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water-table_n of_o the_o wall_n to_o the_o cordeau_n and_o from_o the_o bottom_n of_o the_o dike_a on_o the_o field-side_n level_v with_o the_o ground_n and_o all_o of_o free_a stone_n the_o midst_n of_o the_o bastion_n be_v fill_v up_o with_o earth_n to_o the_o cordeau_n and_o make_v slope_n to_o the_o firm_a ground_n of_o the_o town_n and_o their_o rampire_n and_o brest-work_n be_v raise_v fifteen_o yard_n above_o the_o cordeau_n chap._n vii_o of_o the_o superficies_n contain_v in_o the_o first_o plare_fw-la since_o the_o principle_n of_o geometry_n be_v the_o very_a groundwork_n of_o the_o art_n of_o fortification_n i_o judge_v it_o convenient_a to_o begin_v this_o abstract_n by_o the_o demonstration_n of_o such_o superficies_n as_o be_v most_o common_o use_v in_o the_o practice_n of_o this_o art_n superficie_fw-la of_o superficie_fw-la a_o superficie_fw-la be_v proper_o any_o kind_n of_o form_n demonstrate_v upon_o paper_n or_o upon_o the_o ground_n enclose_v with_o three_o line_n 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they_o be_v not_o useful_a in_o this_o art_n all_o angle_n derive_v from_o the_o conjunction_n of_o two_o line_n in_o one_o point_n angle_n of_o angle_n whether_o they_o be_v straight_o circulary_a diagonall_a or_o mix_v with_o any_o one_o of_o these_o as_o figure_n 4._o be_v call_v a_o plain_a angle_n because_o two_o strait_a level_n line_n from_o one_o and_o the_o same_o distance_n meet_v at_o one_o point_n and_o a_o strait_a perpendicular_a and_o a_o level_n line_n fall_v in_o the_o midst_n of_o a_o strait_a line_n produce_v a_o obtus_fw-la and_o a_o strait_a angle_n as_o in_o figure_n 7._o and_o two_o circulary_a line_n meet_v at_o one_o point_n make_v a_o circulary_a angle_n as_o in_o figure_n 5._o and_o a_o strait_a and_o a_o circulary_a line_n meet_v at_o one_o point_n produce_v a_o mix_a angle_n as_o in_o figure_n 6._o there_o be_v also_o divers_a other_o sort_n of_o angle_n some_o of_o which_o we_o shall_v have_v occasion_n to_o speak_v of_o in_o another_o place_n the_o eight_o figure_n demonstrate_v how_o to_o make_v a_o perpendicularie_a line_n cut_v a_o strait_a line_n give_v figure_n of_o the_o eight_o figure_n and_o how_o you_o may_v cut_v that_o perpendiculary_a line_n in_o three_o part_n by_o three_o demicircle_n to_o make_v up_o the_o line_n give_v three_o parallel_a line_n without_o alter_v the_o compass_n to_o any_o other_o distance_n but_o the_o first_o first_o let_v a_o strait_a line_n be_v give_v then_o take_v the_o just_a distance_n of_o that_o line_n with_o your_o compass_n and_o set_v one_o of_o the_o point_n of_o it_o upon_o the_o right_a hand_n end_n of_o the_o line_n and_o with_o the_o other_o point_n make_v a_o small_a demicircle_n above_o then_o remove_v the_o point_n of_o your_o compass_n upon_o the_o left_a hand_n side_n of_o the_o line_n give_v and_o with_o the_o other_o point_n make_v another_o small_a circle_n cut_v the_o first_o and_o where_o these_o two_o small_a circle_n cut_v one_o another_o set_v your_o rule_n and_o draw_v a_o strait_a line_n to_o the_o line_n give_v and_o it_o will_v cut_v the_o same_o with_o a_o strait_a perpendicularie_n line_n now_o to_o avoid_v all_o error_n that_o may_v proceed_v from_o the_o mis-placing_a of_o your_o rule_n upon_o the_o line_n give_v make_v but_o two_o other_o small_a demicircle_n below_o the_o line_n as_o you_o do_v above_o and_o where_o these_o cut_v one_o another_o put_v one_o end_n of_o your_o rule_n and_o the_o other_o upon_o the_o upper_a cut_n of_o the_o two_o small_a demicircle_n and_o draw_v a_o strait_a line_n and_o of_o necessity_n the_o perpendicularie_a line_n will_v be_v straight_o and_o without_o error_n now_o to_o cut_v that_o perpendicular_a line_n in_o two_o equal_a part_n more_o to_o make_v up_o the_o line_n give_v three_o parallel_a
line_n set_v your_o compass_n upon_o the_o just_a distance_n of_o the_o length_n of_o the_o perpendicularie_a line_n than_o set_v one_o of_o the_o point_n of_o the_o compass_n upon_o the_o low_a end_n of_o the_o perpendicularie_a line_n and_o make_v a_o demi_a circle_n upward_o then_o remove_v your_o compass_n upon_o the_o upper_a end_n of_o the_o perpendicularie_a line_n and_o make_v a_o demy-circle_n downward_o then_o remove_v the_o point_n of_o your_o compass_n upon_o the_o center-point_n where_o the_o perpendicularie_a line_n do_v cut_v at_o the_o first_o the_o line_n give_v and_o make_v the_o three_o demy-circle_n upward_o and_o this_o demy-circle_n will_v cut_v the_o other_o two_o in_o four_o place_n and_o where_o they_o cut_v set_v your_o rule_n and_o draw_v two_o strait_a line_n and_o these_o will_v make_v up_o the_o line_n give_v three_o perfect_a parallel_n line_n as_o it_o be_v clear_o demonstrate_v in_o figure_n 8._o the_o nine_o figure_n show_v how_o to_o draw_v as_o many_o parallel_a line_n as_o you_o please_v upon_o two_o line_n figure_n of_o the_o nine_o figure_n that_o