hollow_a part_n it_o be_v call_v a_o concave_a if_o it_o be_v plane_n and_o unite_a it_o be_v call_v a_o plane_n b._n a_o convex_a superficies_n c._n a_o concave_a superficies_n a._n a_o plane_n superficies_n the_o first_o part_n teach_v only_o the_o construction_n a_o frame_n of_o plain_a superficies_n a_o term_n or_o bind_v be_v the_o extremity_n of_o any_o thing_n the_o point_n be_v the_o term_n or_o bind_v of_o the_o line_n the_o line_n be_v the_o term_n of_o the_o superficies_n and_o the_o superficies_n be_v the_o term_n or_o bind_v of_o a_o body_n here_o end_v the_o four_o of_o diamond_n see_v the_o five_o of_o diamoud_n of_o superficies_n or_o figure_n rectilinear_n the_o superficies_n do_v take_v particular_a name_n according_a to_o the_o number_n of_o their_o side_n as_o a._n a_o trigone_n or_o triangle_n a_o figure_n of_o 3_o side_n b._n a_o tetragone_fw-mi or_o square_a a_o figure_n of_o 4_o side_n c._n a_o pentagone_fw-mi or_o figure_n of_o 5_o side_n d._n a_o exagone_fw-mi 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on_o the_o side_n of_o the_o end_n d._n let_v the_o pen_n run_v by_o the_o ruler_n towards_o e._n demand_v iii_o draw_v a_o circle_n from_o the_o point_n a._n and_o the_o interval_n a_o b._n the_o practice_n set_a one_o of_o the_o point_n of_o the_o compass_n at_o the_o point_n give_v a._n open_v the_o other_o unto_o the_o point_n b._n turn_v the_o compass_n upon_o the_o point_n a._n and_o draw_v it_o from_o the_o point_n b._n describe_v the_o circle_n demand_v b_o c_o d._n demand_v iu._n from_o the_o point_n give_v e_o and_z f._n make_v a_o section_n the_o practice_n open_v the_o compass_n at_o pleasure_n yet_o in_o such_o manner_n that_o the_o open_n of_o two_o point_n may_v be_v great_a than_o the_o half_a of_o the_o distance_n which_o be_v between_o the_o two_o point_n propound_v e_z and_z f._n by_o this_o open_n of_o the_o compass_n from_o the_o point_n e_o draw_v the_o arch_n l_o m._n from_o the_o point_n f._n draw_v the_o arch_n h_o i._n the_o section_n g._n shall_v be_v the_o 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e._n the_o angle_n b_o a_o e._n shall_v be_v equal_a to_o the_o angle_n c_o d_o g._n the_o which_o be_v propound_v to_o be_v do_v proposition_n ix_o to_o divide_v a_o straight_a line_n give_v into_o as_o many_o equal_a part_n as_o one_o will_n the_o position_n let_v a_o b._n be_v the_o line_n propose_v to_o be_v divide_v into_o six_o equal_a part_n the_o practice_n from_o the_o end_n a._n draw_v at_o discretion_n the_o line_n a_o c._n from_o the_o end_n b._n draw_v the_o line_n b_o d._n parallel_n to_o the_o line_n a_o c._n from_o the_o point_n a_o and_o b._n and_o upon_o the_o line_n a_o c_o b_o d._n bring_v at_o discretion_n six_o equal_a part_n viz._n e_o f_o g_o h_o i_o l._n upon_o the_o line_n a_o c._n r_o q_o p_o o_o n_o m_o upon_o the_o line_n b_o d._n draw_v the_o line_n e_o n._n f_o o._n g_o p._n h_n q._n l_o r._n the_o line_n a_o b._n shall_v be_v divide_v into_o six_o equal_a part_n by_o the_o section_n s_o t._n v._o x._o y._n proposition_n x._o from_o a_o point_n give_v to_o draw_v a_o straight_a line_n which_o touch_v a_o circle_n propound_v the_o position_n let_v a._n be_v the_o point_n from_o which_o we_o must_v draw_v a_o line_n which_o touch_v the_o circle_n d_o o_fw-fr p._n the_o practice_n from_o the_o centre_n of_o the_o circle_n b._n draw_v the_o line_n secant_fw-la b_o a._n divide_v this_o line_n b_o a._n into_o two_o equal_o in_o c._n from_o the_o point_n c._n and_o the_o interval_n c_o a._n draw_v the_o
demicircle_n a_o d_o b._n cut_v the_o circle_n in_o d._n from_o the_o point_n give_v a._n draw_v the_o right_a line_n a_o e._n by_o the_o point_n d._n this_o right_a line_n a_o e._n shall_v be_v the_o line_n touch_v the_o require_v here_n end_v the_o ace_n of_o hart_n see_v the_o deuce_n of_o heart_n proposition_n xi_o to_o draw_v a_o right_a line_n which_o touch_v a_o circle_n at_o a_o point_n propound_v the_o position_n let_v a_o b_o c._n be_v the_o circle_n give_v within_o the_o circumference_n of_o which_o be_v the_o point_n a._n propound_v the_o practice_n from_o the_o point_n or_o centre_n d._n draw_v the_o line_n d_o f._n by_o the_o point_n propound_v a._n at_o the_o point_n propound_v a._n and_o upon_o the_o line_n d_o f._n draw_v the_o perpendicular_a a_o h._n prolong_v towards_o i._o this_o line_n tangent_fw-la h_n i._n shall_v touch_v the_o circle_n at_o the_o point_n propound_v a._n the_o which_o be_v require_v by_o the_o proposition_n proposition_n xii_o a_o circle_n be_v give_v and_o a_o straight_a line_n that_o touch_v it_o to_o find_v the_o point_n where_o it_o touch_v the_o position_n let_v a_o b_o c._n be_v the_o circle_n touch_v by_o the_o line_n g_o h._n we_o must_v find_v the_o point_n where_o it_o touch_v the_o practice_n from_o the_o centre_n of_o the_o circle_n f._n let_v down_o the_o perpendicular_a f_o c._n upon_o the_o line_n touch_v d_o e._n the_o section_n c._n shall_v be_v the_o point_n of_o touch_v demand_v proposition_n xiii_o to_o describe_v a_o line_n spiral_n upon_o a_o straight_a line_n give_v the_o position_n let_v i_o l._n be_v the_o line_n upon_o which_o we_o will_v describe_v a_o line_n spiral_n the_o practice_n provide_v the_o half_a of_o the_o line_n i_o l._n into_o as_o many_o equal_a part_n 18._o page_n 18._o as_o you_o will_v describe_v the_o revolution_n example_n if_o you_o will_v divide_v it_o into_o four_n divide_v the_o half_a b_o i._n into_o four_o equal_a part_n b_o c_o e_o g_o i._n divide_v also_o b_o c._n into_o two_o equal_o at_o a._n 12._o page_n 12._o from_o the_o point_n a._n draw_v the_o demicircle_n b_o c._n d_o e._n f_o g._n h_n i._n from_o the_o point_n b._n draw_v the_o demicircle_n c_o d._n e_o f._n g_z h._n ay_o l._n and_o you_o shall_v have_v the_o spiral_n line_n require_v here_o end_v the_o deuce_n of_o heart_n proposition_n fourteen_o between_o the_o point_n give_v to_o find_v two_o other_o direct_o interpose_v the_o position_n let_v a_o and_o b._n be_v the_o point_v give_v between_o the_o which_o we_o must_v find_v two_o other_o point_n direct_o interpose_v by_o the_o mean_n whereof_o we_o may_v draw_v a_o straight_a line_n from_o the_o point_n a._n to_o the_o point_n b._n with_o a_o short_a ruler_n the_o practice_n from_o the_o point_n a_o and_o b._n make_v the_o section_n c_o and_z d._n from_o the_o point_n c_o and_z d._n make_v the_o section_n g_o and_z h._n the_o point_n g_o and_z h._n shall_v be_v the_o demand_v by_o the_o mean_n of_o which_o one_o may_v draw_v three_o way_n a_o right_a line_n from_o the_o point_n a._n to_o the_o point_n