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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be_v cut_v into_o two_o equal_o the_o practice_n from_o the_o angle_n a._n draw_v at_o pleasure_n the_o arch_n d_o e._n from_o the_o point_n d_o and_z e._n make_v the_o section_n o._n draw_v the_o line_n a_o o._n this_o line_n a_o o._n shall_v divide_v the_o angle_n give_v b_o a_o c._n into_o two_o equal_o here_o end_v the_o knave_n of_o diamond_n see_v the_o ace_n of_o hart_n proposition_n viii_o at_o the_o end_n of_o a_o right_a line_n to_o make_v a_o angle_n rectilinear_n equal_a to_o a_o angle_n rectilinear_n propose_v the_o position_n let_v a._n be_v the_o end_n of_o the_o line_n a_o b._n to_o which_o we_o must_v make_v a_o angle_n equal_a to_o the_o angle_n rectilinear_n give_v c_o d_o g._n the_o practice_n from_o the_o angle_n d._n draw_v at_o pleasure_n the_o arch_n c_o g._n without_o change_v the_o open_n of_o the_o compass_n from_o the_o point_n or_o end_v a._n draw_v the_o arch_n h_o o._n make_v the_o arch_n h_o e._n equal_a to_o the_o arch_n c_o g._n draw_v the_o line_n a_o 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 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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 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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
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
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 the_o practice_n circumscribe_v the_o circle_n a_o b_o c_o d_o e._n from_o the_o point_n a._n and_o from_o the_o interval_n of_o the_o half_a diameter_n a_o f._n describe_v the_o arch_n f_o l._n divide_v this_o arch_n f_o l._n into_o two_o equal_o in_o n._n draw_v the_o line_n a_o n_n i._o from_o the_o point_n a._n and_o from_o the_o interval_n a_o i._o describe_v the_o arch_n i_o o_o h._n draw_v the_o line_n a_o h_n h_o 1._o a_o h_o i_o shall_v be_v the_o triangle_n demand_v proposition_n xv._o to_o inscribe_v a_o square_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 square_n the_o practice_n draw_v the_o right_a line_n b_o e._n let_v down_o the_o perpendicular_a e_o t._n at_o the_o end_n of_o b_o e._n make_v this_o perpendicular_a e_z t._n equal_a to_o the_o line_n b_o e._n draw_v the_o line_n e_o t._n from_o the_o section_n o._n bring_v the_o line_n o_o p._n parallel_n to_o the_o side_n c_o d._n at_o 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
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 may_v be_v proportional_a to_o they_o the_o practice_n make_v at_o discretion_n the_o angle_n d_o n_o e._n divide_v the_o part_n n_o h._n equal_a to_o the_o line_n a_o b._n divide_v the_o part_n n_o o._n equal_a to_o the_o line_n a_o c._n divide_v also_o h_n d._n equal_a to_o the_o line_n a_o c._n bring_v the_o line_n h_o o._n draw_v the_o line_n d_o e._n parallel_n to_o the_o line_n h_o o._n e_o o_o shall_v be_v three_o proportional_a demand_v here_o end_v the_o nine_o of_o spade_n see_v the_o ten_o of_o spade_n proposition_n vi._n to_o find_v a_o four_o proportional_a a_o b_o c_o be_v the_o three_o line_n propose_v we_o must_v find_v a_o four_o which_o may_v be_v to_o the_o three_o as_o the_o second_o to_o the_o first_o the_o practice_n make_v at_o discretion_n the_o angle_n g_o d_o h._n divide_v the_o part_n d_o e._n equal_a to_o the_o line_n a._n divide_v the_o part_n d_o f_o equal_a to_o the_o line_n b._n divide_v the_o part_n e_o g._n equal_a to_o the_o line_n c._n bring_v the_o 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