be_v the_o upper_a and_o the_o low_a line_n in_o this_o figure_n first_o let_v two_o equal_a line_n be_v give_v one_o below_o &_o another_o above_o according_a to_o the_o length_n you_o desire_v to_o have_v your_o parallel_a line_n then_o divide_v these_o two_o line_n with_o your_o compass_n in_o as_o many_o part_n as_o you_o desire_v to_o have_v parallel_a line_n and_o at_o every_o division_n make_v a_o point_n and_o from_o every_o point_n make_v a_o demy-circle_n above_o the_o upper_a and_o the_o low_o give_v line_n than_o set_v your_o rule_n upon_o they_o one_o after_o another_o and_o draw_v as_o many_o line_n as_o there_o be_v demy-circle_n and_o these_o will_v all_o be_v perfect_a parallel_n line_n now_o if_o you_o will_v double_v these_o parallel_a line_n it_o be_v but_o to_o divide_v with_o your_o compass_n the_o just_a breadth_n of_o they_o in_o two_o part_n and_o at_o every_o division_n to_o make_v a_o point_n and_o to_o set_v your_o compass_n again_o upon_o its_o first_o distance_n and_o to_o cut_v your_o first_o demy-circle_n with_o it_o and_o to_o draw_v as_o many_o line_n and_o you_o shall_v have_v as_o many_o more_o perfect_a parallel_n line_n as_o you_o have_v before_o as_o it_o be_v clear_o demonstrate_v in_o figure_n 9_o figure_n of_o the_o ten_o figure_n the_o ten_o figure_n show_v how_o to_o make_v a_o square_a and_o a_o triangle_n upon_o a_o line_n give_v suppose_v the_o base_a of_o the_o square_n be_v the_o line_n give_v now_o to_o make_v a_o perfect_a square_n and_o four_o scalene_n triangle_n of_o this_o line_n you_o be_v to_o set_v your_o compass_n upon_o the_o just_a distance_n of_o the_o line_n and_o to_o set_v one_o point_n of_o it_o on_o the_o right_a hand_n end_n of_o it_o and_o to_o make_v a_o demy-circle_n and_o to_o remove_v your_o compass_n upon_o the_o left_a end_n and_o to_o make_v another_o demy-circle_n and_o where_o they_o cut_v one_o another_o set_v your_o rule_n and_o draw_v a_o perpendicularie_a line_n then_o turn_v your_o rule_n and_o draw_v a_o strait_a top_n line_n and_o join_v the_o base_a line_n with_o this_o top_n line_n by_o two_o strait_a line_n and_o you_o will_v have_v a_o perfect_a square_n and_o four_o perfect_a square_a scalene_n triangle_n as_o it_o be_v demonstrate_v in_o figure_n 10._o the_o eleven_o figure_n show_v how_o to_o make_v a_o paralograme_n out_o of_o two_o circle_n figure_n of_o the_o eleven_o figure_n first_o set_v your_o compass_n upon_o the_o demi_a breadth_n of_o the_o paralograme_n that_o you_o intend_v to_o make_v and_o then_o make_v a_o circle_n on_o the_o right_a hand_n then_o remove_v your_o compasse-point_n upon_o the_o extreme_a of_o the_o first_o circle_n on_o the_o left_a hand_n upon_o a_o strait_a line_n and_o with_o the_o other_o point_n mark_v the_o center-point_n of_o the_o other_o circle_n then_o turn_v your_o compass_n round_o and_o you_o will_v have_v two_o circle_n who_o extreme_n will_v touch_v one_o another_o than_o set_v your_o rule_n upon_o the_o four_o extreme_n of_o these_o two_o circle_n and_o draw_v four_o strait_a line_n and_o where_o these_o line_n cut_v one_o another_o there_o be_v the_o four_o angle_n of_o the_o paralograme_n as_o it_o be_v demonstrate_v in_o figure_n 11._o the_o twelve_o figure_n show_v how_o to_o make_v a_o perfect_a square_n by_o the_o outside_n of_o a_o circle_n figure_n of_o the_o twelfth_n figure_n make_v choice_n of_o your_o centre_n and_o set_v your_o compass_n upon_o the_o very_a diameter_n that_o you_o intend_v to_o have_v your_o square_n then_o make_v a_o circle_n and_o draw_v four_o strait_a line_n upon_o the_o extreme_n of_o it_o and_o it_o will_v produce_v a_o perfect_a square_n as_o be_v demonstrate_v in_o figure_n 12._o chap._n viii_o of_o the_o superficies_n contain_v in_o the_o second_o plate_n among_o all_o other_o superficies_n there_o be_v none_o so_o useful_a for_o the_o art_n of_o fortification_n as_o the_o triangle_n and_o yet_o mr._n ward_n in_o his_o animadversion_n of_o war_n make_v mention_v but_o of_o three_o sort_n yet_o there_o be_v six_o principal_a sort_n of_o triangle_n from_o which_o all_o other_o mix_v triangle_n be_v derive_v and_o these_o six_o sort_n may_v be_v all_o set_v out_o by_o the_o circle_n as_o it_o be_v demonstrate_v in_o this_o plate_n the_o equilateral_a triangle_n deserve_v the_o precedency_n same_o 1_o of_o the_o equilateral_a triangle_n and_o the_o best_a way_n to_o set_v out_o the_o same_o because_o it_o be_v the_o only_a triangle_n that_o can_v be_v fortify_v by_o bastion_n for_o all_o her_o side_n be_v equal_a the_o best_a way_n to_o set_v it_o out_o be_v to_o set_v your_o compass_n upon_o the_o just_a distance_n of_o the_o base_a of_o it_o and_o to_o make_v two_o point_n with_o your_o compass_n and_o to_o draw_v two_o demy-circle_n one_o from_o the_o right_a point_n and_o the_o other_o from_o the_o left_a hand_n point_n and_o where_o these_o circle_n cut_v one_o another_o make_v the_o three_o point_n and_o draw_v three_o strait_a