b._n the_o which_o can_v not_o be_v do_v in_o one_o with_o a_o ruler_n which_o shall_v be_v short_a than_o the_o space_n between_o a_o and_o a._n the_o second_o part_n of_o the_o construction_n of_o plain_a figure_n proposition_n i._n to_o frame_v a_o triangle_n equilateral_a upon_o a_o right_a line_n give_v and_o bound_v the_o position_n let_v a_o b._n be_v the_o line_n give_v upon_o the_o which_o we_o must_v frame_v a_o triangle_n equilateral_a the_o practice_n from_o the_o end_n a._n and_o the_o interval_n a_o b._n describe_v the_o arch_n b_o d._n from_o the_o end_n b._n and_o the_o see_v the_o trey_n of_o hart_n interval_n b_o a._n describe_v the_o arch_n a_o e._n from_o the_o section_n c._n draw_v the_o line_n c_o a._n c_o b._n a_o b_o c_o shall_v be_v the_o triangle_n equilateral_a demand_v proposition_n ii_o to_o make_v a_o triangle_n of_o three_o straight_a line_n equal_a to_o three_o straight_a line_n give_v the_o position_n let_v a_o b_o c_o be_v the_o three_o line_n give_v we_o must_v make_v a_o triangle_n of_o three_o right_a line_n equal_a to_o they_o the_o practice_n draw_v the_o right_a line_n d_o e._n equal_a to_o the_o line_n a_o a._n from_o the_o point_n d._n and_o from_o the_o interval_n b_o b._n describe_v the_o arch_n g_o f._n from_o the_o point_n f._n and_o the_o interval_n c_o c._n describe_v the_o arch_n h_o i._n from_o the_o section_n o._n draw_v the_o line_n o_fw-fr e_o o_o d._n the_o triangle_n d_o e_o o._n shall_v be_v comprise_v of_o three_o right_a line_n equal_a to_o the_o three_o right_a line_n give_v a_o a_o b_o b_o c_o c._n here_n end_v the_o trey_n of_o hart_n see_v the_o four_o of_o heart_n proposition_n iii_o to_o frame_v a_o square_a upon_o one_o right_a line_n give_v and_o bound_v the_o position_n let_v a_o b_o be_v the_o right_a line_n give_v and_o bound_v upon_o the_o which_o we_o must_v frame_v a_o square_n the_o practice_n elevate_v the_o perpendicular_a a_o c._n from_o the_o point_n a._n 4._o page_n 4._o describe_v the_o arch_n b_o c._n from_o the_o point_n b_o and_z c._n and_o from_o the_o interval_n a_o b._n make_v the_o section_n d._n from_o the_o point_n d._n draw_v the_o line_n d_o c_o d_o b._n a_n b_o c_o d_o shall_v be_v the_o square_n demand_v frame_v upon_o the_o right_a line_n give_v a_o b._n proposition_n iu._n to_o frame_v a_o pentagone_fw-mi regular_n upon_o a_o right_a line_n give_v the_o position_n let_v a_o b_o be_v the_o line_n give_v upon_o the_o which_o we_o must_v frame_v a_o pentagone_fw-mi the_o practice_n from_o the_o end_n a._n and_o from_o the_o interval_n a_o b._n describe_v the_o arch_n b_o d_o f._n elevate_v the_o perpendicular_a a_o c._n divide_v the_o arch_n b_o c._n into_o five_o equal_a part_n i_o d_o l_o m._n draw_v the_o right_a line_n a_o d._n cut_v the_o base_n a_o b._n into_o two_o equal_o in_o o._n elevate_v the_o perpendicular_a o_o e._n from_o the_o section_n e._n and_o from_o the_o interval_n e_o a._n describe_v the_o circle_n a_o b_o f_o g_o h._n bring_v five_o time_n the_o line_n a_o b._n within_o the_o circumference_n of_o the_o circle_n and_o you_o shall_v have_v a_o pentagone_n regular_n equiangle_n equilateral_a a_o b_o f_o g_o h._n proposition_n v._n to_o frame_v a_o exagone_fw-mi regular_n upon_o a_o right_a line_n give_v the_o position_n let_v a_o b._n be_v the_o right_a line_n upon_o the_o which_o we_o must_v frame_v a_o exagone_fw-mi the_o practice_n from_o the_o end_n a_o and_o b._n and_o from_o the_o interval_n a_o b._n describe_v the_o arch_n a_o c_o b_o c._n from_o the_o section_n c._n describe_v the_o circle_n a_o b_o e_o f_o g._n bring_v six_o time_n the_o line_n give_v a_o b_o within_z the_o circumference_n and_o you_o shall_v have_v a_o exagone_fw-mi regular_n a_o b_o e_o f_o g_o d_o frame_v upon_o the_o line_n give_v a_o b._n here_n end_v the_o four_o of_o heart_n see_v the_o five_o of_o heart_n proposition_n vi._n upon_o a_o right_a line_n give_v to_o describe_v such_o a_o poligone_n as_o you_o will_v have_v from_o the_o exagone_fw-mi unto_o the_o dodecagone_fw-mi the_o position_n let_v a_o b_o be_v the_o line_n upon_o the_o which_o that_o must_v frame_v a_o exagone_fw-mi or_o a_o eptagone_fw-mi or_o a_o octogone_n etc._n etc._n the_o practice_n cut_a the_o line_n a_o b._n into_o two_o equal_o in_o o._n 10._o page_n 10._o elevate_v the_o perpendicular_a o_o i._n from_o the_o point_n b._n describe_v the_o arch_n a_o c._n divide_v a_o c._n into_o six_o equal_a part_n m_o n._n p_o q_o r._n this_o may_v make_v a_o eptagone_fw-mi if_o you_o will_n from_o the_o point_n c._n and_o the_o interval_n of_o one_o part_n c_o m._n describe_v the_o arch_n m_o d._n d_o shall_v be_v the_o centre_n to_o describe_v a_o circle_n capable_a of_o contain_v seven_o time_n the_o line_n a_o b._n if_o you_o will_v make_v a_o octogone_fw-mi from_o the_o point_n c._n and_o the_o interval_n of_o two_o part_n c_o n._n describe_v the_o arch_n n_o e._n e._n shall_v be_v the_o centre_n to_o describe_v a_o circle_n capable_a of_o contain_v eight_o time_n the_o line_n a_o b_o if_o you_o will_v make_v a_o enneagone_fw-mi you_o must_v take_v the_o three_o part_n c_o p._n and_o so_o likewise_o of_o other_o always_o augment_v it_o by_o one_o part_n here_n end_v the_o five_o of_o heart_n see_v the_o six_o of_o heart_n proposition_n vii_o upon_o a_o right_a line_n give_v to_o frame_v such_o a_o poligone_n as_o one_o will_v have_v from_o 12_o to_o 24_o side_n the_o position_n let_v a_o b._n be_v the_o line_n upon_o the_o which_o
one_o will_v frame_v any_o poligone_n the_o practice_n divide_v the_o arch_n a_o c._n into_o twelve_o equal_a part_n from_o the_o point_n c._n take_v as_o many_o part_n upon_o c_o a._n as_o there_o be_v need_n above_o the_o twelve_o to_o have_v as_o many_o part_n as_o one_o require_v of_o the_o side_n example_n if_o you_o make_v a_o figure_n of_o fifteen_o side_n from_o the_o point_n c._n and_o from_o the_o interval_n of_o three_o part_n c_o e._n describe_v the_o arch_n e_o o._n a_o c_o of_o 12_o c_z o_o of_o 3_o will_v make_v together_o 15._o from_o the_o point_n o_o and_o the_o interval_n o_o b._n describe_v the_o arch_n b_o e._n from_o the_o point_n f._n and_o the_o interval_n f_o a._n describe_v a_o circumference_n it_o shall_v contain_v fifteen_o time_n the_o line_n give_v a_o b._n and_o so_o of_o other_o poligone_n here_n end_v the_o six_o of_o heart_n see_v the_o seven_o of_o hart_n proposition_n viii_o upon_o a_o right_a line_n give_v to_o describe_v a_o portion_n of_o a_o circle_n capable_a of_o a_o angle_n equal_a to_o a_o angle_n give_v the_o position_n let_v a_o b._n be_v a_o line_n bound_v upon_o that_o