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 requisite_a to_o know_v that_o it_o operate_v there_o just_a in_o the_o manner_n as_o if_o it_o draw_v the_o weight_n along_o a_o plane_n circular_o incline_v and_o that_o the_o inclination_n of_o each_o of_o the_o point_n of_o this_o circular_a plane_n be_v to_o be_v measure_v by_o that_o of_o the_o right_a line_n that_o touch_v the_o circle_n in_o this_o point_n as_o for_o example_n when_o the_o force_n be_v at_o the_o point_n b_o for_o to_o find_v the_o proportion_n that_o it_o ought_v to_o have_v with_o the_o ponderosity_n of_o the_o weight_n which_o be_v at_o that_o time_n at_o the_o point_n g_o its_o necessary_a to_o draw_v the_o tangent_fw-la line_n g_z m_o and_o to_o account_v that_o the_o ponderosity_n of_o the_o weight_n be_v to_o the_o force_n which_o be_v require_v to_o draw_v it_o along_o this_o plane_n and_o consequent_o to_o raise_v it_o according_a to_o the_o circle_n f_o g_o h_n as_o the_o line_n g_o m_o be_v to_o s_o m._n again_o for_o as_o much_o as_o b_o o_fw-fr be_v triple_a of_o o_o g_o the_o force_n in_o b_o needs_o to_o be_v to_o the_o weight_n in_o g_o but_o as_o the_o â…“_n of_o the_o line_n s_o m_o be_v to_o the_o whole_a line_n g_o m._n in_o the_o self_n same_o manner_n when_o the_o force_n be_v at_o the_o point_n d_o to_o know_v how_o much_o the_o weight_n weigh_v at_o i_o its_o necessary_a to_o draw_v the_o contingent_a line_n betwixt_o i_o and_o p_o and_o the_o right_a line_n i_o n_n perpendicular_a to_o the_o horizon_n and_o from_o the_o point_n p_o take_v at_o discression_n in_o the_o line_n i_o p_o provide_v that_o it_o be_v below_o the_o point_n i_o you_o must_v draw_v p_o n_n parallel_n to_o the_o same_o horizon_n to_o the_o end_n you_o may_v have_v the_o proportion_n that_o be_v betwixt_o the_o line_n i_o p_o and_o the_o â…“_n of_o i_o n_o for_o that_o which_o be_v betwixt_o the_o ponderosity_n of_o the_o weight_n and_o the_o force_n that_o ought_v to_o be_v at_o the_o point_n d_o for_o the_o move_n of_o it_o and_o so_o of_o other_o where_o nevertheless_o you_o must_v except_v the_o point_n h_n at_o which_o the_o contingent_a line_n be_v perpendicular_a to_o the_o horizon_n the_o weight_n can_v be_v no_o other_o than_o triple_a the_o force_n which_o ought_v to_o be_v in_o c_o for_o the_o move_n of_o it_o in_o the_o point_v f_o and_o k_n at_o which_o the_o contingent_a line_n be_v parallel_n to_o the_o horizon_n itself_o the_o least_o force_n that_o one_o can_v assign_v be_v sufficient_a to_o move_v the_o weight_n moreover_o that_o you_o may_v be_v perfect_o exact_a you_o must_v observe_v that_o the_o line_n s_o m_n and_o p_o n_n ought_v to_o be_v part_n of_o a_o circle_n that_o have_v for_o their_o centre_n that_o of_o the_o earth_n and_o g_z m_n and_o i_o p_o part_n of_o spiral_n draw_v between_o two_o such_o circle_n and_o last_o that_o the_o right_a line_n s_o m_n and_o i_o n_o both_o tend_v towards_o the_o centre_n of_o the_o earth_n be_v not_o exact_o parallel_n and_o furthermore_o that_o the_o point_n h_n where_o i_o suppose_v the_o contingent_a line_n to_o be_v perpendicular_a to_o the_o horizon_n aught_o to_o be_v some_o small_a matter_n near_o to_o the_o point_n f_o than_o to_o k_o at_o the_o which_o f_o and_o k_n the_o contingent_a line_n be_v parallel_n to_o the_o say_a horizon_n this_o do_v we_o may_v easy_o resolve_v all_o the_o difficulty_n of_o the_o balance_n and_o show_v that_o then_o when_o its_o most_o exact_a and_o for_o instance_n suppose_v its_o centre_n at_o o_o by_o which_o it_o be_v sustain_v to_o be_v no_o more_o but_o a_o indivisible_a point_n like_v as_o i_o have_v suppose_v here_o for_o the_o leaver_n if_o the_o arm_n be_v decline_v one_o way_n or_o the_o other_o that_o which_o shall_v be_v the_o lowermost_a aught_o evermore_o to_o be_v adjudge_v the_o heavy_a so_o that_o the_o centre_n of_o gravity_n be_v not_o fix_v and_o immovable_a in_o each_o several_a body_n as_o the_o ancient_n have_v suppose_v which_o no_o person_n that_o i_o know_v of_o have_v 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suppose_v