o one_o be_v erect_v perpendicular_o to_o a_o certain_a plain_a superficies_n the_o other_o also_o by_o the_o 8._o of_o the_o eleven_o be_v erect_v perpendicular_o to_o the_o self_n same_o superficies_n wherefore_o the_o line_n gh_o be_v erect_v perpendicular_o to_o the_o plainâ—_n superficies_n wherein_o the_o line_n ed_z and_o da_z be_v wherefore_o also_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n gh_o be_v erect_v perpendicular_o to_o all_o the_o right_a line_n which_o touch_v it_o and_o be_v in_o the_o plain_a superficies_n wherein_o the_o line_n ed_z and_o ad_fw-la be_v but_o the_o line_n of_o touch_v it_o be_v in_o the_o superficies_n wherein_o the_o line_n ed_z and_o ad_fw-la be_v by_o the_o â—_o of_o this_o book_n wherefore_o the_o line_n gh_o be_v erect_v perpendicular_o to_o the_o line_n fa._n wherefore_o also_o the_o line_n favorina_n be_v erect_v perpendicular_o to_o the_o line_n gh_o and_o the_o line_n of_o be_v also_o erect_v perpendicular_o to_o the_o line_n de._n wherefore_o of_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n hg_o and_o de._n but_o if_o a_o right_a line_n be_v erect_v perpendicular_o from_o the_o common_a section_n of_o two_o right_a line_n cut_v the_o one_o the_o other_o it_o shall_v also_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o say_v two_o line_n by_o the_o 4._o of_o the_o eleven_o wherefore_o the_o line_n of_o be_v erect_v perpendicular_o to_o that_o superficies_n wherein_o the_o line_n ed_z and_o gh_o be_v but_o the_o superficies_n wherein_o the_o line_n ed_z and_o gh_o be_v be_v the_o ground_n superficies_n wherefore_o the_o line_n of_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n wherefore_o from_o a_o point_n give_v on_o high_a namely_o from_o the_o point_n a_o be_v draw_v to_o the_o ground_n superficies_n a_o perpendicular_a line_n which_o be_v require_v to_o be_v do_v in_o this_o figure_n shall_v you_o much_o more_o plain_o see_v both_o the_o case_n of_o this_o former_a demonstration_n for_o as_o touch_v the_o first_o case_n you_o must_v erect_v perpendicular_o to_o the_o ground_n superficies_n the_o superficies_n wherein_o be_v draw_v the_o line_n ad_fw-la and_o compare_v it_o with_o the_o demonstration_n and_o it_o will_v be_v clear_a unto_o you_o for_o the_o second_o case_n you_o must_v erect_v perpendicular_o unto_o the_o ground_n superficies_n the_o superficies_n wherein_o be_v draw_v the_o line_n of_o and_o unto_o it_o let_v the_o other_o superficies_n wherein_o be_v draw_v the_o line_n ad_fw-la incline_v so_o that_o the_o point_n a_o of_o the_o one_o may_v concur_v with_o the_o point_n a_o of_o the_o other_o and_o so_o with_o your_o figure_n thus_o order_v compare_v it_o with_o
the_o demonstration_n and_o there_o will_v be_v in_o it_o no_o hardness_n at_o all_o ¶_o the_o 2._o problem_n the_o 12._o proposition_n unto_o a_o plain_a superficies_n give_v and_o from_o a_o point_n in_o it_o give_v to_o raise_v up_o a_o perpendicular_a line_n svppose_v that_o there_o be_v a_o ground_n plain_a superficies_n give_v and_o let_v the_o point_n in_o it_o give_v be_v a._n it_o be_v require_v from_o the_o point_n a_o to_o raise_v up_o unto_o the_o ground_n plain_a superficies_n a_o perpendicular_a line_n understand_v some_o certain_a point_n on_o high_a and_o let_v the_o same_o be_v b._n construction_n and_o from_o the_o point_n b_o draw_v by_o the_o 11._o of_o the_o eleven_o a_o perpendicular_a line_n to_o the_o ground_n superficies_n and_o let_v the_o same_o be_v bc._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n a_o draw_v unto_o the_o line_n bc_o a_o parallel_a line_n da._n now_o forasmuch_o as_o there_o be_v two_o parallel_n right_a line_n ad_fw-la and_o cb_o the_o one_o of_o they_o namely_o demonstration_n cb_o be_v erect_v perpendicular_o to_o the_o 〈◊〉_d superficies_n wherefore_o the_o other_o line_n also_o namely_o ad_fw-la be_v 〈◊〉_d perpendicular_o to_o the_o same_o ground_n superficies_n by_o the_o eight_o of_o 〈…〉_o leventh_o wherefore_o unto_o a_o plain_a superficies_n give_v and_o 〈◊〉_d point_n in_o it_o give_v namely_o a_o be_v raise_v up_o a_o perpendicular_a lynâ—â—required_v to_o be_v do_v in_o this_o second_o figure_n you_o may_v consider_v plain_o the_o demonstration_n of_o the_o former_a proposition_n if_o you_o erect_v perpendicular_o the_o superficies_n wherein_o be_v draw_v the_o line_n ad_fw-la and_o cb._n ¶_o the_o 11._o theorem_a the_o 13._o prâ—position_n from_o one_o and_o the_o self_n point_n and_o to_o one_o and_o the_o self_n same_o plain_a superficies_n can_v not_o be_v erect_v two_o perpendicular_a right_a line_n on_o one_o and_o the_o self_n same_o side_n for_o if_o it_o be_v possible_a from_o the_o point_n a_o let_v there_o be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n two_o righâ—_n line_n ab_fw-la and_o ac_fw-la on_o one_o and_o the_o self_n same_o side_n impossibility_n and_o extend_v the_o superficies_n wherein_o be_v the_o line_n ab_fw-la and_o ac_fw-la mathematical_a and_o it_o shall_v make_v at_o length_n a_o common_a section_n in_o the_o ground_n superficies_n which_o common_a section_n shall_v be_v a_o right_a line_n and_o shall_v pass_v by_o the_o point_n a_o let_v that_o common_a section_n be_v the_o line_n dae_n wherefore_o by_o the_o 3._o of_o the_o eleven_o the_o line_n ab_fw-la ac_fw-la and_o dae_n 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 ca_n be_v erect_v perpendicular_o to_o the_o ground_n superficies_n therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o it_o make_v right_a angle_n with_o all_o the_o right_a line_n that_o touch_v it_o and_o be_v in_o the_o ground_n superficies_n but_o the_o line_n dae_n touch_v it_o be_v in_o the_o ground_n superficies_n wherefore_o the_o angle_n caesar_n be_v a_o right_a angle_n and_o by_o the_o same_o reason_n also_o the_o angle_n bae_o be_v a_o right_a angle_n wherefore_o by_o the_o 4_o petition_n the_o angle_n caesar_n be_v equal_a to_o the_o angle_n bae_o the_o less_o to_o the_o more_o both_o angle_n be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n which_o be_v impossible_a wherefore_o from_o one_o and_o the_o self_n same_o point_n and_o to_o one_o and_o the_o self_n same_o plain_a superficies_n can_v not_o be_v erect_v two_o perpendicular_a right_a line_n on_o one_o &_o the_o self_n same_o side_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o figure_n if_o you_o erect_v perpendicular_o the_o superficies_n wherein_o be_v draw_v the_o line_n â—a_o and_o ca_n to_o the_o ground_n superficies_n wherein_o be_v draw_v the_o line_n dae_n and_o so_o compare_v it_o with_o the_o the_o demonstration_n of_o the_o former_a proposition_n it_o will_v be_v clear_a unto_o you_o m._n do_v you_o his_o annotation_n euclides_n word_n in_o this_o 13._o proposition_n admit_v two_o case_n one_o if_o thâ—_z 〈…〉_o in_o the_o plain_a superficies_n as_o common_o the_o demonstration_n suppose_v the_o other_o if_o the_o point_n assign_v be_v any_o where_o without_o the_o say_v plain_a superficies_n to_o which_o the_o perpendicular_n fall_v be_v consider_v contrary_n to_o either_o of_o which_o if_o the_o adversary_n affirm_v admit_v from_o one_o point_n two_o right_a line_n perpendicular_n to_o one_o and_o the_o self_n same_o plain_a superficies_n and_o on_o one_o and_o the_o same_o side_n thereof_o by_o the_o 6._o of_o the_o eleven_o he_o may_v be_v bridle_v which_o will_v â—ore_v he_o to_o confess_v his_o two_o perpendicular_n to_o be_v also_o parallel_v but_o by_o supposition_n agree_v one_o they_o concur_v at_o one_o and_o the_o same_o poynâ—_n which_o by_o the_o definition_n of_o parallel_n iâ—_z impossible_a therefore_o our_o adversary_n must_v recant_v aâ—d_n yield_v to_o out_o proposition_n ¶_o the_o 12._o theorem_a the_o 14._o proposition_n to_o whatsoever_o plain_a superficiece_n one_o and_o the_o self_n same_o right_a line_n be_v erect_v perpendicular_o those_o superficiece_n be_v parallel_n the_o one_o to_o the_o other_o svppose_v that_o a_o right_a line_n ab_fw-la be_v erect_v perpendicular_o to_o either_o of_o these_o plain_a superficiece_n cd_o and_o ef._n then_o i_o say_v that_o these_o superficiece_n cd_o and_o of_o be_v parallel_n the_o one_o to_o the_o other_o impossibility_n for_o if_o not_o then_o if_o they_o be_v extend_v they_o will_v at_o the_o length_n meet_v let_v they_o meet_v if_o it_o be_v possible_a now_o than_o their_o common_a section_n shall_v by_o the_o 3._o of_o the_o eleven_o be_v a_o right_a line_n let_v that_o common_a section_n be_v gh_o and_o in_o the_o line_n gh_o take_v a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v k._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n king_n and_o a_o other_o from_o the_o point_n b_o to_o the_o point_n k._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o therefore_o the_o line_n ab_fw-la be_v also_o erect_v perpendicular_o to_o the_o line_n bk_o which_o be_v in_o the_o extend_a superficies_n ef._n wherefore_o the_o angle_n abk_n be_v a_o right_a angle_n and_o by_o the_o same_o reason_n also_o the_o angle_n back_o be_v a_o right_a angle_n wherefore_o in_o the_o triangle_n abk_n these_o two_o angle_n abk_n &_o bak_n be_v equal_a to_o two_o right_a angle_n which_o by_o the_o 17._o of_o the_o first_o be_v impossible_a wherefore_o these_o superficiece_n cd_o and_o of_o be_v extend_v meet_v not_o together_o wherefore_o the_o superficiece_n cd_o and_o of_o be_v parallel_n wherefore_o to_o what_o soever_o plain_a superficiece_n one_o and_o the_o self_n same_o right_a line_n iâ—_z erect_v perpendicular_o those_o superficies_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 in_o this_o figure_n may_v you_o plain_o see_v the_o former_a demonstration_n if_o you_o erect_v the_o three_o superficiece_n gd_o ge_z and_o klm_v perpendiculary_a to_o the_o ground_n plain_a superficies_n but_o yet_o in_o such_o sort_n that_o the_o two_o superficiâ—â—_n 〈…〉_o may_v concur_v in_o the_o common_a line_n g_o 〈…〉_o the_o demonstration_n a_o corollary_n add_v by_o campane_n if_o a_o right_a line_n be_v erect_v perpendicular_o to_o one_o of_o those_o superficies_n it_o 〈…〉_o erect_v perpendicular_o to_o the_o other_o for_o if_o it_o shall_v not_o be_v erect_v perpendicular_o to_o the_o other_o than_o it_o fall_v upon_o that_o other_o shall_v make_v with_o some_o one_o line_n thereof_o a_o angle_n less_o than_o a_o right_a angle_n which_o line_n shall_v by_o the_o 5._o petition_n of_o the_o first_o concur_v with_o some_o one_o line_n of_o that_o superficies_n whereunto_o it_o be_v perpendicular_a so_o that_o those_o superficiece_n shall_v not_o be_v parallel_n which_o be_v contrary_a to_o the_o supposition_n for_o they_o be_v suppse_v to_o be_v parallel_n ¶_o the_o 13._o theorem_a the_o 15._o proposition_n if_o two_o right_a line_n touch_v the_o one_o the_o other_o be_v parallel_n to_o two_o other_o right_a line_n touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o the_o self_n same_o plain_a superficies_n with_o the_o two_o first_o the_o plain_a superficiece_n extend_v by_o those_o right_a line_n be_v also_o parallel_n the_o one_o to_o the_o other_o svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o bc_o touch_v the_o one_o the_o other_o be_v parallel_n to_o these_o two_o right_a line_n de_fw-fr &_o of_o touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o the_o self_n same_o plain_a superficies_n with_o
the_o right_a line_n ab_fw-la and_o bc._n then_o i_o say_v that_o the_o plain_a superficiece_n by_o the_o line_n ab_fw-la and_o bc_o and_o the_o line_n de_fw-fr and_o of_o be_v extend_v shall_v not_o meet_v together_o that_o be_v they_o be_v equedistant_a and_o parallel_n construction_n from_o the_o point_n b_o draw_v by_o the_o 11._o of_o the_o eleven_o a_o perpendicular_a line_n to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o and_o let_v that_o perpendicular_a line_n be_v bg_o and_o by_o the_o point_n g_o in_o the_o plain_a superficies_n pass_v by_o de_fw-fr and_o of_o draw_v by_o the_o 31._o of_o the_o first_o unto_o the_o line_n ed_z a_o parallel_a line_n gh_o and_o likewise_o by_o that_o point_n g_o draw_v in_o the_o same_o superficies_n unto_o the_o line_n of_o a_o parallel_a line_n gk_o demonstration_n and_o forasmuch_o as_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o it_o be_v also_o erect_v perpendicular_o to_o all_o the_o right_a line_n which_o touch_v it_o and_o be_v in_o the_o self_n same_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o ef._n but_o either_o of_o these_o line_n gh_o and_o gk_o touch_v it_o and_o be_v also_o in_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o therefore_o either_o of_o these_o angle_n bgh_o and_o bgk_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n basilius_n be_v a_o parallel_n to_o the_o line_n gh_o that_o the_o line_n gh_o and_o gk_o be_v parallel_n unto_o the_o line_n ab_fw-la and_o bc_o it_o be_v manifest_a by_o the_o 9_o of_o this_o book_n therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n gba_n and_o bgh_n be_v equal_a to_o two_o right_a angle_n but_o the_o angle_n bgh_n be_v by_o construction_n a_o right_a angle_n therefore_o also_o the_o angle_n gba_n be_v a_o right_a angle_n therefore_o the_o line_n gb_o be_v erect_v perpendicular_o to_o the_o line_n basilius_n and_o by_o the_o same_o reason_n also_o may_v it_o be_v prove_v that_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o line_n bc._n now_o forasmuch_o as_o the_o right_a line_n bg_o be_v erect_v perpendicular_o to_o these_o two_o right_a line_n basilius_n and_o bc_o touch_v the_o one_o the_o other_o therefore_o by_o the_o 4._o of_o the_o eleven_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n basilius_n and_o be_v and_o it_o be_v also_o erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n gh_o and_o gk_o but_o the_o superficies_n wherein_o be_v the_o line_n gh_o and_o gk_o be_v that_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o wherefore_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o ef._n wherefore_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o and_o to_o the_o superficies_n wherein_o be_v the_o line_n ab_fw-la and_o bc._n but_o if_o one_o and_o the_o self_n same_o right_a line_n be_v erect_v perpendicular_o to_o plain_a superficiece_n those_o superficiece_n be_v by_o the_o 14._o of_o the_o eleven_o parallel_v the_o one_o to_o the_o other_o wherefore_o the_o superficies_n wherein_o be_v the_o line_n ab_fw-la and_o bc_o be_v a_o parallel_n to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o ef._n if_o therefore_o two_o right_a line_n touch_v the_o one_o the_o other_o be_v parallel_n to_o two_o other_o right_a line_n touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o the_o self_n same_o plain_a superficies_n with_o the_o two_o first_o the_o plain_a superficiece_n extend_v by_o those_o right_a line_n be_v also_o parallel_v the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v by_o this_o figure_n here_o put_v you_o may_v more_o clear_o see_v both_o the_o former_a 15._o proposition_n and_o also_o the_o demonstration_n thereof_o if_o you_o erect_v perpendicular_o unto_o the_o ground_n superficies_n the_o three_o superficiece_n abc_n khe_n and_o lhbm_n and_o so_o compare_v it_o with_o the_o demonstration_n ¶_o a_o corollary_n add_v by_o flussas_n unto_o a_o plain_a superficies_n be_v give_v to_o draw_v by_o a_o point_n give_v without_o it_o a_o parallel_n plain_a superficies_n suppose_v as_o in_o the_o former_a description_n that_o the_o superficies_n give_v be_fw-mi abc_n &_o let_v the_o point_n give_v without_o it_o be_v g._n now_o then_o by_o the_o point_n g_o draw_v by_o the_o 31._o of_o the_o first_o unto_o the_o line_n ab_fw-la and_o bc_o parallel_v line_n gh_o and_o hk_o and_o the_o superficies_n extend_v by_o the_o line_n gh_o and_o gk_o shall_v be_v parallel_n unto_o the_o superficies_n abc_n by_o this_o 15._o proposition_n the_o 14._o theorem_a the_o 16._o proposition_n if_o two_o parallel_n plain_a superficiece_n be_v cut_v by_o some_o one_o plain_a superficies_n their_o common_a section_n be_v parallel_a line_n svppose_v that_o these_o two_o plain_a superficiece_n ab_fw-la and_o cd_o be_v cut_v by_o this_o plain_a superficies_n efgh_o â—nd_v