line_n to_o these_o three_o point_n and_o it_o will_v produce_v a_o perfect_a equilateral_a triangle_n equal_a of_o all_o side_n some_o set_v it_o out_o by_o the_o circle_n circle_n how_o to_o set_v out_o the_o equilateral_a triangle_n by_o a_o circle_n divide_v the_o diameter_n of_o it_o in_o four_o equal_a part_n and_o draw_v a_o strait_a line_n upon_o the_o first_o quarter_n point_n as_o the_o base_a of_o it_o and_o two_o line_n more_o from_o the_o two_o end_n of_o the_o base_a to_o the_o top_n of_o the_o diametrical_a line_n of_o the_o circle_n but_o this_o way_n be_v not_o so_o perfect_a as_o the_o other_o see_v figure_n 13._o the_o isocele_n triangle_n be_v also_o very_o useful_a in_o the_o art_n of_o fortification_n triangle_n 2_o of_o the_o set_n out_o of_o the_o isocele_n triangle_n for_o all_o the_o regularie_a poligon_n of_o many_o angle_n after_o the_o sexagon_n be_v compose_v of_o isocele_n triangle_n but_o the_o sexagon_n be_v compose_v of_o six_o equilateral_a triangle_n that_o have_v all_o their_o side_n equal_a but_o the_o isocele_n triangle_n have_v always_o two_o side_n equal_a and_o long_o than_o her_o base_a it_o be_v set_v out_o after_o this_o manner_n out_o of_o the_o circle_n after_o your_o circle_n be_v make_v draw_v a_o strait_a perpendicularie_n line_n diametrical_a wise_n upon_o the_o centre_n of_o it_o to_o the_o two_o extreme_n of_o the_o circle_n then_o divide_v that_o last_o line_n into_o eight_o equal_a part_n and_o draw_v a_o strait_a line_n with_o your_o rule_n upon_o the_o first_o point_n of_o the_o eight_o part_n division_n from_o one_o extreme_a of_o the_o circle_n to_o another_o and_o that_o line_n represent_v the_o base_a then_o draw_v two_o strait_a line_n from_o the_o two_o end_n of_o that_o base_a to_o the_o top-point_n of_o the_o diametrical_a line_n of_o the_o circle_n and_o these_o three_o line_n will_v produce_v a_o perfect_a isocele_n triangle_n see_v figur_n 14._o the_o recteligne_a triangle_n be_v also_o useful_a in_o the_o art_n of_o fortification_n to_o set_v out_o half_a moon_n circle_n 3_o of_o the_o rectiligne_a triangle_n and_o how_o it_o be_v to_o be_v set_v out_o by_o the_o circle_n her_o base_a contrary_n to_o the_o isocele_n triangle_n be_v long_a than_o her_o two_o side_n it_o be_v set_v out_o after_o this_o manner_n after_o your_o circle_n be_v make_v draw_v a_o strait_a line_n from_o the_o two_o side_n extremity_n of_o the_o circle_n and_o cut_v with_o that_o line_n the_o center-point_n of_o the_o circle_n and_o this_o line_n represent_v the_o base_a then_o draw_v two_o other_o line_n from_o the_o two_o end_n of_o the_o base_a to_o the_o upper_a point_n of_o the_o circle_n and_o two_o other_o line_n from_o
the_o two_o end_n of_o the_o base_a to_o the_o low_a point_n of_o the_o circle_n and_o it_o will_v produce_v two_o perfect_a recteligne_a triangle_n that_o make_v also_o a_o perfect_a square_a rhomb_n the_o scalene_n triangle_n have_v all_o her_o side_n unequal_a it_o be_v very_o useful_a for_o measure_v of_o land_n circle_n 4_o of_o the_o scalene_n triangle_n and_o how_o it_o may_v be_v set_v out_o by_o the_o circle_n for_o two_o of_o they_o make_v a_o equilateral_a triangle_n and_o four_o of_o they_o a_o perfect_a long_a rhomb_n it_o be_v to_o be_v set_v out_o after_o this_o manner_n after_o your_o circle_n be_v make_v divide_v the_o side_n diametrical_a line_n of_o the_o circle_n into_o eight_o equal_a part_n then_o cut_v with_o a_o strait_a line_n the_o first_o eight_o point_n of_o that_o division_n on_o the_o left_a hand_n point_n and_o then_o divide_v the_o diametrical_a line_n of_o the_o circle_n that_o be_v draw_v perpendiculary-wise_a into_o four_o equal_a part_n and_o cut_v with_o another_o strait_a line_n the_o first_o equal_a point_n of_o that_o division_n towards_o the_o base_a or_o the_o low_a side_n of_o the_o circle_n then_o from_o the_o two_o side_n of_o these_o two_o line_n draw_v a_o strait_a diagonall_a line_n and_o these_o three_o line_n will_v produce_v a_o perfect_a scalene_n triangle_n see_v figure_n 21._o the_o acute_a angle_n circle_n 5_o of_o the_o oxigon_n triangle_n and_o how_o it_o may_v be_v set_v out_o by_o the_o circle_n or_o oxigon_n triangle_n be_v only_o useful_a for_o measure_v of_o land_n all_o her_o side_n be_v also_o unequal_a and_o all_o her_o angle_n acute_a you_o be_v to_o set_v it_o out_o after_o this_o manner_n after_o the_o circle_n be_v make_v draw_v a_o diametrical_a line_n perpendiculary-wise_a that_o may_v cut_v the_o point_n of_o the_o centre_n from_o the_o top_n of_o the_o circle_n to_o the_o bottom_n then_o from_o the_o low_a end_n of_o this_o line_n on_o the_o left_a hand_n make_v a_o point_n at_o the_o four_o part_n of_o the_o circumference_n of_o the_o circle_n and_o another_o point_n on_o the_o right_a hand_n at_o the_o three_o part_n of_o the_o circumference_n of_o the_o circle_n then_o draw_v a_o line_n from_o the_o left_a hand_n point_n to_o the_o right_a point_n and_o two_o