in_o my_o examination_n of_o the_o geostatick_a question_n and_o in_o regard_n that_o if_o it_o be_v not_o true_a all_o the_o rest_n that_o i_o have_v infer_v from_o it_o will_v be_v yet_o less_o true_a i_o will_v not_o only_o to_o day_n defer_v send_v you_o a_o more_o particular_a explication_n it_o be_v requisite_a above_o all_o thing_n to_o consider_v that_o i_o do_v speak_v of_o the_o force_n that_o serve_v to_o raise_v a_o weight_n to_o some_o height_n the_o which_o force_n have_v overmore_o two_o dimension_n and_o not_o of_o that_o which_o serve_v in_o each_o point_n to_o sustain_v it_o which_o have_v never_o more_o than_o one_o dimension_n insomuch_o that_o these_o two_o force_n differ_v as_o much_o the_o one_o from_o the_o other_o as_o a_o superficies_n differ_v from_o a_o line_n for_o the_o same_o force_n which_o a_o nail_n ought_v to_o have_v for_o the_o susstain_n of_o a_o weight_n of_o 100_o l._n one_o moment_n of_o time_n do_v also_o suffice_v for_o to_o 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prove_v touch_v these_o matter_n which_o not_o be_v absolute_o true_a they_o deceive_v the_o more_o the_o more_o true_a they_o seem_v to_o be_v the_o first_o thing_n wherewith_o a_o man_n may_v be_v pre-occupied_n in_o this_o business_n be_v that_o they_o many_o time_n confound_v the_o consideration_n of_o space_n with_o that_o of_o time_n or_o of_o the_o velocity_n so_o that_o for_o example_n in_o the_o leaver_n or_o which_o be_v the_o same_o the_o balance_n a_o b_o c_o d_o have_v suppose_v that_o the_o arm_n a_o b_o be_v double_a to_o b_o c_o and_o the_o weight_n in_o c_o double_v to_o the_o weight_n in_o a_o and_o also_o that_o
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 or_o figure_n of_o 6_o side_n e._n eptagone_n a_o figure_n of_o 7_o side_n f._n octogone_o a_o figure_n of_o 8_o side_n g._n enneagone_n a_o figure_n of_o 9_o side_n h._n decagone_n a_o figure_n of_o 10_o side_n i._o endecagone_n a_o figure_n of_o 11_o side_n l._n dodecagone_n a_o figure_n of_o 12_o side_n all_o these_o figure_n be_v call_v likewise_o by_o one_o general_a name_n poligones_n of_o triangle_n the_o triangle_n be_v also_o distinguish_v by_o the_o quality_n of_o their_o angle_n and_o by_o the_o disposition_n of_o their_o side_n as_o m._n a_o triangle_n rectangle_n which_o have_v a_o right_a angle_n n._n a_o triangle_n ambligone_n which_o have_v a_o obtuse_a angle_n o._n a_o triangle_n oxigone_n which_o have_v three_o angle_n sharp_a p._n a_o triangle_n equilateral_a which_o have_v its_o three_o side_n equal_a q._n a_o triangle_n iso_n feel_v which_o have_v two_o side_n equal_v only_o r._n a_o triangle_n scalene_n which_o have_v his_o three_o side_n unequal_a here_o end_v 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excess_n of_o a_o parallelo-gram_a upon_o another_o parallo-gram_a frame_v upon_o the_o same_o diagonal_a all_o other_o figure_n of_o more_o than_o four_o side_n be_v call_v by_o one_o general_a name_n multi-lateres_a or_o many-size_n here_o end_v the_o six_o of_o diamond_n see_v the_o seven_o of_o diamond_n of_o figure_n crooked_a or_o curvi-linear_a a._n a_o circle_n be_v a_o superficies_n or_o figure_n perfect_o round_a describe_v or_o draw_v from_o a_o centre_n from_o which_o the_o whole_a circumference_n be_v of_o equal_a distance_n a_o b_o c_o d._n the_o circumference_n be_v the_o extremity_n or_o outmost_a part_n of_o the_o circle_n otherwise_o it_o be_v the_o circular-line_n that_o enclose_v it_o b._n a_o oval_a be_v a_o crooked_a figure_n draw_v from_o many_o centre_n and_o which_o all_o the_o diameter_n divide_v into_o two_o equal_o c._n a_o eclipse_n be_v also_o a_o crooked_a figure_n draw_v from_o many_o centre_n but_o in_o shape_n of_o a_o egg_n within_o the_o which_o 