let_v their_o common_a section_n be_v the_o right_a line_n of_o and_o gh_o then_o i_o say_v that_o the_o line_n of_o be_v a_o parallel_n to_o the_o line_n gh_o for_o if_o not_o than_o the_o line_n of_o and_o gh_o be_v produce_v shall_v at_o the_o length_n meet_v together_o either_o on_o the_o side_n that_o the_o point_n fh_o be_v or_o on_o the_o side_n that_o the_o point_n e_o g_o be_v first_o let_v they_o be_v produce_v on_o that_o side_n that_o the_o point_n f_o h_o be_v and_o let_v they_o mete_v in_o the_o point_n k._n and_o forasmuch_o as_o the_o line_n efk_n be_v in_o the_o superficies_n ab_fw-la therefore_o all_o the_o point_n which_o be_v in_o the_o line_n of_o be_v in_o the_o superficies_n ab_fw-la by_o the_o first_o of_o this_o book_n but_o one_o of_o the_o point_n which_o be_v in_o the_o right_a line_n efk_n be_v the_o point_n king_n absurdity_n therefore_o the_o point_n king_n be_v in_o the_o superficies_n ab_fw-la and_o by_o the_o same_o reason_n also_o the_o point_n king_n be_v in_o the_o superficies_n cd_o wherefore_o the_o two_o superficiece_n ab_fw-la and_o cd_o be_v produce_v do_v mete_v together_o but_o by_o supposition_n they_o meet_v not_o together_o for_o they_o be_v suppose_v to_o be_v parallel_n wherefore_o the_o right_a line_n of_o and_o gh_o produce_v shall_v not_o meet_v together_o on_o that_o side_n that_o the_o point_n f_o h_o be_v in_o like_a sort_n also_o may_v we_o prove_v that_o the_o right_a line_n of_o and_o gh_o produce_v meet_v not_o together_o on_o that_o side_n that_o the_o point_n e_o g_o be_v but_o right_a line_n which_o be_v produce_v on_o no_o side_n mete_v together_o be_v parallel_n by_o the_o last_o definition_fw-la of_o the_o first_o wherefore_o the_o line_n of_o be_v a_o parallel_n to_o the_o line_n gh_o if_o therefore_o two_o parallel_n plain_a superficiece_n be_v cut_v by_o some_o one_o plain_a superficies_n their_o common_a section_n be_v parallel_a line_n which_o be_v require_v to_o be_v prove_v this_o figure_n here_o set_v more_o plain_o 〈…〉_o demonstration_n if_o you_o erect_v perpendicular_a 〈…〉_o superficies_n the_o three_o superficiece_n aâ—_z 〈…〉_o and_o so_o compare_v it_o with_o the_o demonstrâ—_n 〈…〉_o a_o corollary_n add_v by_o flussas_n if_o two_o plain_a superficiece_n be_v parallel_n to_o one_o and_o the_o sâ—lfe_a same_o plain_n 〈…〉_o also_o be_v parallel_n the_o one_o to_o the_o other_o or_o they_o shall_v make_v one_o and_o the_o self_n same_o plain_a sup_n 〈…〉_o in_o this_o figure_n here_o set_v you_o may_v more_o plain_o see_v the_o forme_a demonstration_n if_o you_o elevate_v to_o the_o ground_n superâ—icieces_v acdi_n the_o three_o superâ—icieces_n ab_fw-la dg_o &_o give_v and_o â—o_o compare_v it_o with_o the_o demonstration_n the_o 15._o theorem_a the_o 17._o proposition_n iâ—_z two_o right_a line_n be_v cut_v by_o plain_a superficiece_n be_v parallel_n the_o part_n oâ—_n the_o line_n divide_v shall_v be_v proportional_a demonstration_n suppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v divide_v by_o these_o plain_a superfiâ—iâ—ces_n be_v parallel_n namely_o gh_o kl_o mn_v in_o the_o point_v a_o e_o b_o c_o f_o d._n then_o i_o say_v that_o as_o the_o right_a line_n ae_n be_v to_o the_o right_a line_n ebb_v so_o be_v the_o right_a line_n cf_o to_o the_o right_a line_n fd._n draw_v these_o right_a line_n ac_fw-la bd_o and_o ad._n and_o let_v the_o line_n ad_fw-la and_o the_o superficies_n kl_o concur_v in_o the_o point_n x._o and_o draw_v a_o right_a line_n from_o the_o point_n e_o to_o the_o point_n x_o and_o
a_o other_o from_o the_o point_n x_o to_o the_o point_n f._n and_o forasmuch_o as_o these_o two_o parallel_a superficiece_n kl_o and_o mn_o be_v cut_v by_o the_o superficies_n ebdx_n therefore_o their_o common_a section_n which_o be_v the_o line_n exit_fw-la and_o bd_o be_v by_o the_o 16._o of_o the_o eleven_o parallel_v the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n also_o forasmuch_o as_o the_o two_o parallel_a superficies_n gh_o and_o kl_o be_v cut_v by_o the_o superficies_n axfc_n their_o common_a section_n ac_fw-la and_o xf_n be_v by_o the_o 16._o of_o the_o eleven_o parallel_v and_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n abdâ—_n namely_o to_o the_o side_n bd_o be_v draw_v a_o parallel_a line_n exit_fw-la therefore_o by_o the_o 2._o of_o the_o six_o proportional_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n axe_n to_o the_o line_n xd_v again_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n adc_o namely_o to_o the_o side_n ac_fw-la be_v draw_v a_o parallel_a line_n xf_n therefore_o by_o the_o 2._o of_o the_o six_o proportional_o as_o the_o line_n axe_n be_v to_o the_o line_n xd_v so_o be_v the_o line_n cf_o to_o the_o line_n fd._n and_o it_o be_v prove_v that_o as_o the_o line_n axe_n be_v to_o the_o line_n xd_v so_o be_v the_o line_n ae_n to_o the_o line_n ebb_n therefore_o also_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n if_o therefore_o two_o right_a line_n â—e_z divide_v by_o plain_a superâ—icieces_n be_v parallel_n the_o part_n of_o the_o line_n divide_v shall_v be_v proportional_a which_o be_v require_v to_o be_v demonstrate_v in_o this_o figure_n it_o be_v more_o easy_a to_o see_v the_o former_a demonstration_n if_o you_o erect_v perpendicular_o unto_o the_o ground_n superficies_n acbd_v the_o three_o superficiece_n gh_o kl_o and_o mn_v or_o if_o you_o so_o â—râ—ct_v they_o that_o thâ—y_v be_v equedistant_a one_o to_o the_o other_o ¶_o the_o 16._o theorem_a the_o 18._o proposition_n if_o a_o right_a line_n be_v erect_v perpendicular_o to_o a_o plain_a superficies_n all_o the_o superficiece_n extend_v by_o that_o right_a line_n be_v erect_v perpendicular_o to_o the_o self_n same_o plain_a superficies_n svppose_v that_o a_o right_a line_n ab_fw-la be_v erect_v perpendicular_o to_o a_o ground_n superficies_n then_o i_o say_v that_o all_o the_o superficiece_n pass_v by_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n extend_v a_o superficies_n by_o the_o line_n ab_fw-la and_o let_v the_o same_o be_v ed_z &_o let_v the_o common_a section_n of_o the_o plain_a superficies_n and_o of_o the_o ground_n superficies_n be_v the_o right_a line_n ce._n and_o take_v in_o the_o line_n ce_fw-fr a_o point_n at_o all_o adventure_n construction_n and_o let_v the_o same_o be_v f_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n fletcher_n draw_v unto_o the_o line_n ce_fw-fr a_o perpendicular_a line_n in_o the_o superficies_n de_fw-fr and_o let_v the_o same_o be_v fg._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o all_o the_o right_a line_n that_o be_v in_o the_o ground_n plain_a superficies_n demonstration_n and_o which_o touch_v it_o wherefore_o it_o be_v erect_v perpendicular_o to_o the_o line_n ce._n wherefore_o the_o angle_n abe_n be_v a_o right_a angle_n and_o the_o angle_n gfb_n be_v also_o a_o right_a angle_n by_o construction_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 fg._n but_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n wherefore_o by_o the_o 8._o of_o the_o eleven_o the_o line_n fg_o be_v also_o erect_v perpendicular_o to_o the_o ground_n superficies_n and_o forasmuch_o as_o by_o the_o 3._o definition_n of_o the_o eleven_o a_o plain_a superficies_n be_v then_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 with_o the_o other_o plain_a superficies_n and_o it_o be_v prove_v that_o the_o line_n fg_o draw_v in_o one_o of_o the_o plain_a superficiece_n namely_o in_o de_fw-fr perpendicular_o to_o the_o common_a section_n of_o the_o plain_a superficiece_n namely_o to_o the_o line_n ce_fw-fr be_v erect_v perpendicular_o to_o the_o ground_n superficies_n wherefore_o the_o plain_a superficies_n de_fw-fr be_v erect_v perpendicular_o to_o the_o ground_n superficies_n in_o like_a sort_n also_o may_v we_o prove_v that_o all_o the_o plain_a superficiece_n which_o pass_v by_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n if_o therefore_o a_o right_a line_n be_v erect_v perpendicular_o to_o a_o plain_a superficies_n all_o the_o superficiece_n pass_v by_o the_o right_a line_n be_v erect_v perpendicular_o to_o the_o self_n same_o plain_a superficies_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o figure_n here_o set_v you_o may_v erect_v perpendicular_o at_o your_o pleasure_n the_o superficies_n wherein_o be_v draw_v the_o line_n dc_o gf_o ab_fw-la and_o he_o to_o the_o ground_n superficies_n wherein_o be_v draw_v the_o line_n cfbe_n and_o so_o plain_o compare_v it_o with_o the_o demonstration_n before_o put_v ¶_o the_o 17._o theorem_a the_o 19_o proposition_n if_o two_o plain_a superficiece_n cut_v the_o one_o the_o other_o be_v erect_v perpendicular_o to_o any_o plain_a superficies_n their_o common_a section_n be_v also_o erect_v perpendicular_o to_o the_o self_n same_o plain_a superficies_n svppose_v that_o these_o two_o plain_a superâ—icieces_n ab_fw-la &_o bc_o cut_v the_o one_o the_o other_o be_v erect_v perpendicular_o to_o a_o ground_n superficies_n and_o let_v their_o common_a section_n be_v the_o line_n bd._n then_o i_o say_v that_o the_o line_n bd_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n impossibility_n for_o if_o not_o then_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n d_o draw_v in_o the_o superficies_n ab_fw-la unto_o the_o right_a line_n da_z a_o perpendicular_a line_n de._n and_o in_o the_o superficies_n cb_o draw_v unto_o the_o line_n dc_o a_o perpendicular_a line_n df._n and_o forasmuch_o as_o the_o superficies_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n and_o in_o the_o plain_a superficies_n ab_fw-la unto_o the_o common_a section_n of_o the_o plain_a superficies_n and_o of_o the_o ground_n superficies_n namely_o to_o the_o line_n dam_n be_v erect_v a_o perpendicular_a line_n de_fw-fr therefore_o by_o the_o converse_n of_o the_o 3._o definition_n of_o this_o book_n the_o line_n de_fw-fr be_v erect_v perpendicular_o to_o the_o ground_n superficies_n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o line_n df_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n wherefore_o from_o one_o and_o the_o self_n same_o point_n namely_o from_o d_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n two_o right_a line_n both_o on_o one_o and_o the_o self_n same_o side_n which_o be_v by_o the_o 15._o of_o the_o eleven_o impossible_a wherefore_o from_o the_o point_n d_o can_v not_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n any_o other_o right_a line_n beside_o bd_o which_o be_v the_o common_a section_n of_o the_o two_o superficiece_n ab_fw-la and_o bc._n if_o therefore_o two_o plain_a superâ—icieces_n cut_v the_o one_o the_o other_o be_v erect_v perpendicular_o to_o any_o plain_a superficies_n their_o common_a section_n be_v also_o erect_v perpendicular_o to_o the_o self_n same_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v here_o have_v i_o set_v a_o other_o figure_n which_o will_v more_o plain_o show_v unto_o you_o the_o former_a demonstration_n if_o you_o erect_v perpendicular_o to_o the_o ground_n superficies_n ac_fw-la the_o two_o superficiece_n ab_fw-la and_o bc_o which_o cut_v the_o one_o the_o other_o in_o the_o line_n bd._n the_o 18._o theorem_a the_o 20._o proposition_n if_o a_o solid_a angle_n be_v contain_v under_o three_o plain_a superficial_a angle_n every_o two_o of_o those_o three_o angle_n which_o then_o so_o ever_o be_v take_v be_v great_a than_o the_o three_o svppose_v that_o the_o solid_a angle_n a_o be_v contain_v under_o three_o plain_a superficial_a angle_n that_o be_v under_o bac_n god_n and_o dab_n then_o i_o say_v that_o two_o of_o these_o superficial_a angle_n how_o so_o ever_o they_o be_v take_v be_v great_a than_o the_o three_o if_o the_o
the_o line_n dc_o wherefore_o the_o superficies_n ac_fw-la be_v a_o parallelogram_n in_o like_a sort_n also_o may_v we_o prove_v that_o every_o one_o of_o these_o superficice_n ce_fw-fr gf_o bg_o fb_o and_o ae_n be_v parallelogram_n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n h_o and_o a_o other_o from_o the_o point_n d_o to_o the_o point_n f._n equal_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v prove_v a_o parallel_n to_o the_o line_n cd_o and_o the_o line_n bh_o to_o the_o line_n cf_o therefore_o these_o two_o right_a line_n ab_fw-la and_o bh_o touch_v the_o one_o the_o other_o be_v parallel_n to_o these_o two_o right_a line_n dc_o and_o cf_o touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n wherefore_o by_o the_o 10._o of_o the_o eleven_o they_o comprehend_v equal_a angle_n wherefore_o the_o angle_n abh_n be_v equal_a to_o the_o angle_n dcf_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bh_o be_v proof_n equal_a to_o these_o two_o line_n dc_o and_o cf_o and_o the_o angle_n abh_n be_v prove_v equal_a to_o the_o angle_n dcfâ—_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ah_o be_v equal_a to_o the_o base_a df_o and_o the_o triangle_n abh_n be_v equal_a to_o the_o triangle_n dcf_n and_o forasmuch_o as_o by_o the_o 41._o of_o the_o first_o the_o parallelogram_n bg_o be_v double_a to_o the_o triangle_n abh_n and_o the_o parallelogram_n ce_fw-fr be_v also_o double_a to_o the_o triangle_n dcf_n therefore_o the_o parallelogram_n bg_o be_v equal_a to_o the_o parallelogram_n ce._n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o parallelogram_n ac_fw-la be_v equal_a to_o the_o parallelogram_n gf_o and_o the_o parallelogram_n ae_n to_o the_o parallelogram_n fb_o if_o therefore_o a_o solid_a or_o body_n be_v contain_v under_o six_o parallel_n plain_a superficiece_n the_o opposite_a plain_a superficiece_n of_o the_o same_o body_n be_v equal_a &_o parallelogram_n which_o be_v require_v to_o be_v demonstrate_v i_o have_v for_o the_o better_a help_n of_o young_a beginner_n describe_v here_o a_o other_o figure_n who_o form_n if_o it_o be_v describe_v upon_o past_v paper_n with_o the_o letter_n place_v in_o the_o same_o order_n that_o it_o be_v here_o and_o then_o if_o you_o cut_v fine_o these_o line_n agnostus_n de_fw-fr and_o cf_o not_o through_o the_o paper_n and_o fold_v it_o according_o compare_v it_o with_o the_o demonstration_n and_o it_o will_v shake_v of_o all_o hardness_n from_o it_o the_o 22._o theorem_a the_o 25._o proposition_n if_o a_o parallelipipedom_n be_v cut_v of_o a_o plain_a superficies_n be_v a_o parallel_n to_o the_o two_o opposite_a plain_a superficiece_n of_o the_o same_o body_n then_o as_o the_o base_a be_v to_o the_o base_a so_o be_v the_o one_o solid_a to_o the_o other_o solid_a i_o have_v for_o the_o better_a sight_n of_o the_o construction_n &_o demonstration_n of_o the_o former_a 25._o proposition_n here_o set_v another_o figure_n who_o form_n if_o you_o describe_v upon_o past_v paper_n and_o fine_o cut_v the_o three_o line_n xi_o b_n and_o tie_v not_o through_o the_o paper_n but_o half_a way_n and_o then_o fold_v it_o according_o and_o compare_v it_o with_o the_o construction_n and_o demonstration_n you_o shall_v see_v that_o it_o will_v geve_v great_a light_n thereunto_o here_o flussas_n add_v three_o corollary_n first_o corollary_n if_o a_o prism_n be_v cut_v of_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a superficiece_n corollary_n the_o session_n of_o the_o prism_n shall_v be_v the_o one_o to_o the_o other_o in_o that_o proportion_n that_o the_o section_n of_o the_o base_a be_v the_o one_o to_o the_o other_o for_o the_o section_n of_o the_o base_n which_o base_n by_o the_o 11._o definition_n of_o this_o book_n be_v parallelogram_n shall_v likewise_o be_v parallelogram_n by_o the_o 16._o of_o this_o book_n when_o as_o the_o superficies_n which_o cut_v be_v parallelel_n to_o the_o opposite_a superâ—icieces_n and_o shall_v also_o be_v equiangle_n wherefore_o if_o unto_o the_o base_n by_o produce_v the_o right_a line_n be_v add_v like_o and_o equal_a base_n as_o be_v before_o show_v in_o a_o parallelipipedon_n of_o those_o section_n shall_v be_v make_v as_o many_o like_a prism_n as_o you_o will._n and_o so_o by_o the_o same_o reason_n namely_o by_o the_o common_a excess_n equality_n or_o want_v of_o the_o multiplices_fw-la of_o the_o base_n &_o of_o the_o section_n by_o the_o 5._o definition_n of_o the_o five_o may_v be_v prove_v that_o every_o section_n of_o