other_o line_n from_o the_o right_a hand_n and_o the_o left_a hand_n point_n to_o the_o top_n point_n of_o the_o diametrical_a line_n and_o these_o three_o line_n will_v produce_v a_o perfect_a oxigon_n triangle_n see_v figure_n 22._o the_o obtus_fw-la triangle_n be_v useful_a in_o fortification_n circle_n 6_o of_o the_o obtus_fw-la triangle_n and_o how_o it_o may_v be_v set_v out_o by_o the_o circle_n and_o for_o measure_v of_o land_n two_o of_o her_o side_n be_v equal_a and_o contain_v but_o the_o two_o three_o part_n of_o her_o base_a her_o base_a be_v long_o than_o her_o side_n by_o a_o three_o part_n it_o be_v to_o be_v set_v out_o after_o this_o manner_n after_o the_o circle_n be_v make_v divide_v the_o diametrical_a line_n of_o it_o into_o three_o equal_a part_n and_o cut_v the_o first_o point_n of_o this_o division_n by_o a_o strait_a line_n and_o this_o line_n represent_v the_o base_a then_o draw_v two_o line_n from_o the_o two_o end_n of_o this_o base_a to_o the_o top_n of_o the_o diametrical_a line_n and_o these_o three_o line_n will_v produce_v a_o perfect_a obtus_fw-la triangle_n see_v figure_n 23._o now_o i_o come_v to_o the_o 24._o superficie_fw-la that_o contain_v a_o oval_a oval_a of_o the_o set_n out_o of_o the_o oval_a and_o in_o that_o oval_n two_o long_v rhombs_n a_o paralograme_n two_o equilateral_a triangle_n and_o four_o scalene_n triangle_n but_o because_o i_o have_v already_o speak_v of_o all_o these_o the_o oval_a and_o the_o two_o long_v rhombs_n except_v i_o will_v begin_v with_o the_o oval_n the_o most_o difficult_a of_o all_o other_o superficies_n to_o be_v right_o set_v out_o upon_o paper_n or_o in_o the_o field_n to_o set_v it_o out_o upon_o paper_n you_o be_v to_o set_v your_o compass_n upon_o the_o scale_n of_o the_o three_o part_n of_o the_o length_n you_o intend_v to_o have_v the_o oval_n and_o this_o three_o part_n be_v to_o be_v divide_v again_o into_o two_o equal_a part_n and_o your_o compass_n set_v upon_o that_o six_o part_n then_o make_v choice_n of_o your_o centre_n and_o cut_v the_o point_n of_o the_o centre_n with_o two_o strait_a line_n one_o of_o they_o perpendiculary-wise_a and_o the_o other_o diametricall-wise_a then_o set_v one_o of_o the_o point_n of_o your_o compass_n upon_o the_o centre_n point_n and_o with_o the_o other_o make_v a_o point_n upon_o the_o diametrical_a line_n on_o the_o right_a hand_n and_o another_o point_n upon_o the_o same_o line_n on_o the_o left_a hand_n than_o set_v the_o compass_n again_o upon_o the_o three_o part_n of_o the_o length_n of_o the_o oval_n and_o set_v one_o of_o the_o point_n of_o it_o upon_o the_o right_a hand_n point_n that_o represent_v the_o end_n of_o the_o three_o part_n 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line_n of_o 250_o yard_n with_o a_o loop_n fasten_v in_o the_o midst_n on_o the_o southeast_n stake_n and_o to_o strain_v it_o to_o the_o south-west_n stake_n if_o the_o stake_n be_v at_o the_o just_a distance_n of_o the_o line_n that_o side_n be_v right_o set_v out_o than_o you_o may_v knock_v in_o a_o stake_n where_o the_o loop_n be_v to_o represent_v the_o middle_n of_o that_o side_n where_o the_o gate_n and_o the_o drawbridge_n of_o the_o fort_n be_v to_o be_v place_v and_o so_o prove_v all_o the_o other_o three_o side_n with_o your_o line_n of_o 250_o yard_n knock_v stake_n in_o the_o midst_n on_o every_o side_n for_o they_o will_v stand_v you_o in_o stead_n when_o you_o come_v to_o divide_v the_o side_n into_o six_o part_n now_o if_o any_o of_o these_o side_n be_v amiss_o and_o agree_v not_o with_o your_o line_n than_o you_o be_v to_o amend_v they_o by_o your_o demi-circle_n and_o your_o line_n which_o be_v easy_o to_o be_v do_v and_o when_o the_o four_o angle_n and_o side_n be_v 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of_o a_o mile_n for_o her_o continent_n i_o will_v not_o stand_v upon_o because_o it_o be_v not_o of_o a_o convenient_a form_n to_o erect_v a_o new_a garrison_n but_o only_o fit_a to_o fortify_v some_o market_n town_n if_o need_n shall_v require_v of_o which_o form_n there_o be_v many_o in_o this_o kingdom_n most_o of_o they_o be_v as_o long_o again_o as_o they_o be_v broad_a yet_o let_v the_o reader_n observe_v that_o incase_o he_o be_v require_v to_o fortify_v a_o town_n that_o may_v be_v reduce_v to_o this_o form_n that_o have_v her_o broad_a side_n of_o 700_o yard_n and_o her_o end_n side_n of_o 350_o yard_n that_o he_o may_v bold_o undertake_v to_o fortify_v the_o same_o after_o this_o manner_n so_o he_o divide_v his_o end_n into_o five_o part_n and_o his_o broad_a side_n into_o ten_o part_n and_o do_v reduce_v the_o rest_n of_o the_o dimension_n of_o the_o face_n breast_n flank_n courtine_n and_o line_n of_o defence_n after_o the_o proportion_n of_o his_o side_n for_o then_o 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the_o same_o to_o the_o north_n and_o by_o west_n angle_n stake_n and_o strain_v it_o as_o before_o and_o knock_v in_o a_o