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e._n figure_n excentrical_a be_v those_o which_o be_v contain_v in_o other_o of_o divers_a centre_n here_o end_v the_o seven_o of_o diamond_n see_v the_o eight_o of_o diamond_n of_o figure_n regular_a and_o irregular_a a._n a_o figure_n regular_n it_o that_o which_o have_v its_o opposite_a part_n like_o a_o equal_a b._n a_o irregular_a figure_n be_v that_o which_o be_v compose_v of_o angle_n and_o side_n unlike_a e_o e._n figure_n alike_a be_v those_o whereof_o all_o the_o part_n of_o one_o be_v proportionable_a to_o all_o the_o part_n of_o the_o other_o although_o the_o one_o be_v great_a or_o equal_a or_o lesser_a than_o the_o other_o f_o f._n figure_n equal_a be_v those_o which_o contain_v equal_o which_o may_v be_v like_a and_o unlike_a c._n the_o figure_n equi-angle_n which_o have_v all_o its_o angle_n equal_a e_o e._n one_o figure_n be_v equi-angle_n to_o another_o when_o as_o all_o the_o angle_n of_o the_o one_o be_v equal_a to_o all_o the_o angle_n of_o the_o other_o c_o d._n a_o figure_n equilateral_a which_o have_v all_o its_o side_n equal_a here_o end_v the_o eight_o of_o diamond_n see_v the_o nine_o of_o diamond_n the_o axiom_n i._o thing_n equal_a to_o the_o one_o and_o the_o same_o be_v equal_a among_o themselves_o the_o line_n a_o c_o a_o c._n which_o be_v equal_a to_o a_o b._n circle_n by_o the_o definition_n of_o a_o circle_n be_v also_o equal_a between_o themselves_o ii_o if_o to_o equal_a thing_n one_o shall_v add_v thing_n equal_a all_o will_v become_v equal_a the_o line_n a_o c_o a_o c_o be_v equal_a the_o add_v c_z d_o c_o d_o be_v equal_a all_o of_o they_o a_o d_o a_o d._n be_v also_o equal_a iii_o if_o from_o equal_a thing_n one_o take_v equal_a thing_n the_o remainder_n shall_v be_v equal_a if_o from_o equal_a line_n a_o d_o a_o d._n one_o take_v equal_a part_n a_o c_o a_o c_o the_o part_n remain_v c_o d_o c_o d._n shall_v be_v also_o equal_a iv_o if_o to_o thing_n unequal_a one_o add_v thing_n equal_a the_o whole_a will_v be_v unequal_a if_o two_o line_n unequal_a d_o e_z d_o e._n one_o do_v add_v the_o equal_a a_o d_o a_o d._n the_o whole_a a_o e_o a_o e._n shall_v be_v unequal_a v._o if_o from_o thing_n unequal_a one_o take_v thing_n equal_a ehe_fw-ge remainder_n shall_v be_v unequal_a if_o from_o the_o line_n unequal_a a_o e_o a_o e._n one_o take_v the_o equal_a a_o d_o a_o d._n the_o remainder_n d_o e_o d_o e._n shall_v be_v unequal_a vi_o the_o thing_n that_o be_v double_a to_o one_o another_o be_v equal_a between_o themselves_o the_o right_a line_n d_o d_o d_o d._n which_o be_v double_a to_o the_o line_n a_o d._n be_v equal_a between_o themselves_o vii_o the_o thing_n which_o be_v the_o half_a of_o the_o one_o and_o the_o same_o or_o of_o thing_n equal_a be_v equal_a among_o themselves_o the_o line_n a_o d_o a_o d_o which_o be_v half_a of_o the_o line_n d_o d_o d_o d_o be_v equal_a between_o themselves_o that_o which_o be_v say_v of_o line_n may_v be_v also_o say_v of_o number_n superficies_n and_o of_o body_n here_n end_v the_o nine_o of_o diamond_n see_v the_o ten_o of_o diamond_n the_o petition_n or_o demand_n serve_v for_o the_o order_v of_o the_o practice_n demand_v i._n draw_v a_o straight_a line_n from_o the_o point_n a._n to_o the_o point_n b._n the_o practice_n apply_v the_o rule_n to_o the_o point_n a_o and_o b._n draw_v the_o line_n demand_v a_o b._n by_o let_v the_o pencil_n or_o draught_n run_v close_o to_o the_o rule_n from_o the_o point_n a._n unto_o the_o point_n b._n demand_v ii_o enlarge_v infinite_o the_o line_n c_o d._n as_o the_o side_n of_o the_o end_n d._n the_o practice_n join_v the_o rule_n to_o the_o line_n c_o d._n continue_v infinite_o the_o say_a line_n c_o d
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 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