the_o prism_n multiply_v by_o any_o multiplication_n whatsoever_o shall_v have_v to_o any_o other_o section_n that_o proportion_n that_o the_o section_n of_o the_o base_n have_v second_o corollary_n solides_n who_o be_v two_o opposite_a superficieâ—es_n be_v poligonon_o figure_n like_o equal_a and_o parallel_n the_o other_o superficies_n which_o of_o necessity_n be_v parallelogram_n column_n be_v cut_v of_o a_o plain_a superficies_n parallel_v to_o the_o two_o opposite_a superficies_n the_o section_n of_o the_o base_a be_v the_o one_o to_o the_o other_o as_o the_o section_n of_o the_o solid_a be_v thâ—_z one_o to_o the_o other_o which_o thing_n be_v manifest_a for_o such_o solid_n be_v divide_v into_o prism_n which_o have_v one_o common_a side_n namely_o the_o axe_n or_o right_a line_n which_o be_v draw_v by_o the_o centre_n of_o the_o opposite_a base_n wherefore_o how_o many_o paâ—allelogrammes_n or_o base_n be_v set_v upon_o the_o opposite_a poligonon_fw-fr figure_n of_o so_o many_o prism_n shall_v the_o whole_a solid_a be_v compose_v for_o those_o polygonon_n figure_n be_v divide_v into_o so_o many_o like_a triangle_n by_o the_o 20._o of_o the_o six_o which_o describe_v prism_n which_o prism_n be_v cut_v by_o a_o superficies_n parallel_v to_o the_o opposite_a superficiece_n the_o section_n of_o the_o base_n shall_v by_o the_o former_a corollary_n be_v proportional_a with_o the_o section_n of_o the_o prism_n wherefore_o by_o the_o â—â—_o of_o the_o five_o as_o the_o section_n of_o the_o one_o be_v the_o one_o to_o the_o other_o so_o be_v the_o section_n of_o the_o whole_a the_o one_o to_o the_o other_o of_o these_o solid_n there_o be_v infinite_a kind_n according_a to_o the_o variety_n of_o the_o opposite_a and_o parallel_v poligonon_o figure_n which_o poligonon_o figure_n do_v alter_v the_o angle_n of_o the_o parallelogram_n set_v upon_o they_o according_a to_o the_o diversity_n oâ—_n their_o situation_n but_o this_o be_v no_o let_v at_o all_o to_o this_o corollary_n for_o that_o which_o we_o have_v prove_v will_v always_o follow_v when_o as_o the_o superficiece_n which_o be_v set_v upon_o the_o opposite_a like_a &_o equal_a polygonon_n and_o parallel_n superficiece_n be_v always_o parallelogram_n three_o corollary_n corollary_n tâ—e_z foresay_a solid_n compose_v of_o prism_n be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a superficiece_n be_v the_o one_o to_o the_o other_o as_o the_o head_n or_o high_a part_n cut_v be_v for_o it_o be_v prove_v that_o they_o be_v the_o one_o to_o the_o other_o as_o the_o base_n be_v which_o base_n forasmuch_o as_o they_o be_v parâ—llelogrammes_n be_v the_o one_o to_o the_o other_o as_o the_o right_a line_n be_v upon_o which_o they_o be_v set_v by_o the_o first_o of_o the_o six_o which_o right_a line_n be_v the_o head_n or_o high_a part_n of_o the_o prism_n the_o 4._o problem_n the_o 26._o proposition_n upon_o a_o right_a line_n give_v and_o at_o a_o point_n in_o it_o give_v to_o make_v a_o solid_a angle_n equal_a to_o a_o solid_a angle_n give_v in_o these_o two_o 〈…〉_o here_o put_v you_o may_v in_o 〈◊〉_d clear_o concern_v the_o â—â—â—â—mer_a construction_n and_o dâ—â—monstration_n if_o you_o erect_v peâ—â—pendicularly_o unto_o the_o ground_n superficies_n the_o triangle_n alb_n and_o dce_n &_o elevate_v the_o triangle_n alh_n and_o dcf_n that_o the_o line_n la_fw-fr and_o cd_o of_o they_o may_v exact_o agree_v with_o the_o lineâ—_z la_fw-fr and_o cd_o of_o the_o â—riangles_n erecâ—edâ—_n for_o so_o order_v they_o if_o you_o compare_v the_o former_a construction_n and_o demonstration_n with_o they_o they_o will_v be_v plaint_n unto_o you_o although_o euclides_n demonstration_n be_v touch_v solid_a angle_n which_o be_v contain_v only_o under_o three_o superficial_a angle_n that_o be_v who_o base_n be_v triangle_n yet_o by_o it_o may_v you_o perform_v the_o problem_n touch_v solid_a angle_n contain_v under_o superficial_a angle_n how_o many_o soever_o that_o be_v have_v to_o their_o base_n any_o kind_n of_o poligonon_n figure_n for_o every_o poligonon_n figure_n may_v by_o the_o 20._o of_o the_o six_o be_v resolve_v into_o like_a tringles_n and_o so_o also_o shall_v the_o solid_a angle_n be_v divide_v into_o so_o many_o solid_a angle_n as_o there_o be_v
superficies_n give_v to_o be_v â—_o and_o the_o rectangle_n parallelipipedon_n give_v to_o be_v am._n upon_o â—_o as_o a_o base_a must_n be_o be_v apply_v that_o be_v a_o rectangle_n parâ—llipipedon_n must_v be_v erect_v upon_o â—_o as_o a_o base_a which_o shall_v be_v equal_a to_o am._n by_o the_o last_o of_o the_o second_o to_o the_o right_n line_v figure_n â—_o let_v a_o equal_a square_n be_v make_v which_o suppose_v to_o be_v frx_n extend_v produce_v one_o side_n of_o the_o base_a of_o the_o parallelipipedom_n be_o which_o let_v be_v ac_fw-la extend_v to_o the_o point_n p._n let_v the_o other_o side_n of_o the_o say_v base_a concur_v with_o ac_fw-la be_v cg_o as_o cg_o be_v to_o fr_z the_o side_n of_o the_o square_a frx_n so_o let_v the_o same_o fr_z be_v to_o a_o line_n cut_v of_o from_o cp_n sufficient_o extend_v by_o the_o 11._o of_o the_o six_o and_o let_v that_o three_o proportional_a line_n be_v cp_n let_v the_o rectangle_n parallelogram_n be_v make_v perfect_a as_o cd_o it_o be_v evident_a that_o cd_o be_v equal_a to_o the_o square_a frx_n by_o the_o 17._o of_o the_o six_o and_o by_o construction_n frx_n be_v equal_a to_o b._n wherefore_o cd_o be_v equal_a to_o â—_o by_o the_o 12._o of_o the_o six_o as_o cp_n be_v to_o ac_fw-la so_o let_v a_o the_o heith_n of_o be_o be_v to_o the_o right_a line_n o._n i_o say_v that_o a_o solid_a perpendicular_o erect_v upon_o the_o base_a â—_o have_v the_o heith_n of_o the_o line_n oh_o be_v equal_a to_o the_o parallelipipedon_n am._n for_o cd_o be_v to_o agnostus_n as_o cp_n be_v to_o aâ—_z by_o the_o first_o of_o the_o six_o and_o â—_o be_v prove_v equal_a to_o cd_o wherefore_o by_o the_o 7._o of_o the_o five_o b_o be_v to_o ag_z as_o cp_n be_v to_o ac_fw-la but_o as_o cp_n be_v to_o ac_fw-la so_o be_v a_o to_o oh_o by_o construction_n wherefore_o b_o be_v to_o ag_z as_o a_o be_v to_o o._n so_o than_o the_o base_n â—_o and_o agnostus_n be_v reciprocal_o in_o proportion_n with_o the_o heithe_n a_o and_o o._n by_o this_o 34_o therefore_o a_o solid_a erect_v perpendicular_o upon_o â—_o as_o a_o base_a have_v the_o height_n oh_o be_v equal_a to_o am._n wherefore_o upon_o a_o right_n line_v plain_n superficies_n give_v we_o have_v apply_v a_o rectangle_n parallelipipedon_n give_v which_o be_v requisite_a to_o be_v do_v a_o problem_n 2._o a_o rectangle_n parallelipipedon_n be_v give_v to_o make_v a_o other_o equal_a to_o it_o of_o any_o heith_n assign_v suppose_v the_o rectangle_n parallelipipedon_n give_v to_o be_v a_o and_o the_o heith_n assign_v to_o be_v the_o right_a line_n â—_o now_o must_v we_o make_v a_o rectangle_n parallelipipedon_n equal_a to_o a_o who_o heith_n must_v be_v equal_a to_o â—_o according_a to_o the_o manner_n before_o use_v we_o must_v frame_v our_o construction_n to_o a_o reciprocal_a proportion_n between_o the_o base_n and_o heithe_n which_o will_v be_v do_v if_o as_o the_o heith_n assign_v bear_v itself_o in_o proportion_n to_o the_o heith_n of_o the_o parallelipipedon_n give_v so_o one_o of_o the_o side_n of_o the_o base_a of_o the_o parallelipipedon_n give_v be_v to_o a_o four_o line_n by_o the_o 12._o of_o the_o six_o find_v for_o that_o line_n find_v and_o the_o other_o side_n of_o the_o base_a of_o the_o give_v parallelipipedon_n contain_v a_o parallelogram_n which_o do_v serve_v for_o the_o base_a which_o only_o we_o want_v to_o use_v with_o our_o give_a heith_n and_o so_o be_v the_o problem_n to_o be_v execute_v note_n euclid_n in_o the_o 27._o of_o this_o eleven_o have_v teach_v how_o of_o a_o right_a line_n give_v to_o describe_v a_o parallepipedom_n like_a &_o likewise_o situate_v to_o a_o parallelipipedom_n give_v ay_o have_v also_o add_v how_o to_o a_o parallepipedon_n give_v a_o other_o may_v be_v make_v equal_a upon_o any_o right_n line_v base_a give_v or_o of_o any_o heith_n assign_v but_o if_o either_o euclid_n or_o any_o other_o before_o our_o time_n answerable_o to_o the_o 25._o of_o the_o six_o in_o plain_v have_v among_o solid_n invent_v this_o proposition_n for_o two_o unequal_a and_o unlike_a parallelipipedon_n be_v give_v to_o describe_v a_o parallelipipedon_n equal_a to_o the_o one_o and_o like_a to_o the_o other_o we_o will_v have_v give_v they_o their_o deserve_a praise_n and_o i_o will_v also_o have_v be_v right_o glad_a to_o have_v be_v ease_v of_o my_o great_a travail_n and_o discourse_n about_o the_o invent_v thereof_o here_o end_n i_o dee_n his_o addition_n upon_o this_o 34._o proposition_n the_o 30._o theorem_a the_o 35._o proposition_n if_o there_o be_v two_o superficial_a angle_n equal_a and_o from_o the_o point_n of_o those_o angle_n be_v elevate_v on_o high_a right_a line_n comprehend_v together_o with_o those_o right_a line_n which_o contain_v the_o superficial_a angle_n equal_a angle_n each_o to_o his_o corespondent_a angle_n and_o if_o in_o each_o of_o the_o elevate_v line_n be_v take_v a_o point_n at_o all_o aventure_n and_o from_o those_o point_n be_v draw_v perpendicular_a line_n to_o the_o ground_n plain_a superficiece_n in_o which_o be_v the_o angle_n give_v at_o the_o beginning_n and_o from_o the_o point_n which_o be_v by_o those_o perpendicular_a line_n make_v in_o the_o two_o plain_a superficiece_n be_v join_v to_o those_o angle_n which_o be_v put_v at_o the_o beginning_n right_a line_n those_o right_a line_n together_o with_o the_o line_n elevate_v on_o high_a shall_v contain_v equal_a angle_n svppose_v that_o these_o two_o rectiline_a superficial_a angle_n bac_n and_o edf_o be_v equal_a the_o one_o to_o the_o other_o construction_n and_o from_o the_o point_n a_o and_o d_o let_v there_o be_v elevate_v upward_o these_o right_a line_n agnostus_n and_o dm_o comprehend_v together_o with_o the_o line_n put_v at_o the_o begin_v equal_a angle_n each_o to_o his_o correspondent_a angle_n that_o be_v the_o angle_n mde_v to_o the_o angle_n gab_n and_o the_o angle_n mdf_n to_o the_o angle_n gac_n and_o take_v in_o the_o line_n agnostus_n and_o dm_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v g_z and_o m._n and_o by_o the_o 11._o of_o the_o eleven_o from_o the_o point_n g_z and_o m_o draw_v unto_o the_o ground_n plain_a superficiece_n wherein_o be_v the_o angle_n bac_n and_o e_o df_o perpendicular_a line_n gl_n and_o mn_o and_o let_v they_o fall_v in_o the_o say_v plain_a superâ—icieces_n in_o the_o point_n n_o and_o l_o and_o draw_v a_o right_a line_n from_o the_o point_n l_o to_o the_o point_n a_o and_o a_o other_o from_o the_o point_n n_o to_o the_o point_n d._n then_o i_o say_v that_o the_o angle_n galliard_n be_v equal_a to_o the_o angle_n mdn._n fron_n the_o great_a of_o the_o two_o line_n agnostus_n and_o dm_o which_o let_v be_v agnostus_n cut_v of_o by_o the_o 3._o of_o the_o first_o the_o line_n ah_o equal_a unto_o the_o line_n dm_o and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n h_o draw_v unto_o the_o line_n gl_n a_o parallel_a line_n and_o let_v the_o same_o be_v hk_a now_o the_o line_n gl_n be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n bal_n wherefore_o also_o by_o the_o 8._o of_o the_o eleven_o the_o line_n hk_o be_v erect_v perpendicular_o to_o the_o same_o ground_n plain_a superficies_n bac_n draw_v by_o the_o 12._o of_o the_o first_o from_o the_o point_n king_n and_o n_o unto_o the_o right_a line_n ab_fw-la ac_fw-la df_o &_o de_fw-fr perpendicular_a right_a line_n and_o let_v the_o same_o be_v kc_o nf_n kb_o ne._n and_o draw_v these_o right_a line_n hc_n cb_o mf_n fe_o now_o forasmuch_o aâ—_z by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ha_o be_v equal_a to_o the_o square_n of_o the_o line_n hk_o and_o ka_o demonstration_n but_o unto_o the_o square_n of_o the_o line_n ka_o be_v equal_a the_o square_n of_o the_o line_n kc_o and_o ca_n wherefore_o the_o square_a of_o the_o line_n ha_o be_v equal_a to_o the_o square_n of_o the_o line_n hk_o kc_o and_o ca._n but_o by_o the_o same_o unto_o the_o square_n of_o the_o line_n hk_o and_o kc_o be_v equal_a the_o square_n of_o the_o line_n hc_n wherefore_o the_o square_a of_o the_o line_n ha_o be_v equal_a to_o the_o square_n of_o the_o line_n hc_n and_o ca_n wherefore_o the_o angle_n hca_n be_v by_o the_o 48._o of_o the_o first_o a_o right_a angle_n and_o by_o the_o same_o reason_n also_o the_o angle_n mfd_n be_v a_o right_a angle_n wherefore_o the_o angle_n hca_n be_v equal_a to_o the_o angle_n mfd_n but_o the_o angle_n hac_fw-la be_v by_o supposition_n equal_a to_o the_o angle_n mdf_n wherefore_o there_o be_v two_o triangle_n mdf_n and_o hac_fw-la 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 each_o to_o his_o correspondent_a angle_n
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 that_o side_n which_o subtend_v one_o of_o the_o equal_a angle_n that_o be_v the_o side_n ha_o be_v equal_a to_o the_o side_n dm_o by_o construction_n wherefore_o the_o side_n remain_v be_v by_o the_o 26._o of_o the_o first_o equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ac_fw-la be_v equal_a to_o the_o side_n df._n in_o like_a sort_n may_v we_o prove_v that_o the_o side_n ab_fw-la be_v equal_a to_o the_o side_n de_fw-fr if_o you_o draw_v a_o right_a line_n from_o the_o point_n h_o to_o the_o point_n b_o and_o a_o other_o from_o the_o point_n m_o to_o the_o point_n e._n for_o forasmuch_o as_o the_o square_n of_o the_o line_n ah_o be_v by_o the_o 47._o of_o the_o first_o equal_v to_o the_o square_n of_o the_o line_n ak_o and_o kh_o and_o by_o the_o same_o unto_o the_o square_n of_o the_o line_n ak_o be_v equal_a the_o square_n of_o the_o line_n ab_fw-la and_o bk_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la bk_o and_o kh_o be_v equal_a to_o the_o square_n of_o the_o line_n ah_o but_o unto_o the_o square_n of_o the_o line_n bk_o and_o kh_o be_v equal_a the_o square_n of_o the_o line_n bh_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n hkb_n be_v a_o right_a angle_n for_o that_o the_o line_n hk_o be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n wherefore_o the_o square_a of_o the_o line_n ah_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bh_o wherefore_o by_o the_o 48._o of_o the_o first_o the_o angle_n abh_n be_v a_o right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n dem_n be_v a_o right_a angle_n now_o the_o angle_n bah_o be_v equal_a to_o the_o angle_n edm_n for_o it_o be_v so_o suppose_a and_o the_o line_n ah_o be_v equal_a to_o the_o line_n dm_o wherefore_o by_o the_o 26._o of_o the_o first_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de._n now_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n df_o and_o the_o line_n ab_fw-la to_o the_o line_n de_fw-fr therefore_o these_o two_o line_n ac_fw-la and_o ab_fw-la be_v equal_a to_o these_o two_o line_n fd_v and_o de._n but_o the_o angle_n also_o cab_n be_v by_o supposition_n equal_a to_o the_o angle_n fde_v wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a bc_o be_v equal_a to_o the_o base_a of_o and_o the_o triangle_n to_o the_o triangle_n and_o the_o rest_n of_o the_o angle_n to_o the_o rest_n of_o the_o angle_n wherefore_o the_o angle_n acb_o be_v equal_a to_o the_o angle_n dfe_n and_o the_o right_a angle_n ack_z be_v equal_a to_o the_o right_a angle_n dfn._n wherefore_o the_o angle_n remain_v namely_o bck_n be_v equal_a to_o the_o angle_n remain_v namely_o to_o efn_n and_o by_o the_o same_o reason_n also_o the_o angle_n cbk_n be_v equal_a to_o the_o angle_n fen_n wherefore_o there_o be_v two_o triangle_n bck_o &_o efn_n 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 each_o to_o his_o correspondent_a angle_n 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 that_o side_n that_o lie_v between_o the_o equal_a angle_n that_o be_v the_o side_n bc_o be_v equal_a to_o the_o side_n of_o wherefore_o by_o the_o 26._o of_o the_o first_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ck_o be_v equal_a to_o the_o side_n fn_o but_o the_o side_n ac_fw-la be_v equal_a to_o the_o side_n df._n wherefore_o these_o two_o side_n ac_fw-la