stake_n then_o remove_v the_o line_n and_o do_v the_o like_a to_o the_o west_n and_o by_o north_n angle_n and_o as_o much_o to_o the_o west_n and_o by_o south_n stake_n strain_v the_o line_n as_o be_v say_v before_o and_o at_o every_o fide_fw-la of_o the_o line_n knock_v in_o a_o stake_n and_o these_o five_o stake_n will_v represent_v the_o five_o out-most_a angle_n or_o point_n of_o the_o five_o avant_n guard_n then_o remove_v that_o line_n and_o fasten_v a_o short_a line_n of_o yard_n of_o the_o short_a slank_n be_v sorcy_n yard_n 40_o yard_n to_o the_o second_o stake_n of_o the_o south_n side_n on_o the_o left_a hand_n south_n west_n and_o strain_v the_o same_o into_o the_o field_n perpendiculary_a wise_a and_o knock_v in_o a_o stake_n at_o the_o end_n of_o the_o line_n and_o this_o line_n will_v represent_v the_o short_a flank_n of_o the_o south-west_n avaunt_o 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thus_o set_v out_o and_o trace_v the_o five_o avant_n guard_n with_o their_o face_n long_a and_o short_a flank_n you_o be_v to_o set_v out_o all_o your_o ditch_n after_o the_o dimension_n describe_v in_o chap._n 17._o cast_v the_o earth_n of_o they_o inward_a to_o erect_v the_o brest-work_n and_o their_o five_o footstep_n of_o that_o height_n breadth_n and_o thickness_n there_o describe_v and_o when_o all_o your_o inward_a work_n and_o the_o great_a ditch_n be_v finish_v you_o be_v to_o set_v out_o the_o cunette_n or_o small_a ditch_n out_o of_o the_o great_a ditch_n and_o to_o erect_v with_o the_o earth_n that_o be_v take_v out_o of_o it_o the_o country_n escarpe_a brest-work_n distant_a from_o the_o
orim_n of_o the_o great_a ditch_n four_o yard_n observe_v the_o rule_n and_o dimension_n describe_v in_o chap._n 17._o and_o also_o five_o place_n of_o twelve_o foot_n apiece_o to_o be_v leave_v open_a in_o the_o brest-work_n of_o the_o country_n escarpe_a to_o set_v five_o turn-pike_n right_a against_o these_o five_o halfmoon_n here_o demonstrate_v which_o halfmoon_n you_o be_v to_o set_v out_o after_o your_o country_n escarpe_a brest-work_n be_v finish_v place_v the_o open_a side_n of_o they_o within_o 16_o foot_n distance_n of_o the_o slope_n of_o your_o country_n escarpe_a that_o there_o may_v be_v a_o clear_a passage_n for_o horse_n and_o foot_n to_o come_v to_o the_o halfmoon_n without_o spoil_v of_o the_o slope_n of_o the_o country_n escarpe_a that_o the_o french_a call_v glasis_n the_o five_o outguard_n of_o this_o figure_n be_v five_o halfmoon_n that_o have_v one_o side_n long_o than_o the_o other_o by_o twenty_o yard_n guard_n yard_n and_o this_o must_v of_o necessity_n be_v observe_v that_o the_o ling_n side_n of_o the_o halt-moone_n may_v cover_v the_o slank_n of_o the_o avant_n guard_n the_o long_a side_n be_v of_o 80_o yard_n and_o the_o short_a side_n of_o 60_o yard_n their_o ditch_n be_v to_o be_v nine_o foot_n broad_a and_o fix_v foot_n deep_a their_o brest-work_n be_v to_o be_v 9_o foot_n broad_a in_o the_o botteme_n with_o the_o foot-step_n and_o six_o foot_n thick_a at_o top_n have_v a_o foot_n slope_n and_o their_o height_n six_o foot_n from_o the_o ground_n and_o four_o foot_n and_o a_o half_a from_o the_o footstep_n the_o gate_n and_o drawbridge_n of_o this_o fort_n be_v to_o be_v in_o the_o middle_n of_o the_o south_n side_n that_o it_o may_v be_v defend_v with_o the_o south-west_n flank_n and_o cover_v with_o the_o south_n halfmoon_n mark_v with_o the_o letter_n a._n chap._n xxiv_o of_o the_o fortification_n of_o the_o demi-sexagon_a fortify_v by_o avant_n guard_n and_o outguard_n demonstrate_v in_o plate_n 18._o this_o sexagon_n be_v of_o a_o large_a extent_n for_o her_o side_n be_v of_o 800_o yard_n apiece_o and_o by_o consequence_n her_o cucumference_n be_v of_o 4800_o yard_n that_o be_v almost_o two_o english_a mile_n and_o a_o half_a and_o therefore_o fit_a to_o fortify_v a_o good_a corporation_n then_o for_o a_o citadel_n or_o fort_n the_o angle_n of_o her_o centre_n be_v of_o 60_o degree_n and_o the_o same_o direction_n that_o be_v give_v you_o for_o to_o set_v out_o the_o sexagon_n demonstrate_v in_o plate_n 12._o fortify_v by_o bastion_n with_o orillon_n will_v suffice_v with_o the_o observation_n contain_v in_o chap._n 19_o viz._n to_o knock_v in_o stake_n at_o every_o hundred_o yard_n when_o side_n or_o diametrical_a or_o diagonall_a line_n exceed_v 250_o yard_n the_o superficie_fw-la of_o this_o sexagon_n be_v then_o set_v out_o and_o full_o trace_v as_o it_o have_v be_v show_v in_o the_o foresay_a plate_n you_o be_v to_o divide_v every_o one_o of_o the_o base_n into_o three_o equal_a part_n that_o will_v be_v of_o 266_o yard_n and_o â…”_n apiece_o and_o to_o knock_v in_o a_o stake_n at_o every_o division_n than_o you_o be_v to_o fasten_v a_o line_n to_o the_o two_o inward_a stake_n one_o after_o another_o of_o 60_o guard_n 60_o 60_o yard_n be_v the_o just_a moiety_n of_o the_o breast_n of_o the_o avant_n guard_n yard_n and_o to_o strain_v the_o same_o