which_o one_o will_v make_v a_o portion_n of_o a_o circle_n capable_a of_o contain_v a_o angle_n equal_a to_o the_o angle_n give_v c._n the_o practice_n make_v the_o angle_n b_o a_o d._n equal_a to_o the_o angle_n c._n 2._o page_n 16._o page_n 2._o page_n 2._o elevate_v upon_o a_o d._n the_o perpendicular_a a_o e._n divide_v the_o line_n a_o b._n into_o two_o equal_o in_o h._n elevate_v the_o perpendicular_a h_n e._n from_o the_o section_n f._n and_o from_o the_o interval_n f_o a._n describe_v the_o portion_n of_o the_o circle_n a_o e_o b._n all_o the_o angle_n that_o you_o shall_v make_v within_o the_o portion_n of_o a_o circle_n and_o upon_o the_o line_n give_v a_o b_o shall_v be_v all_o equal_a to_o the_o angle_n c._n proposition_n ix_o to_o find_v the_o centre_n of_o a_o circle_n give_v the_o position_n let_v a_o b_o c._n be_v a_o circle_n propound_v whereof_o we_o must_v find_v the_o centre_n the_o practice_n draw_v at_o discretion_n the_o right_a line_n a_o b._n bound_v itself_o at_o the_o circumference_n b_o c._n cut_v this_o right_a line_n a_o b._n into_o two_o by_o the_o line_n d_o c_o cut_v also_o this_o right_a line_n c_o d._n into_o two_o equal_o in_o f._n the_o point_n f._n shall_v be_v the_o centre_n demand_v of_o the_o circle_n a_o b_o c._n here_n end_v the_o seven_o of_o hart_n see_v the_o eight_o of_o heart_n proposition_n x._o to_o finish_v a_o circumference_n begin_v whereof_o the_o centre_n be_v lose_v the_o position_n let_v a_o b_o c._n be_v the_o part_n of_o the_o circumference_n give_v we_o must_v find_v the_o centre_n that_o we_o may_v finish_v it_o the_o practice_n place_n at_o discretion_n the_o three_o point_n a_o b_o c._n within_o the_o circumference_n begin_v from_o the_o point_n a_o and_o b._n make_v the_o section_n e_o and_z f._n draw_v the_o right_a line_n e_o f._n from_o the_o point_n b_o and_z c._n make_v the_o section_n g_o and_n h._n draw_v the_o right_a line_n g_o h._n from_o the_o intersection_n and_o centre_n i._o and_o from_o the_o interval_n i_o a._n finish_v the_o circumference_n begin_v proposition_n xi_o to_o describe_v a_o circumference_n by_o three_o point_n give_v the_o position_n let_v a_o b_o c._n be_v the_o three_o point_n by_o the_o which_o one_o will_v pass_v a_o circumference_n the_o practice_n from_o the_o point_n give_v a_o b_o c._n describe_v the_o three_o circle_n d_o e_o h._n d_o e_o f._n f_o g_o l._n of_o the_o same_o interval_n divide_v each_o other_o in_o the_o point_n d_o and_o e_o f_o and_o g._n draw_v the_o right_a line_n d_o e_z f_o g._n until_o that_o they_o meet_v each_o other_o in_o i._o from_o the_o point_n i._o and_o the_o interval_n i_o a._n describe_v the_o circumference_n require_v this_o practice_n be_v like_o the_o former_a here_n end_v the_o eight_o of_o heart_n see_v the_o nine_o of_o heart_n proposition_n xii_o to_o describe_v a_o oval_a upon_o a_o length_n give_v the_o position_n let_v a_o b._n be_v the_o length_n upon_o the_o which_o we_o must_v frame_v a_o oval_a the_o practice_n divide_v the_o length_n give_v a_o b._n into_o three_o part_n equal_a a_o c_o d_o b._n 18._o page_n 18._o from_o the_o point_n c_o and_z d._n and_o from_o the_o interval_n c_o a._n describe_v the_o circle_n a_o e_o f_o b_o e_o f._n from_o the_o section_n e_o and_z f._n and_o from_o the_o interval_n of_o the_o diameter_n e_o h._n describe_v the_o arch_n i_o h_n o_o p._n a_o i_o h_o b_o p_o o._n shall_v be_v the_o oval_a require_v proposition_n xiii_o to_o describe_v a_o oval_a upon_o two_o diameter_n give_v the_o position_n a_o b_o c_o d._n be_v the_o diameter_n upon_o the_o which_o we_o must_v frame_v a_o oval_a the_o practice_n make_v the_o rule_n m_o o._n equal_a to_o the_o great_a half_a diameter_n a_o e._n upon_o the_o which_o you_o shall_v make_v the_o length_n m_o n._n equal_a to_o the_o half_a diameter_n c_o e._n this_o rule_n be_v so_o order_v place_v it_o so_o upon_o the_o demand_n a_o b._n c_o d._n that_o the_o point_n n._n slide_v upon_o the_o line_n a_o b._n the_o end_n o._n may_v never_o leave_v the_o line_n c_o d._n running_n on_o so_o the_o say_a rule_n m_o o._n describe_v the_o oval_a by_o the_o end_n m._n proposition_n fourteen_o to_o find_v the_o centre_n and_o the_o two_o diameter_n of_o a_o oval_a the_o position_n let_v a_o b_o c_o d._n be_v the_o oval_a propound_v whereof_o we_o must_v find_v the_o centre_n and_o the_o diameter_n the_o practice_n within_o the_o oval_n propound_v a_o b_o c_o d._n draw_v at_o discretion_n 10._o pgae_fw-la 10._o the_o line_n parallel_n a_o n._n h_n i._n cut_v these_o line_n a_o n._n h_n i._n into_o two_o equal_o in_o l_o and_z m._n draw_v the_o line_n p_o l_o m_o o._n cut_v it_o into_o two_o equal_o in_o e._n and_o the_o point_n e._n shall_v be_v already_o the_o centre_n from_o the_o point_n e._n describe_v at_o discretion_n the_o circle_n f_o h_o q._n cut_v the_o oval_a in_o f_o and_o g._n from_o the_o section_n f_o and_o g._n draw_v the_o right_a line_n f_o g._n cut_v it_o into_o two_o equal_o in_o r._n draw_v the_o great_a diameter_n b_o d._n by_o the_o point_n e_o r._n from_o the_o centre_n e._n 10._o page_n 10._o draw_v the_o less_o diameter_n a_o e_o c._n parallel_n to_o the_o line_n f_o g._n this_o be_v it_o which_o be_v propound_v here_o end_v the_o nine_o of_o heart_n see_v the_o ten_o of_o heart_n proposition_n xv._o to_o frame_v a_o figure_n rectilinear_n upon_o a_o right_a line_n bound_v like_a unto_o a_o figure_n rectilinear_n propound_v let_v a_o b._n be_v the_o line_n upon_o the_o which_o we_o must_v frame_v a_o figure_n like_a to_o the_o figure_n c_o d_o e_o f._n the_o practice_n draw_v the_o diagonal_a c_o e._n make_v the_o angle_n b_o a_o g._n equal_a to_o the_o angle_n f_o c_o e._n make_v the_o angle_n a_o b_o g._n equal_a to_o the_o angle_n c_o f_o e._n the_o triangle_n a_o b_o g._n shall_v be_v like_a to_o the_o triangle_n c_o f_o e._n the_o same_o make_v the_o triangle_n a_o g_o h._n like_v to_o the_o triangle_n c_o e_o d._n the_o whole_a figure_n a_o b_o g_o h._n shall_v be_v like_a to_o the_o whole_a figure_n c_o d_o e_o f._n proposition_n xvi_o upon_o a_o right_a line_n propound_v to_o frame_v two_o rectangle_v according_a to_o one_o reason_n give_v a_o b_o be_v the_o line_n upon_o which_o we_o must_v frame_v two_o rectangle_v which_o may_v be_v between_o they_o according_a to_o the_o reason_n of_o c_o to_z d._n the_o practice_n cut_a the_o line_n a_o b._n at_o the_o point_n e_o according_o to_z the_o reason_n or_o proportion_n of_o c_o to_z d._n make_v the_o square_a a_o b_o h_o f._n draw_v the_o line_n e_o i._n parallel_n to_o the_o line_n a_o f._n b_o e_o i_o h_o a_o e_o i_o f_o shall_v be_v the_o rectangle_n require_v the_o rectangle_n a_o i._o be_v to_o the_o rectangle_n e._n h._n as_o the_o line_n d._n be_v to_o the_o line_n c._n see_v the_o ten_o of_o heart_n proposition_n i._n within_o a_o circle_n give_v to_o inscribe_v a_o triangle_n equilateral_a a_o exagone_fw-mi and_o a_o dodecagone_fw-mi let_v a_o b_o g._n be_v the_o circle_n within_o which_o we_o must_v inscribe_v a_o triangle_n equilateral_a etc._n etc._n the_o practice_n of_o the_o triangle_n equilateral_a from_o the_o point_n as_o a._n and_o from_o the_o interval_n of_o the_o half_a diameter_n a_o b._n describe_v the_o arch_n c_o b_o d._n draw_v the_o right_a line_n d_o c._n bring_v this_o interval_n c_o d._n from_o the_o