and_o ck_o be_v equal_a to_o these_o two_o side_n df_o and_o fn_o and_o they_o contain_v equal_a angle_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ak_n be_v equal_a to_o the_o base_a dn_o and_o forasmuch_o as_o the_o line_n ah_o be_v equal_a to_o the_o line_n dm_o therefore_o the_o square_a of_o the_o line_n ah_o be_v equal_a to_o the_o square_n of_o the_o line_n dm_o but_o unto_o the_o square_n of_o the_o line_n ah_o be_v equal_a the_o square_n of_o the_o line_n ak_o and_o kh_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n akh_n be_v a_o right_a angle_n and_o to_o the_o square_n of_o the_o line_n dm_o be_v equal_a the_o square_n of_o the_o line_n dn_o and_o nm_o for_o the_o angle_n dnm_n be_v a_o right_a angle_n wherefore_o the_o square_n of_o the_o line_n ak_o and_o kh_o be_v equal_a to_o the_o square_n of_o the_o line_n dn_o and_o nm_o of_o which_o two_o the_o square_a of_o the_o line_n ak_o be_v equal_a to_o the_o square_n of_o the_o line_n dn_o for_o the_o line_n ak_o be_v prove_v equal_a to_o the_o line_n a_fw-la wherefore_o the_o residue_n namely_o the_o square_a of_o the_o line_n kh_o be_v equal_a to_o the_o residue_n namely_o to_o the_o square_n of_o the_o line_n nm_o wherefore_o the_o line_n hk_o be_v equal_a to_o the_o line_n mn_v and_o forasmuch_o as_o these_o two_o line_n ha_o and_o ak_o be_v equal_a to_o these_o two_o line_n md_o and_o dn_o the_o one_o to_o the_o other_o and_o the_o base_a hk_n be_v equal_a to_o the_o base_a mn_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n hak_n be_v equal_a to_o the_o angle_n mdn._n if_o therefore_o there_o be_v two_o superficial_a angle_n equal_a and_o from_o the_o point_n of_o those_o angle_n be_v elevate_v on_o high_a right_a line_n comprehend_v together_o with_o those_o right_a line_n which_o be_v put_v at_o the_o beginning_n equal_a angle_n each_o to_o his_o corespondent_a angle_n and_o if_o in_o each_o of_o the_o erect_a line_n be_v take_v a_o point_n at_o all_o adventure_n and_o from_o those_o point_n be_v draw_v perpendicular_a line_n to_o the_o plain_a superficiece_n in_o which_o be_v the_o angle_n give_v at_o the_o beginning_n and_o frâ—â—_n the_o point_n which_o be_v by_o the_o perpendicular_a line_n make_v in_o the_o two_o plain_a superficiece_n be_v join_v right_a line_n to_o those_o angle_n which_o be_v put_v at_o the_o beginning_n those_o right_a line_n shall_v together_o with_o the_o line_n elevate_v on_o high_a make_v equal_a angle_n which_o be_v require_v to_o be_v prove_v because_o the_o figure_n of_o the_o former_a demonstration_n be_v somewhat_o hard_a to_o conceive_v as_o they_o be_v there_o draw_v in_o a_o plain_a by_o reason_n of_o the_o line_n that_o be_v imagine_v to_o be_v elevate_v on_o high_a i_o have_v here_o set_v other_o figure_n wherein_o you_o must_v erect_v perpendicular_o to_o the_o ground_n superficiece_v the_o two_o triangle_n bhk_v and_o emn_a and_o then_o elevate_v the_o triangle_n dfm_n &_o ache_n in_o such_o sort_n that_o the_o angle_n m_o and_o h_o of_o these_o triangle_n may_v concur_v with_o the_o angle_n m_o and_o h_o of_o the_o other_o erect_v triangle_n and_o then_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n g_o of_o the_o line_n agnostus_n to_o the_o point_n l_o in_o the_o ground_n superficies_n compare_v it_o with_o the_o former_a construction_n &_o demonstration_n and_o it_o will_v make_v it_o very_o easy_a to_o conceive_v ¶_o corollary_n by_o this_o it_o be_v manifest_a that_o if_o there_o be_v two_o rectiline_v superficial_a angle_n equal_a and_o upon_o those_o angle_n be_v elevate_v on_o high_a equal_a right_a line_n contain_v together_o with_o the_o right_a line_n put_v at_o the_o beginning_n equal_a angle_n perpendicular_a line_n draw_v from_o those_o elevate_v line_n to_o the_o ground_n plain_a superficiece_n wherein_o be_v the_o angle_n put_v at_o the_o beginning_n be_v equal_a the_o one_o to_o the_o other_o for_o it_o be_v manifest_a that_o the_o perpendicular_a line_n hk_o &_o mn_a which_o be_v draw_v from_o the_o end_n of_o the_o equal_a elevate_v line_n ah_o and_o dm_o to_o the_o ground_n superficiece_n be_v equal_a ¶_o the_o 31._o theorem_a the_o 36._o proposition_n if_o there_o be_v three_o right_a line_n proportional_a a_o parallelipipedon_n describe_v of_o those_o three_o right_a line_n be_v equal_a to_o the_o parallelipipedon_n describe_v of_o the_o middle_a line_n so_o that_o it_o consist_v of_o equal_a side_n and_o also_o be_v equiangle_n to_o the_o foresay_a parallelipipedon_n svppose_v that_o these_o three_o line_n a_o b_o c_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v b_o be_v to_o c._n then_o i_o say_v that_o the_o parallelipipedon_n make_v of_o the_o line_n a_o b_o c_o be_v equal_a to_o the_o parallelipipedon_n make_v of_o the_o line_n b_o so_fw-mi that_o the_o solid_a make_v of_o the_o line_n b_o consist_v of_o equal_a side_n and_o be_v also_o equiangle_n to_o the_o solid_a make_v of_o the_o line_n a_o b_o c._n describe_v by_o the_o 23._o of_o the_o
by_o his_o motion_n describe_v the_o round_a conical_a superficies_n about_o the_o cone_n and_o as_o the_o circumference_n of_o the_o semicircle_n describe_v the_o round_a spherical_a superficies_n about_o the_o sphere_n in_o this_o example_n it_o be_v the_o superficies_n describe_v of_o the_o line_n dc_o corollary_n by_o this_o definition_n it_o be_v plain_a that_o the_o two_o circle_n or_o base_n of_o a_o cilind_a be_v ever_o equal_a and_o parallel_n for_o that_o the_o line_n move_v which_o produce_v they_o remain_v always_o equal_a and_o parallel_n also_o the_o axe_n of_o a_o cilind_a be_v ever_o a_o erect_a line_n unto_o either_o of_o the_o base_n for_o with_o all_o the_o line_n describe_v in_o the_o base_n and_o touch_v it_o it_o make_v right_a angle_n sphere_n campane_n vitellâ—o_o with_o other_o late_a writer_n call_v this_o solid_a or_o body_n a_o round_a columnâ—_n or_o pillar_n and_o campane_n add_v unto_o this_o definition_n this_o as_o a_o corollary_n that_o of_o a_o round_a column_n of_o a_o sphere_n and_o of_o a_o circle_n the_o centre_n be_v one_o and_o the_o self_n same_o campane_n that_o be_v as_o he_o himself_o declare_v it_o &_o prove_v the_o same_o where_o the_o column_n the_o sphere_n and_o the_o circle_n have_v one_o diameter_n 20_o like_o cones_fw-la and_o cilinder_n be_v those_o definition_n who_o axe_n and_o diameter_n of_o their_o base_n be_v proportional_a the_o similitude_n of_o cones_fw-la and_o cilinder_n stand_v in_o the_o proportion_n of_o those_o right_a line_n of_o which_o they_o have_v their_o original_n and_o spring_n for_o by_o the_o diameter_n of_o their_o base_n be_v have_v their_o length_n and_o breadth_n and_o by_o their_o axe_n be_v have_v their_o height_n or_o deepness_n wherefore_o to_o see_v whether_o they_o be_v like_a or_o unlike_a you_o must_v compare_v their_o axe_n together_o which_o be_v their_o depth_n and_o also_o their_o diameter_n together_o which_o be_v their_o length_n &_o breadth_n as_o if_o the_o axe_n â—g_z of_o the_o cone_n abc_n be_v to_o to_o the_o axe_n ei_o of_o the_o cone_n def_n as_o the_o diameter_n ac_fw-la of_o the_o cone_n abc_n be_v to_o the_o diameter_n df_o of_o the_o cone_fw-it def_n then_o aâ—e_v the_o cones_fw-la abc_n and_o def_n like_o cones_fw-la likewise_o in_o the_o cilinder_n if_o the_o axe_n ln_o of_o the_o cilind_a lhmn_n have_v that_o proportion_n to_o the_o axe_n oq_fw-la of_o the_o cilind_a ropq_n which_o the_o diameter_n he_o have_v to_o the_o diameter_n rp_n then_o be_v the_o cilinder_n hlmn_n and_o ropq_n like_o cilinder_n and_o so_o of_o all_o other_o 21_o a_o cube_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o six_o equal_a square_n definition_n as_o be_v a_o die_v which_o have_v six_o side_n and_o each_o of_o they_o be_v a_o full_a and_o perfect_a square_n as_o limit_n or_o border_n under_o which_o it_o be_v contain_v and_o as_o you_o may_v conceive_v in_o a_o piece_n of_o timber_n contain_v a_o foot_n square_v every_o way_n or_o in_o any_o such_o like_a so_o that_o a_o cube_n be_v such_o a_o solid_a who_o three_o dimension_n be_v equal_a the_o length_n be_v equal_a to_o the_o breadth_n thereof_o and_o each_o of_o they_o equal_a to_o the_o depth_n here_o be_v as_o it_o may_v be_v in_o a_o plain_a superficies_n set_v a_o image_n thereof_o in_o these_o two_o figure_n whereof_o the_o first_o be_v as_o it_o be_v common_o describe_v in_o a_o plain_n the_o second_o which_o be_v in_o the_o begin_n of_o the_o other_o side_n of_o this_o leaf_n be_v draw_v as_o it_o be_v describe_v by_o art_n upon_o a_o plain_a superficies_n to_o show_v somewhat_o bodilike_o and_o in_o deed_n the_o latter_a description_n be_v for_o the_o sight_n better_o they_o the_o first_o but_o the_o first_o for_o the_o demonstration_n of_o euclides_n proposition_n in_o the_o five_o book_n follow_v be_v of_o more_o use_n for_o that_o in_o it_o may_v be_v consider_v and_o see_v all_o the_o fix_v side_n of_o the_o cube_n and_o so_o any_o line_n or_o 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mathematical_a derivative_n derivative_n from_o the_o principal_n oâ—_n which_o some_o have_v the_o name_n of_o the_o principal_n as_o arithmetic_n vulgar_a which_o consider_v arithmetic_n of_o most_o usual_a whole_a number_n and_o of_o fraction_n to_o they_o appertain_v arithmetic_n of_o proportion_n arithmetic_n circular_a arithmetic_n of_o radical_a number_n simple_a compound_v mix_v and_o of_o their_o fraction_n arithmetic_n of_o cossike_n number_n with_o their_o fraction_n and_o the_o great_a art_n of_o algiebar_n geometry_n vulgar_a which_o teach_v measure_v at_o hand_n all_o lengthe_n mecometrie_n all_o plain_n as_o land_n borde_n glass_n etc_n etc_n embadometrie_n all_o solid_n as_o timber_n stone_n vessel_n etc_n etc_n stereometry_n with_o distance_n from_o the_o thing_n measure_v as_o how_o far_o from_o the_o measurer_n any_o thing_n be_v of_o he_o see_v on_o land_n or_o water_n call_v apomecometrie_n of_o which_o be_v grow_v the_o feat_n &_o art_n of_o geodesie_n more_o cunning_o to_o measure_n and_o survey_v land_n wood_n water_n etc_n etc_n geographie_n chorography_n hydrography_n stratarithmetrie_n how_o high_a or_o deep_a from_o the_o level_a of_o the_o measurer_n stand_v any_o thing_n be_v see_v of_o he_o on_o land_n or_o water_n call_v hypsometrie_n of_o which_o be_v grow_v the_o feat_n &_o art_n of_o geodesie_n more_o cunning_o to_o measure_n and_o survey_v land_n wood_n water_n etc_n etc_n geographie_n chorography_n hydrography_n stratarithmetrie_n how_o broad_a a_o thing_n be_v which_o be_v in_o the_o measurers_n view_n so_o it_o be_v situate_v on_o land_n or_o water_n call_v platometrie_n of_o which_o be_v grow_v the_o feat_n &_o art_n of_o geodesie_n more_o cunning_o to_o measure_n and_o survey_v land_n wood_n water_n etc_n etc_n geographie_n chorography_n hydrography_n stratarithmetrie_n proper_a name_n as_o perspective_n which_o demonstrate_v the_o manner_n and_o property_n of_o all_o radiation_n direct_v break_a and_o reflect_v astronomy_n which_o demonstrate_v the_o distance_n magnitude_n and_o all_o natural_a motion_n apparence_n and_o passion_n proper_a to_o the_o planet_n and_o fix_a star_n fâ—r_v any_o time_n past_a present_a and_o to_o come_v in_o respect_n of_o a_o certain_a horizon_n or_o without_o respect_n of_o any_o horizon_n music_n which_o demonstrate_v by_o reason_n and_o teach_v by_o sense_n perfect_o to_o judge_v and_o order_v the_o diversity_n of_o sound_v hiâ—_z or_o lâ—w_n cosmography_n which_o whole_o and_o perfect_o make_v description_n of_o the_o heavenly_a and_o also_o elemental_a part_n of_o the_o world_n and_o of_o these_o part_n make_v hâ—mâ—lâ—gall_a application_n and_o mutual_a collation_n necessary_a astrology_n which_o reasonable_o demonstrate_v the_o operation_n and_o effect_n of_o the_o natural_a beaâ—es_n of_o light_n and_o 〈◊〉_d influence_n of_o the_o planet_n and_o fix_v star_n 〈◊〉_d every_o element_n and_o elemental_a body_n at_o all_o time_n in_o any_o horizon_n assign_v statike_n which_o demonstrate_v the_o cause_n of_o heaviness_n and_o lightness_n of_o all_o thing_n and_o of_o the_o motion_n and_o property_n to_o heaviness_n and_o lightness_n belong_v anthropographie_n which_o describe_v the_o number_n measure_n waight_n figure_n situation_n and_o colour_n of_o every_o diverse_a thing_n contain_v in_o the_o perfectâ—_n body_n of_o â—â—_o a_fw-la and_o give_v certain_a knowledge_n of_o the_o figure_n symmetric_a waight_n characterization_n &_o due_a local_a motion_n of_o any_o pâ—rcell_n of_o the_o say_a body_n assign_v and_o of_o number_n to_o the_o say_a pâ—rcell_n appertain_v trochilic_a which_o demonstrate_v the_o property_n of_o all_o circular_a motion_n simple_a and_o compound_v helicosophie_n which_o demonstrate_v the_o design_v of_o all_o spiral_n line_n in_o plain_a on_o cylinder_n coâ—â—_n sphere_n câ—nâ—id_n and_o spharoâ—d_v and_o their_o property_n pneumatithmie_n which_o demonstrate_v by_o close_a hollow_a geometrical_a figure_n regular_a and_o irregular_a the_o strange_a property_n in_o motion_n or_o stay_n of_o the_o water_n air_n smoke_n and_o fire_n in_o their_o continuitiâ—_n and_o as_o they_o be_v join_v to_o the_o element_n next_o they_o menadrie_a which_o demonstrate_v how_o about_o nature_n virtue_n and_o power_n simple_a virtue_n and_o force_n may_v be_v multiply_v and_o so_o to_o directe_v to_o lifâ—_n to_o pull_v to_o aâ—d_z to_o put_v or_o cast_v fro_o any_o multiply_v or_o simple_a determine_a virtue_n waight_n or_o force_n natural_o not_o so_o directible_a or_o movable_a hypogeiodie_n which_o demonstrate_v how_o under_o the_o spharicall_a superficies_n of_o the_o earth_n at_o â—ny_a depth_n to_o any_o perpendicular_a line_n assign_v who_o distance_n from_o the_o perpendicular_a of_o the_o entrance_n and_o the_o azimuth_n likewise_o 〈◊〉_d resperse_n of_o the_o say_a entrance_n be_v know_v certain_a way_n may_v be_v prescribe_v and_o gâ—ne_v etc_n etc_n hydragogie_n which_o demonstrâ—teth_v the_o possible_a lead_n of_o water_n by_o nature_n lâ—â—_n and_o by_o artificial_a help_n frâ—â—_n any_o head_n be_v spring_n stand_v or_o run_v water_n to_o any_o other_o place_n assign_v horometrie_n which_o demonstrate_v how_o at_o all_o time_n appoint_v the_o precise_a usual_a denomination_n of_o time_n â—â—y_o â—e_a knowâ—n_n for_o any_o place_n assign_v zographie_n which_o demonstrate_v and_o teach_v how_o the_o intersection_n of_o all_o usual_a 〈…〉_o assign_a the_o centre_n distancâ—_n and_o light_n bâ—ing_v determine_v may_v be_v by_o line_n and_o proper_a colour_n repreâ—â—â—â—_n architecture_n which_o be_v a_o sciâ—â—â—â—_n garnish_v with_o many_o doctrine_n and_o 〈…〉_o be_v judge_v navigation_n thaumaturgike_a archemastrie_n ¶_o the_o first_o book_n of_o euclides_n element_n in_o this_o first_o book_n be_v entreat_v of_o the_o most_o simple_a book_n easy_a and_o first_o matter_n and_o ground_n of_o geometry_n as_o namely_o of_o lynes_n angle_n triangle_n parallel_n square_n and_o parallelogram_n first_o of_o their_o definition_n show_v what_o they_o be_v after_o that_o it_o teachâ—th_v how_o to_o draw_v parallel_n line_n and_o how_o to_o form_v diverse_o figure_n of_o three_o side_n &_o four_o side_n according_a to_o the_o variety_n of_o their_o side_n and_o angle_n &_o