first_o on_o the_o right_a hand_n and_o upon_o the_o side_n trace_v and_o the_o end_n of_o the_o line_n you_o be_v to_o knock_v in_o a_o stake_n than_o you_o be_v to_o wind_v the_o say_a line_n about_o on_o the_o left_a hand_n and_o to_o strain_v the_o same_o upon_o a_o strait_a line_n upon_o the_o sidetraced_a and_o at_o the_o end_n of_o the_o line_n you_o be_v to_o knock_v in_o a_o stake_n than_o you_o be_v to_o remove_v the_o line_n and_o to_o fasten_v the_o same_o to_o the_o other_o in_o ward_n stake_n of_o that_o side_n and_o to_o do_v as_o you_o do_v before_o and_o to_o knock_v in_o two_o stake_n at_o the_o end_n of_o the_o line_n and_o these_o four_o stake_n represent_v the_o breadth_n broad_a breadth_n that_o be_v of_o 120_o yard_n broad_a of_o the_o two_o breast_n of_o these_o two_o avant_n guard_n and_o the_o middle_a stake_n represent_v the_o centre_n of_o the_o avant_n guard_n then_o you_o be_v to_o set_v out_o the_o breast_n of_o the_o other_o ten_o avant_n guard_n of_o the_o other_o five_o side_n after_o the_o very_a same_o manner_n as_o you_o do_v these_o last_o and_o that_o be_v do_v you_o be_v to_o fasten_v a_o line_n of_o 120_o yard_n to_o every_o one_o of_o the_o exterior_a stake_n that_o represent_v the_o breadth_n of_o the_o breast_n of_o the_o avant_n gnard_n and_o to_o strain_v the_o same_o into_o the_o field_n upon_o a_o strait_a diagonall_a line_n derive_v from_o the_o centre_n stake_n of_o the_o sexagon_n that_o you_o be_v to_o guide_v by_o the_o sight_n of_o your_o demi-circle_n stand_v at_o the_o centre_n stake_n while_o the_o man_n that_o carry_v your_o line_n strain_v the_o foresay_a line_n into_o the_o field_n and_o at_o the_o end_n of_o it_o the_o man_n that_o carry_v your_o stake_n be_v to_o knock_v in_o a_o stake_n and_o this_o stake_n represent_v the_o utmost_a point_n of_o the_o long_a flank_n of_o your_o first_o avant_n guard_n and_o as_o you_o have_v set_v out_o this_o long_a strain_v long_a note_v and_o observe_v this_o course_n be_v to_o be_v take_v lecause_fw-mi the_o demi_fw-fr diagonal_a line_n of_o this_o sexagon_n from_o the_o centre_n stake_n to_o the_o utmost_a point_n of_o the_o long_a flank_n be_v of_o 680_o yard_n that_o be_v too_o long_a a_o line_n to_o be_v convenient_o strain_v flank_n set_v out_o the_o other_o eleven_o that_o remain_v and_o that_o be_v do_v you_o be_v to_o fasten_v a_o line_n of_o 60_o yard_n to_o the_o inward_a stake_n of_o that_o avant_n guard_n that_o represent_v the_o breadth_n of_o the_o breast_n of_o it_o and_o to_o strain_v the_o same_o into_o the_o field_n parallel-wise_a to_o that_o of_o the_o long_a flank_n and_o at_o the_o end_n of_o it_o you_o be_v to_o knock_v in_o a_o stake_n and_o this_o stake_n represent_v the_o utmost_a point_n of_o the_o short_a flank_n of_o that_o avaunt_o guard_n but_o observe_v apiece_o observe_v these_o long_a flank_n be_v of_o 120_o yard_n apiece_o to_o trace_v the_o long_a and_o short_a flank_n before_o you_o remove_v the_o line_n to_o another_o flake_n to_o avoid_v mistake_n and_o as_o you_o have_v set_v out_o this_o short_a apiece_o short_a these_o short_a flank_n be_v of_o 60_o yard_n apiece_o flank_n set_v out_o the_o eleven_o that_o remain_v that_o be_v do_v knock_v fix_v stake_n in_o the_o middle_n of_o the_o six_o side_n of_o this_o s_n xigon_n than_o you_o be_v to_o fasten_v a_o line_n to_o every_o one_o of_o they_o one_o oft_o another_o of_o 240_o yard_n and_o to_o strain_n and_o fasten_v the_o same_o first_o on_o the_o right_a hand_n stake_n that_o represent_v the_o utmost_a point_n of_o the_o long_a flank_n and_o this_o line_n will_v cut_v the_o end_n of_o the_o line_n of_o the_o short_a flank_n and_o set_v out_o the_o face_n of_o that_o avant_n guard_n and_o withal_o do_v represent_v the_o line_n of_o defence_n aprece_n defence_n the_o line_n of_o defence_n be_v take_v from_o the_o midst_n of_o the_o courtine_n and_o the_o extent_n of_o it_o in_o this_o figure_n be_v oâ—_n 240_o yard_n and_o the_o extent_n or_o all_o the_o courtine_n be_v oâ—_n 186_o yard_n â…“_n aprece_n but_o before_o you_o remove_v this_o line_n you_o be_v to_o trace_v the_o face_n of_o this_o avant_n guard_n that_o extend_v it_o selce_o from_o the_o stake_n of_o the_o short_a flank_n to_o the_o stake_n that_o represent_v the_o utmost_a point_n of_o the_o long_a flank_n then_o you_o be_v to_o wind_v the_o very_a same_o line_n about_o and_o to_o fasten_v it_o to_o the_o left_a hand_n inward_a stake_n that_o represent_v the_o utmost_a point_n of_o that_o long_a flank_n and_o to_o do_v as_o you_o do_v before_o to_o the_o other_o on_o the_o right_a hand_n and_o then_o the_o two_o avant_n guard_n of_o that_o fide_fw-la of_o the_o sexagon_n will_v be_v perfect_v and_o as_o you_o have_v do_v this_o side_n do_v the_o other_o five_o side_n after_o the_o very_a same_o manner_n when_o the_o twelve_o aven_n guard_n be_v thus_o set_v out_o and_o trace_v you_o be_v to_o set_v out_o all_o the_o ditch_n of_o it_o after_o the_o very_a same_o dimension_n set_v down_o in_o chap._n 17._o cast_v all_o the_o earth_n inward_a to_o crect_v the_o brest-work_n and_o their_o five_o footstep_n of_o that_o height_n breadth_n and_o thickness_n describe_v in_o the_o aforesaid_a chapter_n and_o when_o they_o be_v all_o finish_v you_o be_v then_o to_o set_v out_o the_o small_a ditch_n