point_n c._n to_o the_o point_n f._n draw_v the_o line_n f_o c_o f_o d._n c_o d_o f_o shall_v be_v the_o triangle_n require_v of_o the_o exagone_fw-mi bring_v six_o time_n the_o half_a diameter_n a_o b._n within_o the_o circumference_n give_v of_o the_o dodecagone_fw-mi cut_a the_o arch_n of_o the_o exagone_n a_o c._n into_o two_o equal_o in_o o._n a_o o_o shall_v be_v the_o side_n of_o the_o dodecagone_fw-mi here_o end_v the_o ten_o of_o heart_n see_v the_o king_n of_o heart_n proposition_n ii_o within_o a_o circle_n give_v to_o inscribe_v a_o square_a and_o a_o octogone_n let_v a_o b_o c_o d_o be_v the_o circle_n within_o the_o which_o one_o will_v inscribe_v a_o square_n and_o a_o octogone_n the_o practice_n of_o the_o square_n draw_v the_o two_o diameter_n a_o b_o c_o d._n divide_v each_o other_o at_o right_a angle_n that_o be_v to_o say_v draw_v the_o right_a line_n c_o d._n by_o the_o centre_n of_o the_o circle_n o._n from_o the_o point_n or_o end_n c_o and_z d._n make_v the_o section_n i_o and_o l._n draw_v the_o right_a line_n i_o l._n pass_o also_o by_o the_o centre_n o._n the_o line_n or_o diameter_n a_o b_o c_o d._n shall_v divide_v themselves_o at_o right_a angle_n be_v the_o line_n a_o c_o a_o d_o b_o c_o b_o d._n and_o a_o c_o b_o d_o shall_v be_v the_o square_n require_v of_o the_o octogone_n subdivide_v every_o four_o of_o the_o circle_n you_o shall_v make_v the_o octogone_fw-mi proposition_n iii_o within_o a_o circle_n give_v to_o inscribe_v a_o pentagone_fw-mi and_o a_o decagone_fw-mi let_v a_o b_o c_o d_o be_v the_o circle_n propound_v the_o practice_n of_o the_o pentagone_fw-mi draw_v the_o two_o diameter_n a_o b_o c_o d._n divide_v themselves_o at_o right_a angle_n in_o e._n divide_v the_o half_a diameter_n c_o e._n into_o two_o equal_o in_o f._n from_o the_o point_n f._n and_o from_o the_o interval_n f_o a._n describe_v the_o arch_n a_o g._n from_o the_o point_n a._n and_o from_o the_o interval_n a_o g._n describe_v the_o arch_n g_o h._n the_o right_a line_n a_o h._n shall_v divide_v the_o circle_n into_o five_o equal_a part_n of_o the_o decagone_fw-mi subdivide_v every_o part_n of_o the_o circle_n into_o two_o equal_o here_o end_v the_o king_n of_o heart_n see_v the_o queen_n of_o heart_n proposition_n iu._n within_o a_o circle_n give_v to_o inscribe_v a_o eptagone_fw-mi let_v a_o b_o c_o be_v the_o circle_n propound_v within_o the_o which_o we_o must_v make_v a_o eptagone_fw-mi the_o practice_n draw_v the_o half_a diameter_n i_o a._n from_o the_o end_n a._n and_o from_o the_o interval_n a_o i._o describe_v the_o arch_n c_o i_o c._n draw_v the_o right_a line_n c_o c._n bear_v the_o half_a c_o o._n seven_o time_n within_o the_o circumference_n of_o the_o circle_n you_o shall_v have_v the_o eptagone_fw-mi require_v proposition_n v._n within_o a_o circle_n give_v to_o describe_v a_o enneagone_fw-mi let_v b_o c_o d_o be_v a_o circle_n propound_v within_o which_o one_o will_v inscribe_v a_o enneagone_fw-mi the_o practice_n draw_v the_o half_a diameter_n a_o b._n from_o the_o end_n b._n and_o from_o the_o interval_n b_o a._n describe_v the_o arch_n c_o a_o d._n draw_v the_o right_a line_n c_o d._n enlarge_v towards_o f._n make_v the_o line_n e_o f._n equal_a to_o the_o line_n a_o b._n from_o the_o point_n e._n describe_v the_o arch_n f_o g._n from_o the_o point_n f._n describe_v the_o arch_n e_o g._n draw_v the_o right_a line_n a_o g._n d_o h_n shall_v be_v the_o nine_o part_n of_o the_o circumference_n here_o end_v the_o queen_n of_o heart_n see_v the_o knave_n of_o heart_n proposition_n vi._n within_o a_o circle_n give_v to_o inscribe_v a_o endecagone_fw-mi let_v a_o e_o f_o be_v the_o circle_n give_v within_o which_o we_o must_v inscribe_v a_o endecagone_fw-mi the_o practice_n draw_v the_o half_a diameter_n a_o b._n divide_v the_o half_a diameter_n a_o b._n into_o two_o equal_o in_o c._n from_o the_o point_v a_o and_o c._n and_o from_o the_o interval_n a_o c._n describe_v the_o arch_n c_o d_o ay_o a_o d._n from_o the_o point_n i._o and_o from_o the_o interval_n i_o d._n describe_v the_o arch_n d_o o._n the_o interval_n c_o o._n shall_v be_v the_o side_n of_o the_o endecagone_fw-mi require_v very_o punctual_o proposition_n vii_o within_o the_o circle_n give_v to_o inscribe_v such_o a_o poligone_n as_o one_o will_v the_o practice_n draw_v the_o diameter_n a_o b._n describe_v the_o circle_n a_o b_o f._n capable_a to_o contain_v seven_o time_n a_o b._n as_o if_o you_o will_v frame_v upon_o a_o b._n a_o poligone_n like_o to_o that_o which_o you_o shall_v inscribe_v within_o the_o circle_n give_v a_o b_o c._n draw_v the_o diameter_n d_o e._n parallel_n to_o the_o diameter_n a_o b._n draw_v the_o right_a line_n d_o at_fw-fr g_o f_o b_o h._n by_o the_o end_n d_o a_n e_o b._n g_o h_n shall_v divide_v the_o circle_n give_v a_o b_o c._n into_o seven_o equal_a part_n and_o so_o of_o all_o other_o poligone_n here_o end_v the_o knave_n of_o heart_n see_v the_o ace_n of_o spade_n proposition_n viii_o from_o a_o circle_n give_v to_o take_v a_o portion_n capable_a of_o a_o angle_n equal_a to_o a_o angle_n rectilinear_n propound_v let_v a_o c_o e_o be_v the_o circle_n give_v from_o which_o we_o must_v take_v a_o portion_n capable_a to_o contain_v a_o angle_n equal_a to_o the_o angle_n d._n the_o practice_n draw_v the_o half_a diameter_n a_o b._n bring_v the_o line_n touch_v a_o f._n make_v the_o angle_n f_o a_o c._n equal_a to_o the_o angle_n give_v d._n all_o the_o angle_n which_o shall_v be_v frame_v upon_o the_o line_n a_o c._n and_o within_o the_o portion_n a_o e_o c._n shall_v be_v all_o equal_a to_o the_o angle_n give_v d._n so_o the_o portion_n a_o e_o c._n be_v the_o require_v proposition_n ix_o within_o a_o circle_n to_o inscribe_v a_o triangle_n of_o equal_a angle_n to_o a_o triangle_n give_v let_v a_o b_o c_o be_v the_o circle_n within_o the_o which_o we_o must_v