set_v before_o the_o thing_n require_v require_v in_o some_o proposition_n there_o be_v more_o thing_n give_v then_o one_o and_o more_o thing_n require_v then_o one_o in_o some_o there_o be_v nothing_o give_v at_o all_o moreover_o every_o problem_n &_o theorem_a be_v perfect_a and_o absolute_a aught_o to_o have_v all_o these_o part_n namely_o first_o the_o proposition_n proposition_n to_o be_v prove_v exposition_n then_o the_o exposition_n which_o be_v the_o explication_n of_o the_o thing_n give_v after_o that_o follow_v the_o determination_n determination_n which_o be_v the_o declaration_n of_o the_o thing_n require_v then_o be_v set_v the_o construction_n of_o such_o thing_n which_o be_v necessary_a either_o for_o the_o do_v of_o the_o proposition_n construction_n or_o for_o the_o demonstration_n afterwards_o follow_v the_o demonstration_n demonstration_n which_o be_v the_o reason_n and_o proof_n of_o the_o proposition_n and_o last_o of_o all_o be_v put_v the_o conclusion_n conclusion_n which_o be_v infer_v &_o prove_v by_o the_o demonstration_n and_o be_v ever_o the_o proposition_n but_o all_o those_o part_n be_v not_o of_o necessity_n require_v in_o every_o problem_n and_o theorem_a but_o the_o proposition_n demonstration_n and_o conclusion_n be_v necessary_a part_n &_o can_v never_o be_v absent_a the_o other_o part_n may_v sometime_o be_v away_o case_n further_o in_o diverse_a proposition_n there_o happen_v diverse_a case_n which_o be_v nothing_o else_o but_o variety_n of_o delineation_n and_o construction_n or_o change_v of_o position_n as_o when_o point_n line_n superâ—iciesses_n or_o body_n be_v change_v which_o thing_n happen_v in_o diverse_a proposition_n problem_n now_o then_o in_o this_o problem_n the_o thing_n give_v be_v the_o line_n give_v the_o thing_n require_v require_v to_o be_v search_v out_o be_v how_o upon_o that_o line_n to_o describe_v a_o equilater_n triangle_n the_o proposition_n of_o this_o problem_n be_v prâ—position_n upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n exposition_n the_o exposition_n be_v suppose_v that_o the_o right_a line_n give_v be_v ab_fw-la and_o this_o declare_v only_o the_o thing_n give_v the_o determination_n be_v determination_n it_o be_v require_v upon_o the_o line_n ab_fw-la to_o describe_v a_o equilater_n triangle_n for_o thereby_o as_o you_o see_v be_v declare_v only_o the_o thing_n require_v the_o construction_n begin_v at_o these_o woâ—ds_n construction_n now_o therefore_o make_v the_o centre_n the_o point_n a_o &_o the_o space_n ab_fw-la describe_v by_o the_o three_o peâ—ition_n a_o circle_n etc_o etc_o and_o continue_v until_o you_o come_v to_o these_o word_n and_o forasmuch_o at_o the_o point_n a_o etc_n etc_n for_o thetheâ—to_o be_v describe_v circle_n and_o line_n necessary_a both_o for_o the_o do_v of_o the_o proposition_n and_o also_o for_o the_o demonstration_n thereof_o which_o demonstration_n begin_v at_o these_o word_n demonstration_n and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n cbd_v etc_o etc_o and_o so_o proceed_v till_o you_o come_v to_o these_o word_n wherefore_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n abc_n for_o until_o you_o come_v thither_o be_v by_o ground_n before_o set_v and_o construction_n have_v prove_v and_o make_v evident_a that_o the_o triangle_n make_v be_v equilater_n and_o then_o in_o these_o word_n conclusion_n wherefore_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n abc_n be_v put_v the_o first_o conclusion_n for_o there_o be_v common_o in_o every_o proposition_n two_o conclusion_n the_o one_o particular_a the_o other_o universal_a and_o from_o the_o first_o you_o go_v to_o the_o last_o and_o this_o be_v the_o first_o and_o particular_a conclusion_n â—or_a that_o it_o conclude_v that_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n which_o be_v according_a to_o the_o exposition_n after_o it_o câ—nclusâ—â—n_n follow_v the_o last_o and_o universal_a conclusion_n wherefore_o upon_o a_o right_a line_n give_v not_o be_v infinite_a be_v describe_v a_o equilater_n triangle_n for_o whether_o the_o line_n give_v be_v great_a or_o less_o than_o this_o line_n the_o â—ame_n construction_n and_o demonstration_n prove_v the_o same_o conclusion_n last_o of_o all_o be_v add_v this_o clause_n problem_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v whereby_o as_o we_o have_v before_o note_v be_v declare_v that_o this_o proposition_n be_v a_o problem_n and_o not_o a_o theorem_a as_o for_o variety_n of_o case_n in_o this_o proposition_n there_o be_v none_o proposition_n for_o that_o the_o line_n give_v can_v have_v no_o diversity_n of_o position_n as_o you_o have_v in_o this_o problem_n see_v plain_o set_v forth_o the_o thing_n give_v and_o the_o thing_n require_v moreover_o the_o proposition_n exposition_n determination_n construction_n demonstration_n and_o conclusion_n which_o be_v general_a also_o to_o many_o other_o both_o problem_n and_o theorem_n so_o may_v you_o by_o the_o example_n thereof_o distinct_a they_o and_o search_v they_o out_o in_o other_o problemesâ—_n â—_z and_o also_o theorem_n demonstration_n this_o also_o be_v to_o be_v note_v that_o there_o be_v three_o kind_n of_o demonstration_n the_o one_o be_v call_v demonstratio_fw-la a_o priori_fw-la or_o composition_n the_o other_o be_v call_v demonstratiâ—_n a_o posteriori_fw-la or_o resolution_n and_o the_o three_o be_v a_o demonstration_n lead_v to_o a_o impossibility_n a_o demonstration_n a_o priori_fw-la composition_n or_o composition_n be_v when_o in_o reason_v from_o the_o principle_n and_o first_o ground_n we_o pass_v descend_v continual_o till_o after_o many_o reason_n make_v we_o come_v at_o the_o length_n to_o conclude_v that_o which_o we_o first_o chief_o intend_v and_o this_o kind_n of_o demonstration_n use_v euclid_n in_o his_o bookeâ—_n for_o the_o most_o part_n a_o demonstration_n a_o posteriori_fw-la resolution_n or_o resolution_n be_v when_o contrariwise_o in_o reason_v we_o pass_v from_o the_o last_o conclusion_n make_v by_o the_o premise_n and_o by_o the_o premise_n of_o the_o premise_n continual_o ascend_v till_o we_o come_v to_o the_o first_o principle_n and_o ground_n which_o be_v indemonstrable_a and_o for_o their_o simplicity_n can_v suffer_v no_o far_o resolution_n a_o demonstration_n lead_v to_o a_o impossibility_n be_v that_o argument_n who_o conclusion_n be_v impossible_a impossibility_n that_o be_v when_o it_o conclude_v direct_o against_o any_o principle_n or_o against_o any_o proposition_n before_o prove_v by_o principle_n or_o proposition_n beâ—ore_o prove_v premise_n in_o a_o argument_n be_v proposition_n go_v before_o the_o conclusion_n by_o which_o the_o conclusion_n be_v prove_v be_v composition_n pass_v from_o the_o cause_n to_o the_o effect_n or_o from_o thing_n simple_a to_o thing_n more_o compound_v resolution_n contrariwise_o pass_v from_o thing_n compound_v to_o thing_n more_o simple_a or_o from_o the_o effect_n to_o â—he_n cause_n composition_n or_o the_o first_o kind_n of_o demonstration_n which_o pass_v from_o the_o principle_n may_v easy_o be_v see_v in_o this_o first_o proposition_n of_o euclid_n proposition_n the_o demonstration_n whereof_o begin_v thus_o and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n cbd_v therefore_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la this_o reason_n you_o see_v take_v his_o beginning_n of_o a_o principle_n namely_o of_o the_o definition_n of_o a_o circle_n and_o this_o be_v the_o first_o reason_n reason_n again_o forasmuch_o as_o b_o be_v the_o centre_n of_o the_o circle_n caesar_n therefore_o the_o line_n bc_o be_v equal_a to_o the_o line_n basilius_n which_o be_v the_o second_o reason_n reason_n and_o it_o be_v before_o prove_v that_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la wherefore_o either_o of_o these_o line_n ca_n &_o cb_o be_v equal_a to_o the_o line_n ab_fw-la and_o this_o be_v the_o three_o reason_n reason_n but_o thing_n which_o be_v equal_a to_o one_o &_o the_o self_n same_o thing_n be_v also_o equal_a the_o one_o to_o the_o other_o wherefore_o the_o line_n ca_n be_v equal_a to_o the_o line_n cb._n and_o this_o be_v the_o four_o argument_n reason_n wherefore_o these_o three_o line_n ca_n ab_fw-la and_o bc_o be_v equal_a the_o one_o to_o the_o other_o which_o be_v the_o conclusion_n conclusion_n and_o the_o thing_n to_o be_v prove_v you_o may_v also_o in_o the_o same_o first_o proposition_n easy_o take_v a_o example_n of_o resolution_n proposition_n use_v a_o contrary_a order_n pass_a backward_o from_o the_o last_o conclusion_n of_o the_o former_a demonstration_n till_o you_o come_v to_o the_o first_o principle_n or_o ground_n whereon_o it_o begin_v for_o the_o last_o argument_n or_o reason_n in_o composition_n be_v the_o first_o in_o resolution_n &_o the_o first_o in_o composition_n be_v the_o last_o in_o resolution_n thus_o therefore_o must_v you_o proceed_v the_o triangle_n abc_n be_v contain_v of_o three_o equal_a right_a line_n
diameter_n be_v double_a to_o that_o square_n who_o diameter_n it_o be_v corollary_n the_o 34._o theorem_a the_o 48._o proposition_n if_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o let_v the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n there_o namely_o of_o the_o side_n bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la then_o i_o say_v that_o the_o angle_n bac_n be_v a_o right_a angle_n raise_v up_o by_o the_o 11._o proposition_n from_o the_o point_n a_o unto_o the_o right_a line_n ac_fw-la a_o perpendicular_a line_n ad._n and_o by_o the_o third_o proposition_n unto_o the_o line_n ab_fw-la put_v a_o equal_a line_n ad._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o poinâ—_n c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ab_fw-la the_o square_n which_o be_v make_v of_o the_o line_n dam_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la put_v the_o square_n of_o the_o line_n ac_fw-la common_a to_o they_o both_o wherefore_o the_o square_n of_o the_o line_n dam_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la but_o by_o the_o proposition_n go_v before_o the_o square_a of_o the_o line_n dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o ac_fw-la for_o the_o angle_n dac_o be_v a_o right_a angle_n and_o the_o square_a of_o bc_o be_v by_o supposition_n equal_a to_o the_o square_n of_o ab_fw-la and_o ac_fw-la wherefore_o the_o square_a of_o dc_o be_v equal_a to_o the_o square_n of_o bc_o wherefore_o the_o side_n dc_o be_v equal_a to_o the_o side_n bc._n and_o forasmuch_o as_o ab_fw-la be_v equal_a to_o ad_fw-la â—nd_n ac_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o side_n dam_n and_o ac_fw-la be_v equal_a to_o these_o two_o side_n basilius_n and_o ac_fw-la the_o one_o to_o the_o other_o and_o the_o base_a dc_o be_v equal_a to_o the_o base_a be_v wherefore_o by_o the_o 8._o proposition_n the_o angle_n dac_o be_v equal_a to_o the_o angle_n bac_n but_o the_o angle_n dac_o be_v a_o right_a angle_n wherefore_o also_o the_o angle_n bac_n be_v a_o right_a angle_n if_o therefore_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n which_o be_v require_v to_o be_v prove_v former_a this_o proposition_n be_v the_o converse_n of_o the_o former_a and_o be_v of_o pelitarius_n demonstrate_v by_o a_o argument_n lead_v to_o a_o impossibility_n after_o this_o manner_n the_o end_n of_o the_o first_o book_n of_o euclides_n element_n ¶_o the_o second_o book_n of_o euclides_n element_n in_o this_o second_o book_n euclid_n show_v book_n what_o be_v a_o gnomon_n and_o a_o right_a angle_a parallelogram_n also_o in_o this_o book_n be_v set_v forth_o the_o power_n of_o line_n divide_v even_o and_o uneven_o and_o of_o line_n add_v one_o to_o a_o other_o the_o power_n of_o a_o line_n line_n be_v the_o square_a of_o the_o same_o line_n that_o be_v a_o square_a every_o side_n of_o which_o be_v equal_a to_o the_o line_n so_o that_o here_o be_v set_v forth_o the_o quality_n and_o propriety_n of_o the_o square_n and_o right_n line_v figure_n which_o be_v make_v of_o line_n &_o of_o their_o part_n algebra_n the_o arithmetician_n also_o our_o of_o this_o book_n gather_v many_o compendious_a rule_n of_o reckon_n and_o many_o rule_v also_o of_o algebra_n with_o the_o equation_n therein_o use_v the_o ground_n also_o of_o those_o rule_n be_v for_o the_o most_o part_n by_o this_o second_o book_n demonstrate_v book_n this_o book_n moreover_o contain_v two_o wonderful_a proposition_n one_o of_o a_o obtuse_a angle_a triangle_n and_o the_o other_o of_o a_o acute_a which_o with_o the_o aid_n of_o the_o 47._o proposition_n of_o the_o first_o book_n of_o euclid_n which_o be_v of_o a_o rectangle_n triangle_n of_o how_o great_a force_n and_o profit_n they_o be_v in_o matter_n of_o astronomy_n they_o know_v which_o have_v travail_v in_o that_o art_n wherefore_o if_o this_o book_n have_v none_o other_o profit_n be_v side_n only_o for_o these_o 2._o proposition_n sake_n it_o be_v diligent_o to_o be_v embrace_v and_o study_v the_o definition_n 1._o every_o rectangled_a parallelogram_n definition_n be_v say_v to_o be_v contain_v under_o two_o right_a line_n comprehend_v a_o right_a angle_n a_o parallelogram_n be_v a_o figure_n of_o four_o side_n be_v who_o two_o opposite_a or_o contrary_a side_n be_v equal_a the_o one_o to_o the_o other_o there_o be_v of_o parallelogram_n four_o kind_n parallelogram_n a_o square_a a_o figure_n of_o one_o side_n long_o a_o rombus_n or_o diamond_n and_o a_o romboide_n or_o diamond_n like_o figure_n as_o before_o be_v say_v in_o the_o 33._o definition_n of_o the_o first_o book_n of_o these_o four_o sort_n the_o square_a and_o the_o figure_n of_o one_o side_n long_o be_v only_o right_a angle_a parallelogram_n for_o that_o all_o their_o angle_n be_v right_a angle_n and_o either_o of_o they_o be_v contain_v according_a to_o this_o definition_n under_o two_o right_a line_n whiâ—h_o concur_v together_o and_o cause_v the_o right_a angle_n and_o contain_v the_o same_o of_o which_o two_o line_n the_o one_o be_v the_o length_n of_o the_o figure_n &_o the_o other_o the_o breadth_n the_o parallelogram_n be_v imagine_v to_o be_v make_v by_o the_o draught_n or_o motion_n of_o one_o of_o the_o line_n into_o the_o length_n of_o the_o other_o as_o if_o two_o number_n shall_v be_v multiply_v the_o one_o into_o the_o other_o as_o the_o figure_n abcd_o be_v a_o parallelogram_n and_o be_v say_v to_o be_v contain_v under_o the_o two_o right_a line_n ab_fw-la and_o ac_fw-la which_o contain_v the_o right_a angle_n bac_n or_o under_o the_o two_o right_a line_n ac_fw-la and_o cd_o for_o they_o likewise_o contain_v the_o right_a angle_n acd_o of_o which_o 2._o line_n the_o one_o namely_o ab_fw-la be_v the_o length_n and_o the_o other_o namely_o ac_fw-la be_v the_o breadth_n and_o if_o we_o imagine_v the_o line_n ac_fw-la to_o be_v draw_v or_o move_v direct_o according_a to_o the_o length_n of_o the_o line_n ab_fw-la or_o contrary_a wise_a the_o line_n ab_fw-la to_o be_v move_v direct_o according_a to_o the_o length_n of_o the_o line_n ac_fw-la you_o shall_v produce_v the_o whole_a rectangle_n parallelogram_n abcd_o which_o be_v say_v to_o be_v contain_v of_o they_o even_o as_o one_o number_n multiply_v by_o a_o other_o produce_v a_o plain_a and_o right_a angle_a superficial_a number_n as_o you_o see_v in_o the_o figure_n here_o set_v where_o the_o number_n of_o six_o or_o six_o unity_n be_v multiply_v by_o the_o number_n of_o five_o or_o by_o five_o unity_n of_o which_o multiplication_n be_v produce_v 30._o which_o number_n be_v set_v down_o and_o describe_v by_o his_o unity_n represent_v a_o plain_n and_o a_o right_a angle_a number_n wherefore_o even_o as_o equal_a number_n multiple_v by_o equal_a number_n produce_v number_n equal_v the_o one_o to_o the_o other_o so_o rectangle_n parallelogram_n which_o be_v comprehend_v under_o equal_a line_n be_v equal_a the_o one_o to_o the_o other_o definition_n 2._o in_o every_o parallelogram_n one_o of_o those_o parallelogram_n which_o soever_o it_o be_v which_o be_v about_o the_o diameter_n together_o with_o the_o two_o supplement_n be_v call_v a_o gnomon_n those_o particular_a parallelogram_n be_v say_v to_o be_v about_o the_o diameter_n of_o the_o parallelogram_n which_o have_v the_o same_o diameter_n which_o the_o whole_a parallelogram_n have_v and_o supplement_n be_v such_o which_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o of_o the_o parallelogram_n abcd_o the_o partial_a or_o particular_a parallelogram_n gkcf_n and_o ebkh_n be_v parallelogram_n about_o the_o diameter_n for_o that_o each_o of_o they_o have_v for_o his_o diameter_n a_o part_n of_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o ck_v and_o kb_v the_o particular_a diameter_n be_v part_n of_o the_o line_n cb_o which_o be_v the_o diameter_n of_o the_o whole_a parallelogram_n and_o the_o two_o parallelogram_n aegk_n and_o khfd_n be_v supplement_n because_o they_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n namely_o cb._n now_o any_o one_o of_o those_o partial_a parallelogram_n