brest-work_n broad_a and_o eighteen_o inch_n high_a the_o slope_n wall_n slope_n the_o slope_n of_o brest-work_n wall_n of_o the_o inside_n wall_n be_v to_o be_v but_o a_o foot_n in_o six_o foot_n because_o the_o footstep_n serve_v instead_o of_o buttress_n to_o that_o inward_a side_n but_o the_o step_n of_o the_o outside_n wall_n be_v to_o be_v a_o foot_n in_o three_o and_o a_o water_n table_n be_v to_o be_v leave_v of_o eight_o inch_n broad_a if_o the_o earth_n be_v good_a between_o the_o brim_n of_o the_o ditch_n and_o the_o turf_n of_o the_o outside_n wall_n if_o the_o ground_n be_v sandy_a or_o a_o tun_v clay_n it_o be_v to_o be_v eighteen_o inch_n broad_a notwithstanding_o the_o chat_n of_o ignorant_a man_n the_o bottom_n or_o foundation_n of_o the_o brest-work_n be_v to_o be_v thirty_o five_o foot_n in_o the_o bottom_n with_o the_o five_o footstep_n and_o twenty_o foot_n broad_a at_o top_n that_o it_o may_v be_v of_o cannon_n proof_n and_o it_o be_v to_o have_v a_o slope_n 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consequence_n the_o circumference_n of_o it_o be_v of_o two_o thousand_o yard_n that_o be_v a_o english_a mile_n and_o the_o â…•_n part_n of_o a_o mile_n the_o angle_n of_o her_o centre_n be_v of_o 45_o degree_n it_o be_v to_o be_v set_v out_o in_o the_o field_n as_o the_o sexagon_n by_o the_o demi-cirle_a and_o line_n as_o it_o be_v describe_v in_o plate_n twelve_o and_o chapter_n 15._o i_o will_v therefore_o only_o give_v you_o some_o direction_n how_o to_o set_v out_o the_o small_a flanker_n because_o it_o be_v a_o new_a method_n of_o fortification_n that_o we_o have_v not_o as_o yet_o give_v you_o any_o direction_n for_o when_o the_o superficie_fw-la of_o this_o octogon_n have_v be_v set_v out_o and_o full_o trace_v after_o the_o manner_n you_o be_v direct_v to_o set_v out_o the_o sexagon_n you_o be_v to_o divide_v all_o the_o side_n of_o it_o in_o two_o equal_a part_n and_o to_o knock_v in_o a_o stake_n yard_n stake_n from_o these_o middle_a stake_n be_v the_o line_n of_o defence_n draw_v that_o be_v in_o this_o figure_n of_o 150._o yard_n and_o the_o courtine_n 200_o yard_n in_o every_o one_o of_o their_o division_n than_o you_o be_v to_o fasten_v a_o line_n of_o twenty_o five_o yard_n to_o every_o one_o of_o the_o eight_o angle_n stake_v one_o after_o another_o and_o to_o strain_v that_o line_n first_o on_o the_o right_a hand_n and_o then_o at_o the_o left_a hand_n upon_o a_o strait_a line_n upon_o the_o side_n trace_v on_o the_o left_a and_o right_a hand_n of_o that_o stake_n and_o at_o the_o end_n of_o the_o line_n you_o be_v to_o knock_v a_o stake_n and_o these_o two_o stake_n will_v represent_v the_o two_o demi-gorges_n yard_n demi-gorges_n the_o whole_a gorge_n of_o the_o flanker_n be_v to_o be_v 50_o yard_n of_o that_o flanker_n then_o remove_v your_o line_n to_o another_o angle_n stake_n and_o do_v as_o you_o do_v before_o till_o you_o have_v set_v out_o all_o the_o demi-gorge_n after_o that_o manner_n then_o set_v by_o that_o line_n and_o fasten_v another_o line_n of_o fifty_o flanker_n fifty_o the_o line_n of_o fifty_o yard_n represent_v the_o distance_n from_o the_o centre_n to_o the_o point_n of_o the_o flanker_n yard_n to_o every_o anglestake_n of_o the_o side_n and_o strain_v the_o same_o into_o the_o field_n upon_o a_o strait_a diagonall_a line_n stand_v at_o the_o center-stake_n with_o your_o demi-circle_n to_o guide_v the_o same_o because_o the_o demi-diagonall_a line_n from_o the_o center-stake_n to_o the_o utmost_a point_n of_o the_o flank_n be_v of_o 380_o yard_n in_o this_o figure_n and_o therefore_o too_o long_o to_o be_v strain_v convenient_o and_o at_o the_o end_n of_o this_o line_n of_o fifty_o yard_n drive_v in_o a_o stake_n and_o this_o stake_n will_v represent_v the_o utmost_a angle_n or_o point_n of_o the_o say_a flanker_n and_o as_o you_o have_v do_v this_o set_v out_o the_o other_o seven_o angle_n of_o the_o flanker_n after_o this_o very_a manner_n than_o you_o be_v to_o fasten_v a_o line_n of_o 28_o flank_n 28_o 28_o yard_n be_v the_o extent_n of_o the_o slope_n flank_n yard_n to_o every_o one_o of_o the_o sixteen_o stake_n one_o after_o another_o that_o represent_v the_o demi-gorges_n of_o the_o flanker_n and_o to_o strain_v it_o three_o yard_n more_o slope_n than_o upon_o a_o strait_a perpendicularie_n line_n towards_o the_o other_o flanker_n of_o that_o side_n of_o the_o octogon_n and_o at_o the_o end_n of_o the_o line_n you_o be_v to_o drive_v in_o a_o stake_n the_o reason_n why_o you_o strain_v this_o line_n three_o yard_n more_o slope_n then_o straight_o be_v to_o preserve_v the_o