inscribe_v a_o triangle_n like_o to_o the_o triangle_n d_o e_o f._n the_o practice_n bring_v the_o line_n touch_v g_z h._n from_o the_o point_n of_o the_o touch_v a._n make_v the_o angle_n h_o a_o c._n equal_a to_o the_o angle_n e._n make_v also_o the_o angle_n g._n a_o b._n equal_a to_o the_o angle_n d._n draw_v the_o line_n b_o c._n a_o b_o c_o be_v the_o triangle_n require_v like_a to_o the_o triangle_n give_fw-mi d_o e_o f._n proposition_n x._o to_o inscribe_v a_o circle_n within_o a_o triangle_n give_v let_v a_o b_o c_o be_v the_o triangle_n within_o the_o which_o we_o must_v inscribe_v a_o circle_n the_o practice_n divide_v the_o two_o angle_n b_o and_z c._n each_o into_o two_o equal_o by_o the_o right_a line_n b_o d_o c_o d._n from_o the_o section_n d._n bring_v down_o the_o perpendicular_a d_o f._n from_o the_o section_n or_o centre_n d._n and_o from_o the_o interval_n d_o f._n describe_v the_o circle_n demand_v e_o f_o g._n here_o end_v the_o ace_n of_o spade_n see_v the_o deux_fw-fr of_o spade_n proposition_n xi_o to_o inscribe_v a_o square_a within_o a_o triangle_n give_v let_v a_o b_o c_o be_v the_o triangle_n within_o the_o which_o we_o must_v inscribe_v a_o square_n the_o practice_n elevate_v the_o perpendicular_a a_o d._n at_o the_o end_n of_o the_o basis_n a_o b._n make_v this_o perpendicular_a a_o d._n equal_a to_o the_o basis_n a_o b._n from_o the_o angle_n c._n draw_v the_o line_n c_o e._n parallel_n to_o the_o line_n a_o d._n bring_v the_o oblique_a line_n d_o e._n from_o the_o section_n f._n draw_v the_o line_n f_o g._n parallel_n to_o the_o basis_n a_o b._n draw_v the_o line_n f_o h_n g_o i._n parallel_n to_o the_o line_n c_o e_o f_o g_o h_o i_o shall_v be_v the_o square_n require_v proposition_n xii_o to_o inscribe_v a_o pentagone_n regular_n within_o a_o triangle_n equilateral_a let_v a_o b_o c_o be_v the_o triangle_n within_o the_o which_o one_o will_v inscribe_v a_o pentagone_fw-mi the_o practice_n bring_v down_o the_o perpendicular_a a_o i._o from_o the_o centre_n a._n describe_v the_o arch_n b_o i_o m._n divide_v into_o five_o equal_a part_n the_o arch_n b_o i._n bring_v the_o six_o i_o m._n draw_v the_o line_n a_o m._n divide_v a_o m._n into_o two_o equal_o in_o l._n from_o the_o point_n a._n describe_v the_o arch_n l_o d._n draw_v the_o right_a line_n l_o d_o unto_o h._n make_v the_o part_n a_o g._n equal_a to_o the_o part_n b_o h._n draw_v the_o right_a line_n d_o g_o m_o c._n from_o the_o centre_n d._n and_o from_o the_o interval_n of_o the_o section_n n._n describe_v the_o arch_n n_o o._n from_o the_o point_v n_n and_o o._n describe_v the_o arch_n d_o q_n d_o p._n draw_v the_o line_n o_o p_o
e_o f_o g._n here_o end_v the_o six_o of_o spade_n see_v the_o seven_o of_o spade_n proposition_n x._o about_o a_o square_a to_o circumscribe_v a_o pentagone_fw-mi let_v a_o b_o c_o d_o be_v the_o square_n about_o the_o which_o we_o must_v circumscribe_v a_o pentagone_fw-mi prolong_v the_o side_n c_o b._n towards_o n._n divide_v the_o side_n a_o b._n into_o two_o equal_o in_o r._n elevate_v the_o perpendicular_a r_n v._o from_o the_o point_n b_o d_o c._n and_o from_o the_o same_o interval_n b_o r._n divide_v the_o arch_n r_o n_n s_o t_o s_o t._n divide_v the_o arch_n r_o n._n into_o five_o equal_a part_n r_o h_n g_o f_o e_o n._n make_v the_o angle_n r_o b_o v._n from_o the_o open_n of_o two_o part_n r_o g._n make_v the_o angel_n s_n c_o t_o s_o d_o t._n from_o the_o open_n of_o one_o part_n r_o h._n prolong_v the_o line_n v_o b_o c_o t_o in_o o._n make_v the_o line_n o_o q._n equal_a to_o the_o line_n o_fw-mi v._o draw_v the_o other_o side_n on_o the_o some_o manner_n and_o you_o shall_v have_v the_o demand_v proposition_n i._n to_o find_v a_o line_n which_o may_v be_v a_o mean_a proportional_a between_o two_o other_o let_v a_o and_o b_o be_v the_o line_n between_o the_o which_o we_o must_v find_v a_o three_o which_o may_v be_v proportional_a to_o they_o the_o practice_n draw_v a_o line_n undeterminate_v g_z h._n make_v c_o e._n equal_a to_o the_o line_n a._n make_v e_o d._n equal_a to_o the_o line_n b._n divide_v c_o d._n into_o two_o equal_o in_o i._o from_o the_o point_n i._o and_o from_o the_o interval_n i_o c._n describe_v the_o demy-circle_n c_o f_o d._n elevate_v the_o perpendicular_a e_z f._n this_o line_n e_o f._n shall_v be_v the_o mean_a proportional_a between_o a_o and_o b._n according_a as_o it_o be_v propound_v here_o end_v the_o seven_o of_o spade_n see_v the_o eight_o of_o spade_n proposition_n ii_o there_o be_v give_v the_o sum_n of_o the_o extreme_n and_o the_o mean_v proportional_a to_o discern_v the_o end_n and_o extreme_n let_v a_o b_o be_v the_o sum_n of_o the_o end_n that_o be_v to_o say_v two_o grandeur_n or_o greatness_n the_o one_o at_o the_o end_n of_o the_o other_o without_o distinction_n whereof_o the_o line_n c_o be_v the_o mean_a proportional_a and_o by_o the_o mean_n of_o which_o we_o must_v find_v the_o point_n where_o the_o extreme_n or_o end_n do_v join_v themselves_o the_o practice_n divide_v the_o sum_n or_o the_o line_n a_o b._n into_o two_o equal_o in_o g._n from_o the_o point_n g._n and_o from_o the_o interval_n g_o a._n describe_v the_o demy-circle_n a_o e_o b._n elevate_v the_o perpendicular_a b_o d._n equal_a to_o the_o mean_a c._n draw_v the_o line_n d_o e_o parallel_n to_o the_o line_n a_o b._n from_o the_o section_n e._n draw_v the_o line_n e_o f._n parallel_n to_o the_o line_n b_o d_o f_o shall_v be_v the_o point_n where_o the_o extreme_n join_v themselves_o and_o so_o c_z or_o his_o equal_a e_z f._n shall_v be_v the_o mean_a between_o the_o extreme_n a_o f_o and_o f_o b._n proposition_n iii_o there_o be_v give_v the_o mean_a of_o three_o proportional_n and_o the_o difference_n of_o the_o extreme_n or_o end_n to_o find_v the_o extreme_n let_v g_o h_n be_v the_o mean_a proportional_a and_o a_o b_o the_o difference_n of_o the_o extreme_n we_o must_v find_v the_o length_n of_o the_o extreme_n the_o practice_n elevate_v the_o perpendicular_a b_o c._n at_o the_o end_n of_o the_o difference_n a_o b._n and_o equal_a to_o the_o mean_a g_o h._n divide_v the_o difference_n a_o b._n into_o two_o equal_o in_o d._n prolong_v it_o towards_o e_o and_z f._n from_o the_o point_n d._n and_o from_o the_o interval_n d_o c._n describe_v the_o demy-circle_n e_o c_o f._n b_o e_o b_o f_o shall_v be_v the_o extreme_n demand_v here_o end_v the_o eight_o of_o spade_n see_v the_o nine_o of_o spade_n proposition_n iu._n from_o a_o right_a line_n give_v to_o cut_v off_o a_o part_n which_o may_v be_v a_o mean_a proportional_a between_o the_o rest_n and_o another_o right_a