thereof_o be_v 〈◊〉_d this_o iâ—_z also_o to_o be_v note_v that_o of_o line_n some_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o some_o be_v commensurable_a the_o one_o to_o the_o other_o in_o power_n of_o line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o be_v give_v a_o example_n in_o the_o declaration_n of_o the_o first_o definition_n namely_o the_o line_n a_o and_o b_o which_o be_v commensurable_a in_o length_n one_o and_o the_o self_n measure_n namely_o the_o line_n c_o measure_v the_o length_n of_o either_o of_o they_o of_o the_o other_o kind_n be_v give_v this_o definition_n here_o set_v for_o the_o open_n of_o which_o take_v this_o example_n let_v there_o be_v a_o certain_a line_n namely_o the_o line_n bc_o and_o let_v the_o square_n of_o that_o line_n be_v the_o square_n bcde_v suppose_v also_o a_o other_o line_n namely_o the_o line_n fh_o &_o let_v the_o square_v thereof_o be_v the_o square_a fhik_n and_o let_v a_o certain_a superficies_n namely_o the_o superficies_n a_o measure_v the_o square_n bcde_v take_v 16._o time_n which_o be_v the_o number_n of_o the_o little_a area_n square_n plait_n or_o superficiece_n contain_v and_o describe_v within_o the_o say_v square_n each_o of_o which_o be_v equal_a to_o the_o superficie_fw-la a._n again_o let_v the_o same_o superficies_n a_o measure_n the_o square_a fhik_n 9_o time_n take_v according_a to_o the_o number_n of_o the_o fieldâ—s_n or_o superficiece_n contain_v and_o describe_v in_o the_o same_o you_o see_v they_o that_o one_o and_o the_o self_n same_o superficies_n namely_o the_o superficies_n a_o be_v a_o common_a measure_n to_o both_o these_o square_n and_o by_o certain_a repetition_n thereof_o measure_v they_o both_o wherefore_o the_o two_o line_n bc_o and_o fh_o which_o be_v the_o side_n or_o line_n produce_v these_o square_n and_o who_o power_n these_o square_n be_v be_v by_o this_o definition_n line_n commensurable_a in_o power_n 4_o line_n 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here_o define_v to_o be_v absolute_o incommensurable_a these_o thing_n thus_o stand_v it_o may_v easy_o appear_v that_o if_o a_o line_n be_v assign_v and_o lay_v before_o we_o there_o may_v be_v innumerable_a other_o line_n commensurable_a unto_o it_o and_o other_o incommensurable_a unto_o it_o of_o commensurable_a line_n some_o be_v commensurable_a in_o length_n and_o power_n and_o some_o in_o power_n only_o 5_o and_o that_o right_a line_n so_o set_v forth_o be_v call_v a_o rational_a line_n thus_o may_v you_o see_v how_o to_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n line_n some_o commensurable_a both_o in_o length_n &_o power_n and_o some_o commensurable_a in_o power_n only_o and_o incommensurable_a in_o length_n and_o some_o incommensurable_a both_o in_o power_n &_o in_o length_n and_o this_o first_o line_n so_o set_v whereunto_o and_o to_o who_o square_n the_o other_o line_n and_o their_o square_n be_v compare_v be_v call_v a_o rational_a line_n common_o of_o the_o most_o part_n of_o writer_n line_n but_o some_o there_o be_v which_o mislike_n that_o it_o shall_v be_v call_v a_o rational_a line_n &_o that_o not_o without_o just_a cause_n in_o the_o greek_a copy_n it_o be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d rete_fw-la which_o signify_v a_o thing_n that_o may_v be_v speak_v &_o express_v by_o word_n a_o thing_n certain_a grant_v and_o appoint_v wherefore_o flussates_n a_o man_n which_o bestow_v great_a travel_n and_o diligence_n in_o restore_v of_o these_o element_n of_o euclid_n leave_v this_o word_n rational_a call_v this_o line_n suppose_v and_o first_o set_v a_o line_n certain_a because_o the_o part_n thereof_o into_o which_o it_o be_v divide_v be_v certain_a certain_a and_o know_v and_o may_v be_v express_v by_o voice_n and_o also_o be_v count_v by_o number_n other_o line_n be_v to_o this_o line_n incommensurable_a who_o part_n be_v not_o distinct_o know_v but_o be_v uncertain_a nor_o can_v be_v express_v by_o name_n nor_o assign_v by_o number_n which_o be_v of_o other_o man_n call_v irrational_a he_o call_v uncertain_a and_o surd_a line_n petrus_n montaureus_n although_o he_o do_v not_o very_o well_o like_a of_o the_o name_n yet_o he_o alter_v it_o not_o but_o use_v it_o in_o all_o his_o book_n likewise_o will_v we_o do_v here_o for_o that_o the_o word_n have_v be_v and_o be_v so_o universal_o receive_v and_o therefore_o will_v we_o use_v the_o same_o name_n and_o call_v it_o a_o rational_a line_n for_o it_o be_v not_o so_o great_a a_o matter_n what_o name_n we_o geve_v to_o thing_n so_o that_o we_o full_o understand_v the_o thing_n which_o the_o name_n signify_v this_o rational_a line_n thus_o here_o define_v be_v the_o ground_n and_o foundation_n of_o all_o the_o proposition_n almost_o of_o this_o whole_a ten_o book_n book_n and_o chief_o from_o the_o ten_o proposition_n forward_o so_o that_o unless_o you_o first_o place_n this_o rational_a line_n and_o have_v a_o special_a and_o continual_a 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the_o three_o which_o show_v what_o line_n be_v commensurable_a in_o power_n here_o notâ—_n how_o apt_o &_o natural_o euclid_n in_o this_o place_n use_v these_o word_n commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o because_o that_o all_o line_n which_o be_v commensurable_a in_o length_n be_v also_o commensurable_a in_o pour_v when_o he_o speak_v of_o line_n commensurable_a in_o length_n he_o ever_o add_v and_o in_o power_n but_o when_o he_o speak_v of_o line_n commensurable_a in_o power_n he_o add_v this_o word_n only_o and_o add_v not_o this_o word_n in_o length_n as_o he_o in_o the_o other_o add_v this_o word_n in_o power_n for_o not_o all_o line_n which_o be_v commensurable_a in_o power_n be_v straight_o way_n commensurable_a also_o in_o length_n of_o this_o definition_n take_v this_o example_n let_v the_o first_o line_n rational_a of_o purpose_n which_o be_v suppose_v and_o lay_v forth_o who_o part_n be_v certain_a &_o know_v and_o may_v be_v express_v name_v and_o number_v be_v ab_fw-la the_o quadrate_n whereof_o let_v be_v abcd_o then_o suppose_v again_o a_o other_o line_n namely_o the_o line_n of_o which_o let_v be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o first_o rational_a line_n that_o be_v as_o before_o be_v teach_v let_v one_o line_n measure_v the_o length_n of_o each_o line_n and_o also_o lâ—t_v one_o superficies_n measure_v the_o two_o square_n of_o the_o say_v two_o line_n as_o here_o in_o the_o example_n be_v suppose_v and_o also_o appear_v to_o the_o eye_n then_o be_v the_o line_n e_o fletcher_n also_o a_o rational_a line_n moreover_o if_o the_o line_n of_o be_v commensurable_a in_o power_n only_o to_o the_o rational_a line_n ab_fw-la first_o set_v and_o suppose_v so_o that_o no_o one_o line_n do_v measure_v the_o two_o line_n ab_fw-la and_o of_o as_o in_o example_n yâ—_n see_v to_o be_v for_o
that_o the_o line_n of_o be_v make_v equal_a to_o the_o line_n ad_fw-la which_o be_v the_o diameter_n of_o the_o square_n abcd_o of_o which_o square_v the_o line_n ab_fw-la be_v a_o side_n it_o be_v certain_a that_o the_o â—ide_v of_o a_o square_n be_v incommensurable_a in_o length_n to_o the_o diameter_n of_o the_o same_o square_n if_o there_o be_v yet_o find_v any_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o here_o do_v the_o triangle_n abdella_n or_o the_o triangle_n acd_v note_v in_o the_o square_n abcd_o or_o any_o of_o the_o four_o triangle_n note_v in_o the_o square_a efgh_o as_o appear_v somewhat_o more_o manifest_o in_o the_o second_o example_n in_o the_o declaration_n of_o the_o last_o definition_n go_v before_o the_o line_n of_o be_v also_o a_o rational_a line_n note_v that_o these_o line_n which_o here_o be_v call_v rational_a line_n be_v not_o rational_a line_n of_o purpose_n or_o by_o supposition_n as_o be_v the_o first_o rational_a line_n but_o be_v rational_a only_o by_o reason_n of_o relation_n and_o comparison_n which_o they_o have_v unto_o it_o because_o they_o be_v commensurable_a unto_o it_o either_o in_o length_n and_o power_n or_o in_o power_n only_o far_o here_o be_v to_o be_v note_v that_o these_o word_n length_n and_o power_n and_o power_n only_o be_v join_v only_o with_o these_o wordeâ—_n commensurable_a or_o incommensurable_a and_o be_v never_o join_v with_o these_o word_n rational_a or_o irrational_a so_o that_o no_o line_n can_v be_v call_v rational_a in_o length_n or_o in_o power_n nor_o like_o wise_a can_v they_o be_v call_v irrational_a in_o length_n or_o in_o power_n wherein_o undoubted_o campanus_n be_v deceive_v book_n who_o use_v those_o word_n &_o speech_n indifferent_o cause_v &_o bring_v in_o great_a obscurity_n to_o the_o proposition_n and_o demonstration_n of_o this_o book_n which_o he_o shall_v easy_o see_v which_o mark_v with_o diligence_n the_o demonstration_n of_o campanus_n in_o this_o book_n 7_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n be_v call_v irrational_a definition_n by_o line_n incommensurable_a to_o the_o rational_a line_n suppose_v in_o this_o place_n he_o understand_v such_o as_o be_v incommensurable_a unto_o it_o both_o in_o length_n and_o in_o power_n for_o there_o be_v no_o line_n incommensurable_a in_o power_n only_o for_o it_o can_v be_v that_o any_o line_n shall_v so_o be_v incommensurable_a in_o power_n only_o that_o they_o be_v not_o also_o incommensurable_a in_o length_n what_o so_o ever_o line_n be_v incommensurable_a in_o power_n the_o same_o be_v also_o incommensurable_a in_o length_n neither_o can_v euclid_n here_o in_o this_o place_n mean_a line_n incommensurable_a in_o length_n only_o for_o in_o the_o definition_n before_o he_o call_v they_o rational_a line_n neither_o may_v they_o be_v place_v amongst_o irrational_a line_n wherefore_o it_o remain_v that_o in_o this_o diffintion_n he_o speak_v only_o of_o those_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n first_o give_v and_o suppose_v both_o in_o length_n and_o in_o power_n which_o by_o all_o mean_n be_v incommensurable_a to_o the_o rational_a line_n &_o therefore_o most_o apt_o be_v they_o call_v irrational_a line_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o have_v be_v say_v before_o yet_o for_o the_o more_o plainness_n see_v this_o example_n let_v the_o â—â—rst_a rational_a line_n suppose_v be_v the_o line_n ab_fw-la who_o square_a or_o quadrate_n let_v be_v abcd._n and_o let_v there_o be_v give_v a_o other_o line_n of_o which_o lâ—t_v be_v to_o the_o rational_a line_n incommensurable_a in_o length_n and_o power_n so_o that_o let_v no_o one_o line_n measure_v the_o length_n of_o the_o two_o line_n ab_fw-la and_o of_o and_o let_v the_o square_n of_o the_o line_n of_o be_v efgh_a now_o if_o also_o there_o be_v no_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o be_v suppose_v to_o be_v in_o this_o example_n they_o be_v the_o line_n of_o a_o irrational_a line_n which_o word_n irrational_a as_o before_o do_v this_o word_n rational_a mislike_v many_o learned_a in_o this_o knowledge_n of_o geometry_n flussates_n uncertain_a as_o he_o leave_v the_o word_n rational_a and_o in_o stead_n thereof_o use_v this_o word_n certain_a so_o here_o he_o leave_v the_o word_n irrational_a and_o use_v in_o place_n thereof_o this_o word_n uncertain_a and_o ever_o name_v these_o line_n uncertain_a line_n petrus_n montaureus_n also_o mislike_v the_o word_n irrational_a will_v rather_o have_v they_o to_o be_v call_v surd_a line_n yet_o because_o this_o word_n irrational_a have_v ever_o by_o custom_n and_o long_a use_n so_o general_o be_v receive_n he_o use_v continual_o the_o same_o in_o greek_a such_o line_n be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d alogoi_n which_o signify_v nameless_a unspeakable_a uncertain_a in_o determinate_a line_n and_o with_o out_o proportion_n not_o that_o these_o irrational_a line_n have_v no_o proportion_n at_o all_o either_o to_o the_o first_o rational_a line_n or_o between_o themselves_o but_o be_v so_o name_v for_o that_o their_o proportion_n to_o the_o rational_a line_n can_v be_v express_v in_o number_n that_o be_v undoubted_o very_o untrue_a which_o many_o write_v that_o their_o proportion_n be_v unknown_a both_o to_o we_o and_o to_o nature_n be_v it_o not_o think_v you_o a_o thing_n very_o absurd_a to_o say_v that_o there_o be_v any_o thing_n in_o nature_n and_o produce_v by_o nature_n to_o be_v hide_v from_o nature_n and_o not_o to_o be_v know_v of_o nature_n it_o can_v not_o be_v say_v that_o their_o proportion_n be_v utter_o hide_v and_o unknown_a to_o we_o much_o less_o unto_o nature_n although_o we_o can_v geve_v they_o their_o name_n and_o distinct_o express_v they_o by_o number_n otherwise_o shall_v euclid_n have_v take_v all_o this_o travel_n and_o wonderful_a diligence_n bestow_v in_o this_o bookeâ—_n in_o vain_a and_o to_o no_o useâ—_n in_o which_o he_o do_v nothing_o ellâ—_n but_o teach_v the_o propriety_n and_o passion_n of_o these_o irrational_a linesâ—_n and_o show_v the_o proportion_n which_o they_o have_v the_o one_o to_o the_o other_o here_o be_v also_o to_o be_v note_v which_o thing_n also_o tartalea_n have_v before_o diligent_o notedâ—_n that_o campanus_n and_o many_o other_o writer_n of_o geometryâ—_n over_o much_o â—â—â—ed_a and_o be_v deceive_v in_o that_o they_o write_v and_o teach_v that_o all_o these_o line_n who_o square_n be_v not_o signify_v and_o may_v be_v express_v by_o a_o square_a number_n although_o they_o may_v by_o any_o other_o number_n as_o by_o 11._o 12._o 14._o and_o such_o other_o not_o square_a number_n be_v irrational_a line_n which_o be_v manifest_o repugnant_a to_o the_o ground_n and_o principle_n of_o euclid_n who_o will_v that_o all_o line_n which_o be_v commensurable_a to_o the_o rational_a line_n whether_o it_o be_v in_o length_n and_o power_n or_o in_o power_n only_o shall_v be_v rational_a undoubted_o this_o have_v be_v one_o of_o the_o chief_a and_o great_a cause_n of_o the_o wonderful_a confusion_n and_o darkness_n of_o this_o book_n book_n which_o so_o have_v toss_v and_o turmoil_v the_o wit_n of_o all_o both_o writer_n and_o reader_n master_n and_o scholar_n and_o so_o overwhelm_v they_o that_o they_o can_v not_o with_o out_o infinite_a travel_n and_o sweat_v attain_v to_o the_o truth_n and_o perfect_a understanding_n thereof_o definition_n 8_o the_o square_n which_o be_v describe_v of_o the_o rational_a right_a line_n suppose_v be_v rational_a until_o this_o definition_n have_v euclid_n set_v forth_o the_o nature_n and_o propriety_n of_o the_o first_o kind_n of_o magnitude_n namely_o of_o line_n how_o they_o be_v rational_a or_o irrational_a now_o he_o begin_v to_o â—hew_v how_o the_o second_o kind_n of_o magnitude_n namely_o superficies_n be_v one_o to_o the_o other_o rational_a or_o irrational_a this_o definition_n be_v very_a plain_n suppose_v the_o line_n ab_fw-la to_o be_v the_o rational_a line_n have_v his_o part_n and_o division_n certain_o know_v the_o square_a of_o which_o line_n let_v be_v the_o square_a abcd._n now_o because_o it_o be_v the_o square_a of_o the_o rational_a line_n ab_fw-la it_o be_v also_o call_v rational_a and_o as_o the_o line_n ab_fw-la be_v the_o first_o rational_a line_n unto_o which_o other_o line_n compare_v be_v count_v rational_a or_o irrational_a so_o be_v the_o quadrat_fw-la or_o square_v thereof_o the_o â—irst_n rational_a superficies_n unto_o which_o all_o other_o square_n or_o figure_n compare_v be_v count_v and_o name_v rational_a or_o irrational_a 9_o such_o which_o be_v commensurable_a unto_o it_o be_v rational_a definition_n in_o this_o definition_n where_o it_o be_v say_v such_o as_o be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n be_v not_o understand_v only_o other_o square_n or_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 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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