line_n propound_v let_v a_o a_o be_v the_o line_n from_o the_o which_o we_o must_v cut_v off_o a_o part_n which_o may_v be_v the_o mean_a proportional_a between_o the_o part_n that_o shall_v remain_v and_o the_o line_n propose_v b_o b._n the_o practice_n draw_v the_o line_n indeterminate_v c_o d._n divide_v the_o line_n d_o e_o e_o c._n equal_a to_o the_o line_n a_o a_o and_o b_o b._n describe_v the_o demy-circle_n c_o f_o d._n elevate_v the_o perpendicular_a e_z f._n divide_v the_o line_n c_o e._n into_o two_o equal_o in_o b._n from_o the_o point_n b._n and_o from_o the_o interval_n b_o f._n describe_v the_o arch_n f_o g._n divide_v the_o part_n demand_v a_o g._n equal_a to_o the_o part_n e_o g._n a_o h_n shall_v be_v the_o mean_a proportional_a between_o the_o rest_n h_o i._n and_o the_o other_o line_n propound_v b_o b._n proposition_n v._n there_o be_v give_v two_o right_a line_n to_o find_v a_o three_o proportional_a a_o b_o a_o c_o be_v the_o two_o right_a line_n give_v we_o must_v find_v a_o three_o which_o 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line_n e_o f._n draw_v the_o line_n g_o h._n parallel_n to_o the_o line_n e_o f._n f_o h_n shall_v be_v four_o proportional_a demand_v proposition_n vii_o between_o two_o right_a line_n give_v to_o find_v two_o mean_n proportional_a let_v i_o and_o h_n be_v the_o line_n propound_v between_o the_o which_o we_o must_v find_v two_o mean_v proportional_a the_o practice_n draw_v the_o line_n a_o b._n equal_a to_o the_o line_n h._n let_v down_o the_o perpendicular_n b_o c._n equal_a to_o the_o line_n i._o bring_v the_o line_n a_o c._n divide_v this_o line_n a_o c._n into_o two_o equal_o in_o f._n elevate_v the_o perpendicular_o a_o o_o c_o r._n from_o the_o point_n or_o centre_n f._n describe_v the_o arch_n d_o e._n in_o such_o manner_n that_o the_o cord_n d_o e._n touch_v the_o angle_n b._n a_o d_o c_o e_o shall_v be_v the_o mean_v proportional_a between_o the_o line_n give_v i_o and_o h._n here_o end_v the_o ten_o of_o spade_n proposition_n viii_o to_o divide_v two_o right_a line_n give_v each_o into_o two_o part_n so_o that_o the_o four_o segment_n may_v be_v proportional_a a_o b_o a_o c_o be_v the_o line_n propound_v to_o be_v the_o practice_n make_v the_o right_a angle_n b_o o_fw-fr c._n divide_v the_o line_n b_o o._n equal_a to_o the_o line_n a_o b._n divide_v the_o line_n o_o c._n equal_a to_o the_o line_n a_o c._n bring_v the_o subtendent_fw-la b_o c._n describe_v the_o demy-circle_n b_o d_o o._n from_o the_o section_n d._n bring_v the_o line_n d_o e._n parallel_n to_o the_o line_n c_o o._n the_o line_n d_o f._n parallel_n to_o the_o line_n e_o o._n a_o b_o shall_v be_v divide_v in_o e._n o_o c_o shall_v be_v divide_v in_o f._n so_o that_o b_o e_z shall_v be_v to_o e_z d_o as_o e_z d_o be_v to_z d_o f_o and_o e_o d_o to_o d_o f_o as_o d_o f_o be_v to_o f_o c._n here_o end_v the_o king_n of_o spade_n see_v the_o queen_n of_o spade_n proposition_n ix_o there_o be_v give_v the_o excess_n of_o the_o diagonal_a of_o a_o square_a about_o the_o side_n to_o find_v the_o greatness_n of_o the_o say_a side_n let_v a_o b_o be_v the_o excess_n of_o the_o diagonal_a of_o a_o square_a above_o its_o side_n whereof_o we_o must_v find_v the_o greatness_n the_o practice_n elevate_v the_o perpendicular_a b_o c._n equal_a to_o the_o excess_n b_o a._n draw_v the_o line_n a_o c._n prolong_v towards_o d._n from_o the_o point_n c._n and_o from_o the_o interval_n c_o b._n describe_v the_o arch_n b_o d._n a_o d_o shall_v be_v the_o side_n of_o the_o square_n
whereof_o a_o b_o be_v the_o excess_n of_o the_o diagonal_a a_o e_o above_o the_o say_a side_n a_o d._n proposition_n x._o to_o divide_v a_o right_a line_n terminate_v within_o the_o mean_a and_o extreme_a reason_n let_v a_o b_o be_v the_o line_n which_o we_o must_v divide_v in_o such_o manner_n as_o the_o rectangle_n compose_v of_o the_o whole_a line_n and_o of_o one_o of_o the_o two_o part_n may_v be_v equal_a to_o the_o square_n frame_v upon_o the_o other_o part_n the_o practice_n elevate_v the_o perpendicular_a a_o d._n prolong_v it_o towards_o d._n make_v a_o c._n equal_a to_o the_o half_a of_o a_o b._n from_o the_o point_n c._n and_o from_o the_o interval_n c_o b._n describe_v the_o arch_n b_o d._n from_o the_o point_n a._n and_o from_o the_o interval_n a_o d._n describe_v the_o arch_n e._n the_o line_n a_o b._n shall_v be_v divide_v in_o e._n according_a to_o the_o proposition_n for_o if_o you_o make_v the_o rectangle_n a_o h_n of_o the_o whole_a a_o b_o and_o of_o the_o part_n b_o e_o it_o shall_v be_v equal_a to_o the_o square_a a_o f._n frame_v upon_o the_o other_o part_n a_o e._n here_o end_v the_o queen_n of_o spade_n proposition_n xi_o to_o divide_v a_o right_a line_n terminate_v according_a to_o the_o reason_n give_v let_v a_o b_o be_v the_o line_n propound_v to_o be_v divide_v according_a to_o the_o reason_n c.d.e.f._n the_o practice_n from_o the_o point_n or_o end_n a._n draw_v at_o discretion_n the_o line_n a_o g._n make_v a_o h._n equal_a to_o the_o line_n or_o reason_n c._n make_v h_n i._n equal_a to_o the_o line_n d._n make_v i_o l._n equal_a to_o the_o line_n e._n make_v l_o m._n equal_a to_o the_o line_n f._n draw_v the_o line_n b_o m._n bring_v the_o line_n l_o n_n i_o o_o h_o p._n parallel_n to_o the_o line_n b_o m._n the_o line_n a_o b_o shall_v be_v divide_v in_o the_o point_n p_o o_fw-fr n_n according_a as_o it_o be_v demand_v here_o end_v the_o knave_n of_o spade_n mechanic_n power_n or_o a_o manuscript_n of_o monsi_fw-la des-cartes_n 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foot_n for_o to_o raise_v one_o such_o to_o the_o height_n of_o one_o foot_n only_o this_o same_o aught_o to_o weight_n 200_o l._n for_o it_o be_v the_o same_o thing_n to_o raise_v 100_o l._n to_o the_o height_n of_o one_o foot_n and_o again_o yet_o another_o 100_o l._n to_o the_o height_n of_o one_o foot_n as_o to_o raise_v one_o of_o 200_o l._n to_o the_o height_n of_o one_o foot_n and_o the_o same_o also_o as_o to_o raise_v 100_o l._n to_o the_o height_n of_o two_o foot_n now_o the_o engine_n which_o serve_v to_o make_v this_o application_n of_o a_o force_n which_o act_v at_o a_o great_a space_n upon_o a_o weight_n which_o it_o cause_v to_o be_v raise_v by_o a_o lesser_a be_v the_o pulley_n the_o incline_v plane_n the_o wedge_n the_o capsten_n or_o wheel_n the_o screw_n the_o lever_n and_o some_o other_o for_o if_o we_o will_v not_o apply_v or_o compare_v they_o one_o to_o another_o we_o can_v well_o number_v more_o and_o if_o we_o will_v apply_v they_o we_o need_v not_o instance_n 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ponperosity_n of_o the_o chord_n or_o pulley_n and_o the_o difficulty_n that_o we_o meet_v with_o in_o make_v the_o chord_n to_o slip_v and_o in_o bear_v it_o but_o this_o be_v very_o small_a in_o comparison_n of_o that_o which_o raise_v it_o and_o can_v be_v estimate_v save_v within_o a_o small_a matter_n moreover_o its_o necessary_a to_o observe_v that_o it_o be_v nothing_o but_o the_o redouble_a of_o the_o chord_n and_o not_o the_o pulley_n that_o cause_v this_o force_n for_o if_o we_o fasten_v yet_o another_o pulley_n towards_o a_o about_o which_o we_o pass_v the_o chord_n a_o b_o c_o h_o there_o will_v be_v require_v no_o less_o force_n to_o draw_v h_n towards_o k_o and_o so_o to_o lift_v up_o the_o weight_n e_o than_o there_o be_v before_o to_o draw_v c_o towards_o g._n but_o if_o to_o these_o two_o pulley_n we_o add_v yet_o another_o towards_o d_o to_o which_o we_o fasten_v the_o weight_n and_o in_o which_o we_o make_v the_o chord_n to_o run_v or_o slip_v just_a as_o we_o do_v in_o the_o first_o than_o we_o shall_v need_v no_o more_o force_n to_o lift_v up_o this_o weight_n of_o 200_o l._n than_o to_o lift_v up_o 50_o l._n without_o the_o pulley_n because_o that_o in_o draw_v four_o foot_n of_o chord_n we_o lift_v it_o up_o but_o one_o foot_n and_o so_o in_o multiply_v of_o the_o pulley_n one_o may_v raise_v the_o great_a weight_n with_o the_o least_o force_n it_o be_v requisite_a also_o to_o observe_v that_o a_o little_a more_o force_n be_v always_o necessary_a for_o the_o raise_n of_o a_o weight_n than_o for_o the_o sustain_n of_o it_o which_o be_v the_o reason_n why_o i_o have_v speak_v here_o distinct_o of_o the_o one_o and_o the_o other_o the_o incline_v plane_n if_o not_o have_v more_o force_n than_o suffice_v to_o raise_v 100_o l._n one_o will_v nevertheless_o raise_v this_o body_n f_o that_o weigh_v 200_o l._n to_o the_o height_n of_o the_o line_n b_o a_o there_o need_v no_o more_o but_o to_o draw_v or_o roll_v it_o along_o the_o incline_v plane_n c_o a_o which_o i_o suppose_v to_o be_v twice_o as_o long_o as_o the_o line_n a_o b_o for_o by_o this_o mean_n for_o to_o make_v it_o arrive_v at_o the_o point_n a_o we_o must_v there_o employ_v the_o force_n that_o be_v necessary_a for_o the_o raise_n 100_o l._n twice_o as_o high_a and_o the_o more_o incline_v this_o plane_n shall_v be_v make_v so_o much_o the_o less_o force_n shall_v there_o need_v to_o raise_v the_o weight_n f._n but_o yet_o there_o be_v to_o be_v rebate_v from_o this_o
it_o be_v near_o to_o c._n of_o which_o the_o reason_n be_v that_o the_o weight_n do_v there_o mount_v less_o as_o it_o be_v easy_a to_o understand_v if_o have_v suppose_v that_o the_o line_n c_o o_fw-fr h_n be_v parallel_v to_o the_o horizon_n and_o that_o a_o o_o f_o cut_v it_o at_o right_a angle_n we_o take_v the_o point_n g_o equidistant_n from_o the_o point_n f_o and_o h_n and_o the_o point_n b_o equidistant_n from_o a_o and_o c._n and_o that_o have_v draw_v g_z s_n perpendicular_a to_o f_o o_o we_o observe_v that_o the_o line_n f_o s_n which_o show_v how_o much_o the_o weight_n mount_v in_o the_o time_n that_o the_o force_n operate_v along_o the_o line_n a_o b_o be_v much_o lesser_a than_o the_o line_n s_o o_fw-fr which_o show_v how_o much_o it_o mount_v in_o the_o time_n that_o the_o force_n operate_v along_o the_o line_n b_o c._n and_o to_o measure_v exact_o what_o his_o force_n ought_v to_o be_v in_o each_o point_n of_o the_o curve_v line_n a_o b_o c_o d_o e_o it_o be_v 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be_v the_o line_n upon_o which_o the_o figure_n repose_v itself_o as_o i_o l._n side_n be_v those_o line_n which_o encompass_v a_o figure_n as_o i_o n._n n_n m._n the_o diagonal_a be_v a_o right_a line_n that_o traverse_v a_o figure_n and_o that_o which_o end_v at_o two_o angle_n opposite_a as_o a_o b._n the_o diameter_n be_v a_o right_a line_n which_o traverse_v or_o pass_v through_o a_o circular_a figure_n by_o its_o centre_n and_o which_o end_v at_o the_o circumference_n a_o spiral_n line_n be_v a_o crooked_a line_n which_o part_v from_o its_o centre_n and_o grow_v far_o off_o in_o proportion_n as_o it_o turn_v about_o as_o e_z e._n the_o cord_n or_o subtendant_n be_v a_o right_a line_n which_o be_v join_v to_o a_o arch_n or_o bow_n by_o its_o end_n as_o g_z h._n the_o arch_n or_o bow_n be_v a_o part_n of_o a_o circumference_n as_o g_z ay_o h._n a_o line_n tangent_fw-la be_v that_o which_o touch_v any_o figure_n without_o divide_v it_o and_o without_o be_v able_a to_o divide_v or_o 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more_o or_o less_o open_a it_o receive_v particular_a denomination_n as_o right_o sharp_a obtuse_a or_o blunt_a so_o that_o these_o term_n of_o rectilinear_n curvi-linear_a and_o mix_v be_v in_o respect_n of_o the_o quality_n of_o the_o line_n and_o these_o of_o right_n sharp_a and_o obtuse_a be_v in_o respect_n of_o the_o quantity_n of_o the_o space_n enclose_v within_o the_o say_a line_n it_o be_v a_o right_a angle_n when_o one_o of_o the_o line_n be_v perpendicular_a upon_o the_o other_o as_o e_z d_o f._n the_o angle_n be_v sharp_a when_o as_o it_o be_v less_o open_a than_o the_o right_a angle_n as_o e_z d_o g._n the_o angle_n be_v obtuse_a when_o it_o be_v more_o open_a than_o the_o right_a angle_n as_o f_o d_o g._n the_o letter_n d_o in_o the_o midst_n show_v the_o angle_n here_o end_v the_o trey_n of_o diamond_n see_v the_o four_o of_o diamond_n the_o definition_n of_o the_o superficies_n the_o superficies_n be_v that_o which_o have_v length_n and_o breadth_n without_o depth_n 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p_o q_o n_o q._n d_o o_o p_o q_o n_o shall_v be_v the_o pentagone_fw-mi require_v proposition_n xiii_o to_o inscribe_v a_o triangle_n equilateral_a within_o a_o square_n let_v a_o b_o c_o d_o be_v the_o square_a within_o the_o which_o we_o must_v make_v a_o triangle_n equilateral_a the_o practice_n draw_v the_o diagonal_n a_o c_o b_o d._n from_o the_o centre_n e._n and_o from_o the_o interval_n e_o a._n describe_v the_o circle_n a_o b_o c_o d._n from_o the_o point_n c._n and_o the_o interval_n c_o e._n describe_v the_o arch_n g_o e_o f._n draw_v the_o right_a line_n a_o f_o a_o g._n bring_v the_o right_a line_n h_n 1._o a_o h_o l_o shall_v be_v the_o triangle_n equilateral_a require_v here_o end_v the_o deux_fw-fr of_o spade_n see_v the_o trois_fw-fr of_o spade_n proposition_n fourteen_o to_o inscribe_v a_o triangle_n equilateral_a within_o a_o pentagone_n let_v a_o b_o c_o d_o e_o be_v the_o pentagone_fw-mi within_o the_o which_o