but_o not_o every_o pyramid_n a_o tetrahedron_n and_o in_o deed_n psellus_n in_o number_v of_o these_o five_o solid_n or_o body_n call_v a_o tetrahedron_n a_o pyramid_n in_o manifest_a word_n pyramid_n this_o i_o say_v may_v make_v flussas_n &_o other_o as_o i_o think_v it_o do_v to_o omit_v the_o definition_n of_o a_o tetrahedron_n in_o this_o place_n as_o sufficient_o comprehend_v within_o the_o definition_n of_o a_o pyramid_n give_v before_o but_o why_o then_o do_v he_o not_o count_v that_o definition_n of_o a_o pyramid_n faulty_a for_o that_o it_o extend_v itself_o to_o large_a and_o comprehend_v under_o it_o a_o tetrahedron_n which_o differ_v from_o a_o pyramid_n by_o that_o it_o be_v contain_v of_o equal_a triangle_n as_o he_o not_o so_o advise_o do_v before_o the_o definition_n of_o a_o prism_n definition_n 23_o a_o octohedron_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o eight_o equal_a and_o equilater_n triangle_n as_o a_o cube_n be_v a_o solid_a figure_n contain_v under_o six_o superficial_a figure_n of_o four_o side_n or_o square_n which_o be_v equilater_n equiangle_n and_o equal_v the_o one_o to_o the_o other_o so_o be_v a_o octohedron_n a_o solid_a figure_n contain_v under_o eight_o triangle_n which_o be_v equilater_n and_o equal_a the_o one_o to_o the_o other_o as_o you_o may_v in_o these_o two_o figure_n here_o set_v behold_v whereof_o the_o first_o be_v draw_v according_a as_o this_o solid_a be_v common_o describe_v upon_o a_o plain_a superficies_n the_o second_o be_v draw_v as_o it_o be_v describe_v by_o art_n upon_o a_o plain_a to_o show_v bodilike_o and_o in_o deed_n although_o the_o second_o appear_v to_o the_o eye_n more_o bodilike_o yet_o as_o i_o before_o note_v in_o a_o cube_n for_o the_o understanding_n of_o diverse_a proposition_n in_o these_o five_o book_n follow_v be_v the_o first_o description_n of_o more_o use_n yea_o &_o of_o necessity_n for_o without_o it_o you_o can_v not_o conceive_v the_o draught_n of_o line_n and_o section_n in_o any_o one_o of_o the_o eight_o side_n which_o be_v sometime_o in_o the_o description_n of_o some_o of_o those_o proposition_n require_v wherefore_o to_o the_o consideration_n of_o this_o first_o description_n imagine_v first_o that_o upon_o the_o upper_a face_n of_o the_o superficies_n of_o the_o parallelogram_n abcd_o be_v describe_v a_o pyramid_n have_v his_o four_o triangle_n afb_o afc_n cfd_n and_o dfb_n equilater_n and_o equiangle_n and_o concur_v in_o the_o point_n f._n then_o conceive_v that_o on_o the_o low_a face_n of_o the_o superficies_n of_o the_o former_a parallelogram_n be_v describe_v a_o other_o pyramid_n have_v his_o four_o triangle_n aeb_n aec_o ce_z &_o deb_n equilater_n and_o equiangle_n and_o concur_v in_o the_o point_n e._n for_o so_o although_o somewhat_o gross_o by_o reason_n the_o triangle_n can_v not_o be_v describe_v equilater_n you_o may_v in_o a_o plain_a perceive_v the_o form_n of_o this_o solid_a and_o by_o that_o mean_n conceive_v any_o line_n or_o section_n require_v to_o be_v draw_v in_o any_o of_o the_o say_v eight_o triangle_n which_o be_v the_o side_n of_o that_o body_n 24_o a_o dodecahedron_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o twelve_o equal_a equilater_n definition_n and_o equiangle_n pentagons_n as_o a_o cube_n a_o tetrahedron_n and_o a_o octohedron_n be_v contain_v under_o equal_a plain_a figure_n a_o cube_n under_o square_n the_o other_o two_o under_o triangle_n so_o be_v this_o solid_a figure_n contain_v under_o twelve_o equilater_n equiangle_n and_o equal_a pentagons_n or_o figure_n of_o five_o side_n as_o in_o these_o two_o figure_n here_o set_v you_o may_v perceive_v of_o which_o the_o first_o which_o thing_n also_o be_v before_o note_v of_o a_o cube_n a_o tetrahedron_n and_o a_o octohedron_n be_v the_o common_a description_n of_o it_o in_o a_o plain_a the_o other_o be_v the_o description_n of_o it_o by_o art_n upon_o a_o plain_a to_o make_v it_o to_o appear_v somewhat_o bodilike_o the_o first_o description_n in_o deed_n be_v very_o obscure_a to_o conceive_v but_o yet_o of_o necessity_n it_o must_v so_o neither_o can_v it_o otherwise_o be_v in_o a_o plain_a describe_v to_o understand_v those_o proposition_n of_o euclid_n in_o these_o five_o book_n a_o follow_v which_o concern_v the_o same_o for_o in_o it_o although_o rude_o may_v you_o see_v all_o the_o twelve_o pentagons_n which_o shall_v in_o deed_n be_v all_o equal_a equilater_n and_o equiangle_n and_o now_o how_o you_o may_v somewhat_o conceive_v the_o first_o figure_n describe_v in_o the_o plain_a to_o be_v a_o body_n imagine_v first_o the_o pentagon_n abcde_v â—o_o be_v upon_o a_o ground_n plain_a superficies_n then_o imagine_v the_o pentagon_n fghkl_o to_o be_v on_o high_a opposite_a unto_o the_o pentagon_n abcde_o and_o between_o those_o two_o pentagons_n there_o will_v be_v ten_o pentagons_n pull_v up_o five_o from_o the_o five_o side_n of_o the_o ground_n pentagon_n namely_o from_o the_o side_n ab_fw-la the_o pentagon_n abonm_n from_o the_o side_n bc_o the_o pentagon_n bcqpo_n from_o the_o side_n cd_o the_o pentagon_n cdsrq_n from_o the_o side_n de_fw-fr the_o pentagon_n deut_v from_o the_o side_n ea_fw-la the_o pentagon_n eamxv_n the_o other_o five_o pentagons_n have_v each_o one_o of_o their_o side_n common_a with_o one_o of_o the_o side_n of_o the_o pentagon_n fghkl_o which_o be_v opposite_a unto_o the_o pentagon_n in_o the_o ground_n superficies_n namely_o these_o be_v the_o other_o five_o pentagons_n fgnmx_n ghpon_n hkrqp_n klrst_n lfxut_n so_o here_o you_o may_v behold_v twelve_o pentagons_n which_o if_o you_o imagine_v to_o be_v equal_a equilater_n &_o equiangle_n and_o to_o be_v lift_v up_o you_o shall_v although_o somewhat_o rude_o conceive_v the_o bodily_a form_n of_o a_o pentagon_n and_o some_o light_n it_o will_v geve_v to_o the_o understanding_n of_o certain_a proposition_n of_o the_o five_o book_n follow_v concern_v the_o same_o definition_n 25_o a_o icosahedron_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o twenty_o equal_a and_o equilater_n triangle_n these_o â—ive_a solid_n now_o last_v define_v namely_o a_o cube_n a_o tetrahedron_n a_o octohedron_n a_o dodecahedron_n and_o a_o icosahedron_n be_v call_v regular_a body_n body_n as_o in_o plain_a superficiece_n those_o be_v call_v regular_a figure_n who_o side_n and_o angle_n be_v equal_a as_o be_v equilater_n triangle_n equilater_n pentagon_n hexagon_n &_o such_o like_a so_o in_o solid_n such_o only_o be_v count_v and_o call_v regular_a which_o be_v comprehend_v under_o equal_a plain_a superficiece_n which_o have_v equal_a side_n and_o equal_a angle_n as_o all_o these_o five_o foresay_a have_v as_o manifest_o appear_v by_o their_o definition_n which_o be_v all_o give_v by_o this_o propriety_n of_o equality_n of_o their_o superficiece_n which_o have_v also_o their_o side_n and_o angle_n equal_a and_o in_o all_o the_o course_n of_o nature_n there_o be_v no_o other_o body_n of_o this_o condition_n and_o perfection_n but_o only_o these_o five_o wherefore_o they_o have_v ever_o of_o the_o ancient_a philosopher_n be_v have_v in_o great_a estimation_n and_o admiration_n and_o have_v be_v think_v worthy_a of_o much_o contemplation_n about_o which_o they_o have_v bestow_v most_o diligent_a study_n and_o endeavour_n to_o search_v out_o the_o nature_n &_o property_n of_o they_o they_o be_v as_o it_o be_v the_o end_n and_o perfection_n of_o all_o geometry_n for_o who_o sake_n be_v write_v whatsoever_o be_v write_v in_o geometry_n they_o be_v as_o man_n say_v first_o invent_v by_o the_o most_o witty_a pythagoras_n then_o afterward_o set_v forth_o by_o the_o divine_a plato_n and_o last_o of_o all_o marvellous_o teach_v and_o declare_v by_o the_o most_o excellent_a philosopher_n euclid_n in_o these_o book_n follow_v and_o ever_o since_o wonderful_o embrace_v of_o all_o learned_a philosopher_n body_n the_o knowledge_n of_o they_o contain_v infinite_a secret_n of_o nature_n pithagâ—ras_n timeus_n and_o plato_n by_o they_o search_v out_o the_o composition_n of_o the_o world_n with_o the_o harmony_n and_o preservation_n thereof_o and_o apply_v these_o â—ive_a solid_n to_o the_o simple_a part_n thereof_o the_o pyramid_n or_o tetrahedron_n they_o ascribe_v to_o the_o â—ire_n fire_n for_o that_o it_o ascend_v upward_o according_a to_o the_o figure_n of_o the_o pyramid_n to_o the_o air_n they_o ascribe_v the_o octohedron_n air_n for_o that_o through_o the_o subtle_a moisture_n which_o it_o have_v it_o extend_v itself_o every_o way_n to_o the_o one_o side_n and_o to_o the_o other_o accord_v as_o that_o figure_n do_v unto_o the_o water_n they_o assign_v the_o ikosahedron_n for_o that_o it_o be_v continual_o flow_v and_o move_v water_n and_o as_o it_o be_v make_v angleâ—_n 〈…〉_o â—ide_v according_a to_o that_o figure_n and_o to_o the_o earth_n they_o attribute_v a_o cube_n earth_n as_o to_o a_o thing_n stableâ—_n 〈◊〉_d and_o sure_a as_o the_o figure_n
side_n wherefore_o the_o whole_a parallelipipedon_n ae_n be_v equal_a and_o like_a to_o the_o whole_a parallelipipedon_n yu._n extend_v by_o the_o second_o petition_n the_o line_n doctor_n and_o wv_n until_o they_o concur_v and_o let_v they_o concur_v in_o the_o point_n q._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n t_o draw_v unto_o the_o line_n rq_n a_o parallel_a line_n t_o 4_o and_o extend_v due_o the_o line_n ta_o and_o do_v until_o they_o concur_v and_o let_v they_o concur_v in_o the_o point_n ✚_o and_o make_v perfect_v the_o solid_n qy_n and_o ri._n now_o the_o solid_a qy_n who_o plain_n base_a be_v the_o parallelogram_n ry_o and_o the_o opposite_a side_n unto_o the_o base_a the_o parallelogram_n qb_n be_v equal_a to_o the_o solid_a yv_n who_o base_a be_v the_o parallelogram_n ry_o and_o the_o opposite_a side_n unto_o the_o base_a the_o parallelogram_n we_o for_o they_o consist_v upon_o one_o and_o the_o self_n fame_n base_a namely_o ry_o and_o be_v under_o one_o and_o the_o self_n same_o altitude_n and_o their_o stand_a line_n namely_o rq_n rv_n ta_o tw_n sn_a sd_v yb_n and_o yz_n be_v in_o the_o self_n same_o right_a line_n namely_o qw_o and_o nz_n but_o the_o solid_a yv_n be_v prove_v equal_a to_o the_o solid_a ae_n wherefore_o also_o the_o solid_a yq_n be_v equal_a to_o the_o solid_a ae_n now_o forasmuch_o as_o the_o parallelogram_n ruwt_fw-mi be_v equal_a to_o the_o parallelogram_n qt_n by_o the_o 35._o of_o the_o first_o and_o the_o parallelogram_n ab_fw-la be_v equal_a to_o the_o parallelogram_n rw_n therefore_o the_o parallelogram_n qt_o be_v equal_a to_o the_o parallelogram_n ab_fw-la and_o the_o parallelogram_n cd_o be_v equal_a to_o the_o parallelogram_n ab_fw-la by_o supposition_n wherefore_o the_o parallelogram_n cd_o be_v equal_a to_o the_o parallelogram_n qt_n and_o there_o be_v a_o certain_a other_o superficies_n namely_o dt_n wherefore_o by_o the_o 7._o of_o the_o five_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o base_a qt_n to_o the_o base_a dt_n and_o forasmuch_o as_o the_o whole_a parallelipipedon_n ci_o be_v cut_v by_o the_o plain_a superficies_n rf_n which_o be_v a_o parallel_n to_o either_o of_o the_o opposite_a plain_a superficiece_n therefore_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o solid_a cf_o to_o the_o solid_a ri_n by_o the_o 25._o of_o the_o eleven_o and_o by_o the_o same_o reason_n also_o forasmuch_o as_o the_o whole_a parallelipipedon_n qi_n be_v cut_v by_o the_o plain_a superficies_n ry_o which_o be_v a_o parallel_n to_o either_o of_o the_o opposite_a plain_a superficiece_n therefore_o as_o the_o base_a qt_n be_v to_o the_o base_a dt_n so_o be_v the_o solid_a qy_n to_o the_o solid_a ri._n but_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o base_a qt_n to_o the_o base_a td_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a cf_o be_v to_o the_o solid_a ri_n so_o be_v the_o solid_a qy_n to_o the_o solid_a ri._n wherefore_o either_o of_o these_o solid_n cf_o and_o qy_n have_v to_o the_o solid_a ri_n one_o and_o the_o same_o proportion_n wherefore_o the_o solid_a cf_o be_v equal_a to_o the_o solid_a qy_n but_o it_o be_v prove_v that_o the_o solid_a qy_n be_v equal_a to_o the_o solid_a ae_n wherefore_o also_o the_o solid_a cf_o be_v equal_a to_o the_o solid_a ae_n but_o now_o suppose_v that_o the_o stand_a line_n namely_o agnostus_n hk_o case_n be_v lm_o cx_o open_v df_o and_o r_n be_v not_o erect_v perpendicular_o to_o the_o base_n ab_fw-la and_o cd_o then_o also_o i_o say_v that_o the_o solid_a ae_n be_v equal_a to_o the_o solid_a cf._n draw_v by_o the_o 11._o of_o the_o eleven_o unto_o the_o ground_n plain_a superficiece_n ab_fw-la and_o cd_o from_o these_o point_n king_n e_o c_o m_o p_o f_o x_o saint_n these_o perpendicular_a line_n kn_n et_fw-la gv_n mz_n pw_n fie_o xq_n and_o si_fw-it and_o draw_v these_o right_a line_n nt_v nv_n zv_n zt_n why_o wq_n iq_fw-la and_o jy_n now_o by_o that_o which_o have_v before_o be_v prove_v in_o this_o 31._o proposition_n the_o solid_a kz_o be_v equal_a to_o the_o solid_a piperollo_n for_o they_o consist_v upon_o equal_a you_o base_n namely_o km_o and_o ps_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n who_o stand_a line_n also_o be_v erect_v perpendicular_o to_o the_o base_n but_o the_o solid_a kz_o be_v equal_a to_o the_o solid_a ae_n by_o the_o 29._o of_o the_o eleven_o and_o the_o solid_a piperollo_n be_v by_o the_o same_o equal_a to_o the_o solid_a cf_o for_o they_o consist_v upon_o one_o and_o the_o self_n same_o base_a and_o be_v under_o one_o &_o the_o self_n same_o altitude_n who_o stand_a line_n be_v upon_o the_o self_n same_o right_a line_n wherefore_o also_o the_o solid_a ae_n be_v equal_a to_o the_o solid_a ce._n wherefore_o parallelipipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v the_o demonstration_n of_o the_o first_o case_n of_o this_o proposition_n be_v very_o hard_a to_o conceive_v by_o the_o figure_n describe_v for_o it_o in_o a_o plain_n and_o yet_o before_o m._n do_v you_o invent_v that_o figure_n which_o we_o have_v there_o place_v for_o it_o it_o be_v much_o hard_o for_o both_o in_o the_o greek_a and_o latin_a euclid_n it_o be_v very_o ill_o make_v and_o it_o be_v in_o a_o manner_n impossible_a to_o conceive_v by_o it_o the_o construction_n and_o demonstration_n thereto_o appertain_v wherefore_o i_o have_v here_o describe_v other_o figure_n which_o first_o describe_v upon_o past_v paper_n or_o such_o like_a matter_n and_o then_o order_v they_o in_o manner_n follow_v as_o touch_v the_o solid_a ae_n in_o the_o first_o case_n i_o need_v not_o to_o make_v any_o new_a description_n for_o it_o be_v plain_a enough_o to_o conceive_v as_o it_o be_v there_o draw_v although_o you_o may_v for_o your_o more_o ease_n of_o imagination_n describe_v of_o past_v paper_n a_o parallelipipedoâ—_n have_v his_o side_n equal_a with_o the_o side_n of_o the_o parallelipipedon_n ae_n before_o describe_v and_o have_v also_o the_o six_o parallelogram_n thereof_o contain_v under_o those_o side_n equiangle_n with_o the_o six_o parallelogram_n of_o that_o figure_n each_o side_n and_o each_o angle_n equal_a to_o his_o correspondent_a side_n and_o to_o his_o correspondent_a angle_n but_o concern_v the_o other_o solid_a when_o you_o have_v describe_v these_o three_o figure_n upon_o past_v paper_n where_o note_n for_o the_o proportion_n of_o each_o line_n to_o make_v your_o figure_n of_o past_v paper_n equal_a with_o the_o figure_n before_o describe_v upon_o the_o plain_n let_v your_o line_n open_v cx_o r_n df_o etc_n etc_n namely_o the_o rest_n of_o the_o stand_a line_n of_o these_o figure_n be_v equal_a to_o the_o stand_a line_n open_v cx_o r_n df_o etc_n etc_n of_o that_o figure_n likewise_o let_v the_o line_n oc_fw-fr cr_n rd_o do_v etc_n etc_n namely_o the_o side_n which_o contain_v the_o base_n of_o these_o figure_n be_v equal_a to_o the_o line_n oc_fw-fr cr_n rd_o do_v etc_n etc_n namely_o to_o the_o side_n which_o contain_v the_o base_n of_o that_o figure_n moreover_o let_v the_o line_n px_n xâ—_n sf_n fp_n etc_n etc_n namely_o the_o rest_n of_o the_o line_n which_o contain_v the_o upper_a superficiece_n of_o these_o figure_n be_v equal_a to_o the_o line_n px_n xs_n sf_n fâ—_n etc_n etc_n namely_o to_o the_o rest_n of_o the_o line_n which_o contain_v the_o upper_a superficiece_n of_o that_o figure_n to_o have_v describe_v all_o those_o foresay_a line_n of_o these_o figure_n equal_a to_o all_o the_o line_n of_o that_o figure_n will_v have_v require_v much_o more_o space_n than_o here_o can_v be_v spare_v i_o have_v make_v they_o equal_v only_o to_o the_o half_n of_o those_o line_n but_o by_o the_o example_n of_o these_o you_o may_v if_o you_o will_v describe_v the_o like_a figure_n have_v their_o line_n equal_a to_o the_o whole_a line_n of_o the_o figure_n in_o the_o plain_n each_o line_n to_o his_o correspondent_a line_n when_o i_o say_v you_o have_v as_o before_o be_v teach_v describe_v these_o three_o figure_n cut_v fine_o the_o line_n xc_o sir_n fd_v of_o the_o first_o figure_n and_o the_o line_n sir_n it_o and_o i_o ✚_o of_o the_o second_o figure_n likewise_o the_o line_n â—r_o nq_n zw_n and_o it_o of_o the_o three_o figure_n and_o fold_v these_o figure_n according_o which_o you_o can_v not_o choose_v but_o do_v if_o you_o mark_v well_o the_o letter_n of_o every_o line_n as_o touch_v the_o second_o case_n you_o need_v no_o new_a figure_n for_o it_o be_v plain_a to_o see_v by_o the_o figure_n now_o