we_o must_v inscribe_v a_o triangle_n equilateral_a 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the_o end_n o_o p._n elevate_v the_o perpendicular_o o_o m_n p_o i._n draw_v the_o line_n n_o m._n n_o m_o o_o p_o shall_v be_v the_o square_n require_v here_o end_v the_o trois_fw-fr of_o spade_n see_v the_o four_o of_o spade_n proposition_n i._n about_o a_o triangle_n give_v to_o circumscribe_v a_o circle_n let_v a_o b_o c_o be_v the_o triangle_n about_o the_o which_o one_o will_v circumscribe_v a_o circle_n the_o practice_n describe_v the_o circumference_n a_o b_o c._n by_o the_o three_o point_n a_o b_o c._n and_o you_o shall_v have_v the_o demand_v proposition_n ii_o about_o a_o square_a to_o circumscribe_v a_o circle_n let_v a_o b_o c_o d_o be_v the_o square_n about_o which_o we_o must_v circumscribe_v a_o circle_n the_o practice_n draw_v the_o two_o diagonal_n a_o b_o c_o d._n from_o the_o section_n or_o centre_n g._n and_o from_o the_o interval_n g_o a._n describe_v the_o circle_n demand_v a_o b_o c_o d._n proposition_n iii_o about_o a_o circle_n to_o circumscribe_v a_o triangle_n of_o equal_a angle_n to_o a_o triangle_n give_v let_v d_o e_o v_o be_v the_o circle_n about_o the_o which_o we_o must_v make_v a_o triangle_n which_o may_v be_v like_a to_o the_o triangle_n f_o g_o h._n the_o practice_n draw_v the_o diameter_n a_o b._n by_o the_o centre_n c._n make_v the_o angle_n a_o c_o e._n equal_a to_o the_o angle_n h._n make_v the_o angle_n b_o c_o d._n equal_a to_o the_o angle_n g._n prolong_v the_o line_n e_o c_o d_o c._n towards_o r_n and_o s._n draw_v the_o line_n tangent_fw-la n_n o._n parallel_n to_o the_o line_n d_o r._n draw_v the_o line_n tangent_fw-la o_o i._n parallel_n to_o the_o line_n e_o s._n draw_v also_o the_o line_n touchant_a n_n i._o parallel_n to_o the_o diameter_n a_o b._n i_o n_o o_o shall_v be_v the_o triangle_n demand_v like_v to_o the_o triangle_n f_o g_o h_n circumscribe_v about_o the_o circle_n d_o e_z v._o here_o end_v the_o four_o of_o spade_n see_v the_o five_o of_o spade_n proposition_n iu._n about_o a_o circle_n to_o circumscribe_v a_o square_n let_v a_o b_o c_o d_o be_v the_o circle_n about_o the_o which_o we_o must_v describe_v a_o square_n the_o practice_n draw_v the_o diameter_n a_o b_o c_o d._n divide_v themselves_o at_o right_a angle_n in_o o._n from_o the_o point_n a_o c_o b_o d_o and_o from_o the_o interval_n a_o o._n describe_v the_o demicircle_n h_o o_fw-fr g_o h_o o_o e_o e_o o_o f_o f_o o_o g_o draw_v the_o right_a line_n e_o f_o f_o g_o g_o h_n h_o e_o by_o the_o section_n e_o f_o g_o h._n e_o f_o g_o h_o shall_v be_v the_o square_n demand_v proposition_n v._n about_o a_o circle_n give_v to_o circumscribe_v a_o pentagone_fw-mi the_o practice_n let_v a_o b_o c_o d_o e_o be_v the_o circle_n give_v about_o the_o which_o one_o will_v describe_v a_o pentagone_fw-mi inscribe_v the_o pentagone_fw-mi a_fw-it b_o c_o d_o e._n from_o the_o centre_n f._n and_o by_o the_o midst_n of_o each_o of_o the_o side_n draw_v the_o line_n f_o o_fw-fr f_o p_o f_o q_o f_o r_o f_o s._n bring_v the_o line_n f_o a._n draw_v the_o line_n tangent_fw-la p_o q._n by_o the_o point_n a._n from_o the_o centre_n f._n and_o from_o the_o interval_n f._n p._n describe_v the_o circle_n o_o p_o q_o r_o s._n draw_v the_o side_n of_o the_o pentagone_fw-mi demand_v by_o the_o section_n o_o p_o q_o r_o s_o proposition_n vi._n about_o a_o poligone_n regular_n to_o circumscribe_v the_o same_o poligone_n let_v b_o c_o d_o e_o f_o g_o be_v the_o poligone_n give_v about_o the_o which_o we_o must_v circumscribe_v another_o poligone_n like_a the_o practice_n prolong_v two_o side_n as_o b_o g_o e_o f._n unto_o the_o point_n of_o the_o meeting_n h._n draw_v the_o line_n a_o h._n draw_v the_o line_n f_o i._n divide_v the_o angle_n g_o f_o h._n into_o two_o equal_o from_o the_o centre_n a._n and_o from_o the_o interval_n a_o i._o describe_v the_o circle_n i_o m_o o._n draw_v the_o ray_n a_o l_o a_o m_o a_o n_o a_o o_o by_o the_o midst_n of_o each_o side_n draw_v the_o side_n of_o the_o outward_a poligone_n demand_v by_o the_o section_n i_o l_o m_o n_o o_o p._n proposition_n vii_o about_o a_o triangle_n equilateral_a to_o circumframe_v a_o square_n let_v a_o b_o c_o be_v a_o triangle_n equilateral_a about_o the_o which_o we_o must_v circumscribe_v a_o square_n the_o practice_n divide_v the_o basis_n b_o c._n into_o two_o equal_o in_o e._n prolong_v this_o basis_n b_o c._n the_o one_o part_n and_o the_o other_o towards_o d_o and_z d._n make_v the_o line_n e_o d_o e_o d._n equal_a to_o the_o line_n e_o a._n from_o the_o point_n e._n and_o from_o the_o interval_n e_o c._n describe_v the_o demy-circle_n b_o f_o c._n draw_v the_o line_n a_o e_o f._n from_o the_o point_n f._n draw_v the_o line_n f_o c_o g_o f_o b_o g._n a_o g_o f_o g_o shall_v be_v the_o square_n demand_v here_o end_v the_o five_o of_o spade_n see_v the_o six_o of_o spade_n proposition_n viii_o about_o a_o triangle_n give_v equilateral_a to_o circumscribe_v a_o pentagone_fw-mi let_v a_o b_o c_o be_v the_o triangle_n give_v about_o the_o which_o we_o must_v describe_v a_o pentagone_fw-mi the_o practice_n from_o the_o point_n or_o angle_v a_o b_o c._n and_o with_o the_o same_o open_n of_o the_o compass_n describe_v at_o discretion_n the_o arch_n d_o e_o l_o p._n divide_v the_o arch_n d_o o._n into_o five_o equal_a part_n 1_o 2_o 3_o 4_o 5._o from_o the_o centre_n or_o section_n o._n and_o from_o the_o interval_n of_o 4_o part_n o_o n._n describe_v the_o arch_n n_o m_o e._n draw_v the_o right_a line_n a_o e_o f._n divide_v the_o arch_n m_o p._n equal_a to_o the_o arch_n e_o n._n draw_v the_o right_a line_n f_o p_o c_o g._n equal_a to_o the_o line_n f_o a._n make_v the_o arch_n d_o h._n equal_a to_o the_o arch_n d_o e._n draw_v the_o side_n a_o i_o i_o r._n equal_a to_o the_o side_n a_o f_o f_o g._n the_o side_n g_o r_n shall_v finish_v the_o pentagone_n demand_v proposition_n ix_o about_o a_o square_a to_o circumscribe_v a_o triangle_n of_o equal_a angle_n to_o a_o triangle_n give_v let_v d_o e_o f_o g_o be_v the_o square_n about_o the_o which_o we_o must_v circumscribe_v a_o triangle_n like_o to_o the_o triangle_n a_o b_o c._n the_o practice_n make_v the_o angle_n e_o f_o m._n equal_a to_o the_o angle_n a._n make_v the_o angle_n m_o e_o f._n equal_a to_o the_o angle_n b._n prolong_v the_o line_n m_o e_o m_o f_o d_o g_o towards_z i_o and_o h._n m_o i_o h_n shall_v be_v the_o triangle_n require_v like_a to_o the_o triangle_n a_o b_o c._n and_o circumscribe_v about_o the_o square_n give_v d_o
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