shall_v be_v proportional_a and_o iâ—_z the_o parallelipipedon_n describe_v of_o they_o be_v like_a and_o in_o like_a sort_n describe_v be_v proportional_a those_o right_a line_n also_o shall_v be_v proportional_a svppose_v that_o these_o four_o right_a line_n ab_fw-la cd_o of_o and_o gh_o be_v proportional_a as_o ab_fw-la be_v to_o cd_o so_o let_v of_o be_v to_o gh_o and_o upon_o the_o line_n ab_fw-la cd_o of_o and_o gh_o describe_v these_o parallelipipedon_n ka_o lc_n menander_n and_o ng_z be_v like_a and_o in_o like_a sort_n describe_v then_o i_o say_v that_o as_o the_o solid_a ka_o be_v â—o_o the_o solid_a lc_n so_o be_v the_o solid_a menander_n to_o the_o solid_a ng_z part_n for_o forasmuch_o as_o the_o parallelipipedon_n ka_o be_v like_a to_o the_o parallelipipedon_n lc_n therefore_o by_o the_o 33._o of_o the_o eleven_o the_o solid_a ka_o be_v to_o the_o solid_a lc_n in_o treble_a proportion_n of_o that_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n cdâ—_n and_o by_o the_o same_o reason_n the_o parallelipipedon_n menander_n be_v to_o the_o parallelipipedon_n ng_z in_o treble_a proportion_n of_o that_o which_o the_o side_n of_o be_v to_o the_o side_n gh_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o parallelipipedon_n ka_o be_v to_o the_o parallelipipedon_n lc_n so_o be_v the_o parallelipipedon_n menander_n tâ—_n the_o parallelipipedon_n ng_z but_o now_o suppose_v that_o as_o the_o parallelipipedon_n ka_o be_v to_o the_o parallelipipedon_n lc_n so_o be_v the_o parallelipipedon_n menander_n to_o the_o parallelipipedon_n ng_z then_o i_o say_v part_n that_o as_o the_o right_a line_n ab_fw-la be_v to_o the_o right_a line_n cd_o so_o be_v the_o right_a liâ—â—_n of_o to_o the_o right_a line_n gh_o for_o again_o forasmuch_o as_o the_o solid_a ka_o be_v to_o the_o solid_a lc_n in_o treble_a proportion_n of_o that_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n cd_o and_o the_o solid_a menander_n also_o be_v to_o the_o solid_a ng_z in_o treble_a proportion_n of_o that_o which_o the_o line_n of_o be_v to_o the_o line_n gh_o and_o as_o the_o solid_a ka_o be_v to_o the_o solid_a lc_n so_o be_v the_o solid_a menander_n to_o the_o solid_a ng_z wherefore_o also_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n gh_o if_o therefore_o there_o be_v four_o right_a line_n proportional_a the_o parallelipipedon_n describe_v of_o those_o line_n be_v like_a &_o in_o like_a sort_n describe_v shall_v be_v proportional_a and_o if_o the_o parallelipipedon_n describe_v of_o they_o and_o be_v like_a and_o in_o like_a sort_n describe_v be_v proportional_a those_o right_a line_n also_o shall_v be_v proportional_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 33._o theorem_a the_o 38._o proposition_n if_o a_o plain_a superficies_n be_v erect_v perpendicular_o to_o a_o plain_a superficies_n and_o from_o a_o point_n take_v in_o one_o of_o the_o plain_a superficiece_n be_v draw_v to_o the_o other_o plain_a superficies_n a_o perpendicular_a line_n that_o perpendicular_a line_n shall_v fall_v upon_o the_o common_a section_n of_o those_o plain_a superficiece_n svppose_v that_o the_o plain_a superficies_n cd_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n ab_fw-la and_o let_v their_o common_a section_n be_v the_o line_n dam_fw-ge and_o in_o the_o superficies_n cd_o take_v a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v e._n then_o i_o say_v that_o a_o perpendicular_a line_n draw_v from_o the_o point_n e_o to_o the_o plain_a superficies_n ab_fw-la shall_v fall_v upon_o the_o right_a line_n da._n for_o if_o not_o then_o let_v it_o fall_v without_o the_o line_n dam_fw-ge as_o the_o line_n of_o do_v and_o let_v it_o fall_v upon_o the_o plain_a superficies_n ab_fw-la in_o the_o point_n f._n impossibility_n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n fletcher_n draw_v unto_o the_o line_n dam_fw-ge be_v in_o the_o superficies_n ab_fw-la a_o perpendicular_a line_n fg_o which_o line_n also_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n cd_o by_o the_o three_o definition_n by_o reason_n we_o presuppose_v cd_o and_o ab_fw-la to_o be_v perpendicular_o erect_v each_o to_o other_o draw_v a_o right_a line_n from_o the_o point_n e_o to_o the_o point_n g._n and_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n cd_o and_o the_o line_n eglantine_n touch_v it_o be_v in_o the_o superficies_n cd_o wherefore_o the_o angle_n fge_n be_v by_o the_o 2._o definition_n of_o the_o eleven_o a_o right_a angle_n but_o the_o line_n of_o be_v also_o erect_v perpendicular_o to_o the_o superficies_n ab_fw-la wherefore_o the_o angle_n efg_o be_v a_o right_a angle_n now_o therefore_o two_o angle_n of_o the_o triangle_n efg_o be_v equal_a to_o two_o right_a angle_n which_o by_o the_o 17._o of_o the_o first_o be_v impossible_a wherefore_o a_o perpendicular_a line_n draw_v from_o the_o point_n e_o to_o the_o superficies_n ab_fw-la fall_v not_o without_o the_o line_n da._n wherefore_o it_o fall_v upon_o the_o line_n dam_fw-ge which_o be_v require_v to_o be_v prove_v ¶_o note_n campane_n make_v this_o as_o a_o corollary_n follow_v upon_o the_o 13_o and_o very_o well_o with_o small_a aid_n of_o other_o proposition_n he_o prove_v itâ—_n who_o demonstration_n there_o flussas_n have_v in_o this_o place_n and_o none_o other_o though_o he_o say_v that_o campane_n of_o such_o a_o proposition_n as_o of_o euclides_n make_v no_o mention_n in_o this_o figure_n you_o may_v more_o full_o see_v the_o former_a proposition_n and_o demonstration_n if_o you_o erect_v perpendicular_o unto_o the_o ground_n plain_a superficies_n ab_fw-la the_o superficies_n cd_o and_o imagine_v a_o line_n to_o be_v extend_v from_o the_o point_n e_o to_o the_o point_n fletcher_n instead_n whereof_o you_o may_v extend_v if_o you_o will_v a_o thread_n ¶_o the_o 34._o theorem_a the_o 39_o proposition_n if_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 common_a 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 shall_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n svppose_v that_o of_o be_v a_o parallelipipedon_n and_o let_v the_o opposite_a side_n thereof_o cf_o and_o ah_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n king_n l_o m_o n_o and_o likewise_o let_v the_o opposite_a side_n ad_fw-la and_o gf_o be_v divide_v into_o two_o equal_a part_n in_o the_o pointâ—s_v x_o pea_n oh_o r_o and_o by_o those_o section_n extend_v these_o two_o plain_a superficiece_n kn_n &_o xr_n and_o let_v the_o common_a section_n of_o those_o plain_a superficiece_n be_v the_o line_n we_o and_o let_v the_o diagonal_a line_n of_o the_o solid_a ab_fw-la be_v the_o line_n dg_o then_o i_o say_v that_o the_o line_n we_o and_o dg_o do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n construction_n that_o be_v that_o the_o line_n vt_fw-la be_v equal_a to_o the_o line_n it_o be_v and_o the_o line_n dt_n to_o the_o line_n tg_n draw_v these_o right_a line_n dv_o we_o bs_n and_o sg_n demonstration_n now_o forasmuch_o as_o the_o line_n dx_o be_v a_o parallel_n to_o the_o line_n oe_z therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n dxv_o and_o who_n be_v alternate_a angle_n be_v equal_a the_o one_o to_o the_o other_o and_o forasmuch_o as_o the_o line_n dx_o be_v equal_a to_o the_o line_n oe_z and_o the_o line_n xv_o to_o the_o line_n vo_o and_o they_o comprehend_v equal_a angle_n wherefore_o the_o base_a dv_o be_v equal_a to_o the_o base_a we_o by_o the_o 4._o of_o the_o first_o and_o the_o triangle_n dxv_o be_v equal_a to_o the_o triangle_n who_n and_o the_o rest_n of_o the_o angle_n to_o the_o rest_n of_o the_o angle_n wherefore_o the_o angle_n xud_n be_v equal_a to_o the_o angle_n ove._n wherefore_o due_a be_v one_o right_a line_n and_o by_o the_o same_o reason_n bsg_n be_v also_o one_o right_a line_n and_o the_o line_n bs_n be_v equal_a to_o the_o line_n sg_n and_o forasmuch_o as_o the_o line_n ca_n be_v equal_a to_o the_o line_n db_o and_o be_v unto_o it_o a_o parallel_n but_o the_o line_n ca_n be_v equal_a to_o the_o line_n ge_z and_o be_v unto_o it_o also_o a_o parallel_n wherefore_o by_o the_o firsâ—_o common_a sentence_n the_o line_n db_o be_v equal_a to_o the_o line_n ge_z &_o be_v also_o a_o parallel_n unto_o it_o but_o the_o right_a line_n de_fw-fr and_o bg_o do_v join_v these_o parallel_a line_n together_o wherefore_o by_o the_o 33._o of_o the_o first_o the_o line_n de_fw-fr be_v a_o parallel_n unto_o the_o line_n bg_o and_o in_o either_o of_o these_o line_n be_v take_v point_n
first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n this_o proposition_n in_o discreet_a quantity_n answer_v to_o the_o 23._o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n this_o and_o the_o eleven_o proposition_n follow_v declare_v the_o passion_n and_o property_n ofâ—_n prime_a number_n demonstration_n lead_v to_o a_o impossibility_n this_o be_v the_o converse_n of_o the_o former_a proposition_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n lead_v to_o a_o absurdity_n demonstration_n of_o the_o second_o part_n which_o be_v the_o conâ—câ—se_n of_o the_o first_o lean_v also_o to_o a_o absurdity_n demonstrasion_n lead_v to_o a_o absurdity_n demonstrasion_n a_o corollary_n â—â—ded_v by_o campave_n demonstration_n lâ—ading_v to_o a_o impossibility_n an_o other_o demonstration_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n add_v by_o campane_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o caseâ—_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n leaâ—iâ—g_v â—o_o a_o absurdity_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n construction_n demonstration_n leâ—ding_v to_o a_o absurdity_n a_o corollary_n adâ—ed_v by_o campane_n how_o to_o â—inde_v out_o the_o second_o least_o number_n and_o the_o three_o and_o so_o â—orth_o infinite_o how_o to_o siâ—â—_n out_o the_o least_o â—â—mâ—_n a_o conâ—ayâ—â—g_n â—â—e_z paâ—â—s_n of_o part_n the_o arguâ—â—â—â—_n of_o the_o eight_o book_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n this_o proposition_n be_v the_o â—â—uerse_a of_o the_o first_o demonstrationâ—_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n demonstration_n this_o proposition_n in_o number_n answer_v to_o the_o of_o the_o six_o touch_v parellelogramme_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n demonstrate_v construction_n the_o first_o part_n of_o this_o prâ—position_n demonstrate_v the_o second_o part_n demonstrate_v construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proportion_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proposition_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o fiâ—st_a part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n this_o proposition_n be_v the_o converse_n of_o the_o 18._o proposition_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o 19_o proposition_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n another_o corollary_n add_v by_o flussates_n the_o argument_n of_o the_o niâ—th_o book_n demonstration_n this_o proposition_n be_v the_o converse_n oâ—_n tâ—e_z formâ—â—_n demonstration_n a_o corollary_n aâ—ded_v by_o campane_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o campane_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n the_o second_o part_n demonstrate_v demostration_n of_o the_o three_o part_n demostration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o pâ—rt_n demonstrate_v demonstration_n of_o the_o first_o part_n leave_v to_o a_o absurdity_n demonstration_n of_o the_o â—â—cond_a pâ—â—â—_n lead_v alâ—o_o to_o a_o absurdity_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n aâ—ter_n flussates_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n after_o campane_n demonstration_n lead_v to_o a_o absurdity_n a_o proposition_n add_v by_o campane_n construction_n demonstration_n demonstration_n to_o prove_v that_o the_o number_n a_o and_o c_o be_v prime_a to_o b._n demonstratiou_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstration_n this_o answer_v to_o the_o 2._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 3._o of_o the_o three_o demonstration_n this_o answer_v to_o thâ—_z 4._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 5._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 6._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 7._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 8._o of_o the_o second_o demonstratition_n this_o answer_v to_o thâ—_z 9_o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 10._o oâ—_n the_o second_o demonstration_n a_o negative_a proposition_n demonstration_n leaâ—ing_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o abjurditie_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n divert_v case_n â—n_v this_o proposition_n the_o first_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o proposition_n add_v by_o campane_n a_o other_o add_v by_o he_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n demonstration_n demonstration_n this_o proposition_n teachâ—th_v how_o to_o find_v out_o a_o perfect_a number_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n the_o argument_n of_o the_o ten_o book_n difference_n between_o number_n and_o magnitude_n a_o line_n be_v not_o make_v of_o point_n as_o number_n be_v make_v of_o unity_n this_o book_n the_o hard_a to_o understand_v of_o all_o the_o book_n of_o euclid_n in_o this_o book_n be_v entreat_v of_o a_o strange_a manner_n of_o matter_n then_o in_o the_o former_a many_o even_o of_o the_o well_o learn_v have_v think_v that_o this_o book_n can_v not_o well_o be_v understand_v without_o algebra_n the_o nine_o former_a book_n &_o the_o principle_n of_o this_o â—ooke_n well_o understand_v this_o book_n will_v not_o be_v hard_a to_o understand_v the_o fâ—rst_a definition_n the_o second_o definition_n contraries_n make_v manifest_a by_o the_o compare_v of_o the_o one_o to_o the_o other_o the_o third_o definition_n what_o the_o power_n of_o a_o line_n be_v the_o four_o definition_n unto_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n why_o some_o mislike_n that_o the_o line_n first_o set_v shall_v be_v call_v a_o rational_a line_n flussates_n call_v this_o line_n a_o line_n certain_a this_o rational_a line_n the_o ground_n in_o a_o manner_n of_o all_o the_o proposition_n in_o this_o ten_o book_n note_n the_o line_n rational_a of_o purpose_n the_o six_o definition_n campâ—nus_n â—ath_z cause_v much_o obscurity_n in_o this_o ten_o book_n the_o seven_o definition_n flussates_n in_o steed_n of_o this_o word_n irrational_a use_v this_o word_n uncertain_a why_o they_o be_v call_v irrational_a line_n the_o cause_n of_o the_o obscurity_n and_o confusedness_n in_o this_o book_n the_o eight_o definition_n the_o nine_o definition_n the_o ten_o definition_n the_o eleven_o definition_n construction_n demonstration_n a_o corollary_n construction_n demonstration_n this_o proposition_n teach_v that_o incontinuall_a quantity_n which_o the_o first_o of_o the_o seven_o teach_v in_o discrete_a quantity_n construction_n demonstration_n lead_v to_o a_o absurdity_n two_o case_n in_o this_o proposition_n the_o first_o case_n this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 2._o of_o the_o sâ—â—ith_v teach_v in_o number_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 3._o of_o the_o second_o teach_v in_o number_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n a_o leâ—ma_n necessary_a