flank_n from_o the_o enemy_n battery_n that_o be_v otherwise_o too_o much_o expose_v to_o be_v ruinate_v by_o it_o when_o the_o line_n of_o the_o flank_n be_v draw_v upon_o a_o strait_a perpendicularie_n line_n you_o be_v also_o to_o observe_v that_o the_o strait_a perpendicularie_n line_n of_o the_o flank_n be_v but_o five_o and_o twenty_o yard_n and_o that_o these_o three_o yard_n be_v add_v to_o it_o that_o the_o flank_n may_v be_v three_o yard_n more_o slope_n then_o straight_o because_o a_o slope_n line_n in_o five_o and_o twenty_o yard_n extent_n come_v to_o be_v three_o yard_n long_o than_o a_o strait_n and_o as_o you_o have_v set_v out_o this_o flank_n set_v out_o the_o other_o sixteen_o after_o this_o manner_n and_o drive_v in_o stake_n in_o they_o all_o that_o be_v do_v you_o be_v to_o fasten_v a_o line_n of_o 150_o yard_n to_o the_o middle_a stake_n of_o every_o side_n and_o strain_v it_o first_o on_o the_o right_a hand_n and_o then_o on_o the_o left_a and_o as_o you_o strain_v it_o make_v it_o touch_v the_o flank-stake_n and_o fasten_v it_o to_o the_o stake_n that_o represent_v the_o utmost_a point_n of_o your_o flanker_n and_o this_o line_n do_v represent_v the_o line_n of_o defence_n and_o the_o shoulder_n the_o the_o face_n of_o the_o flanker_n be_v to_o be_v 25_o yard_n beside_o the_o three_o yard_n add_v to_o thicken_v the_o angle_n of_o the_o shoulder_n face_n of_o that_o flanker_n and_o before_o you_o remove_v the_o line_n have_v the_o face_n trace_v by_o the_o pionier_n and_o this_o face_n extend_v itself_o from_o the_o flank-stake_n to_o the_o stake_n that_o represent_v the_o utmost_a angle_n of_o the_o flanker_n and_o as_o you_o have_v set_v out_o this_o face-stake_n set_v out_o all_o the_o fifteen_o other_o and_o they_o be_v set_v out_o and_o trace_v your_o eight_o flanker_n will_v be_v perfect_v then_o you_o be_v to_o set_v out_o all_o your_o dike_n after_o the_o dimension_n describe_v in_o chap._n 25._o and_o these_o direction_n for_o the_o set_n out_o of_o the_o flanker_n of_o all_o the_o other_o ensue_a superficies_n shall_v suffice_v once_o for_o all_o to_o avoid_v repetition_n this_o octogon_n thus_o set_v out_o and_o
the_o second_o stake_n of_o your_o side_n one_o time_n after_o another_o and_o strain_v the_o same_o strait_a perpendiculary-wise_a till_o they_o cut_v the_o other_o line_n that_o represent_v the_o face_n and_o the_o line_n of_o defence_n and_o where_o they_o cut_v one_o another_o knock_v in_o six_o stake_n and_o they_o will_v set_v out_o the_o six_o flank_n of_o the_o three_o bastion_n and_o the_o three_o courtine_n then_o have_v all_o these_o thing_n trace_v by_o divers_a pionier_n after_o you_o have_v fasten_v line_n as_o near_o the_o ground_n as_o you_o can_v to_o they_o all_o and_o that_o be_v do_v you_o be_v to_o set_v out_o the_o six_o orillon_n of_o the_o three_o bastion_n after_o this_o manner_n divide_v the_o line_n of_o the_o flank_n into_o three_o equal_a part_n and_o allow_v one_o part_n for_o the_o flank_n and_o in_o that_o part_n knock_v in_o a_o stake_n and_o the_o other_o two_o part_n for_o the_o turn_n of_o the_o orillon_n than_o you_o be_v to_o advance_v the_o 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poligon_n be_v not_o to_o be_v divide_v into_o eight_o part_n but_o only_o into_o six_o part_n whether_o you_o intend_v to_o erect_v bastion_n with_o or_o without_o orillon_n because_o the_o courtine_n be_v not_o shorten_v at_o all_o by_o the_o form_n of_o the_o orillon_n for_o the_o dike_a run_v between_o the_o rampire_n of_o the_o courtine_n and_o the_o inward_a side_n of_o the_o orillon_n as_o it_o be_v clear_o demonstrate_v in_o this_o triangle_n fortify_v by_o bastion_n with_o orillon_n the_o courtine_n have_v the_o same_o extent_n as_o the_o other_o triangle_n have_v that_o be_v fortify_v by_o bastion_n without_o orillon_n except_o you_o please_v to_o imitate_v as_o i_o have_v say_v the_o best_a italian_a method_n that_o be_v perfect_o set_v out_o in_o a_o demi_a sexagon_n in_o the_o last_o plate_n of_o this_o abstract_n where_o you_o shall_v find_v the_o base_a to_o be_v divide_v first_o into_o ten_o part_n four_o of_o which_o be_v allow_v for_o the_o two_o demi_a gorge_n and_o the_o other_o six_o part_n for_o the_o gourtine_n which_o courtine_n be_v again_o divide_v into_o eight_o equal_a part_n that_o the_o line_n of_o defence_n may_v be_v take_v from_o the_o first_o division_n of_o the_o eight_o equal_a part_n and_o this_o they_o do_v to_o increase_v the_o breadth_n of_o the_o breast_n and_o of_o the_o gorge_n of_o their_o bastion_n that_o they_o may_v be_v more_o capable_a of_o a_o second_o or_o a_o three_o reintrenchment_n when_o the_o flank_n or_o the_o point_n of_o their_o bastion_n be_v beat_v down_o but_o these_o bastion_n erect_v out_o of_o a_o base_a divide_v into_o six_o equal_a part_n be_v repute_v here_o to_o have_v their_o breast_n and_o their_o gorge_n already_o too_o spacious_a and_o therefore_o it_o be_v superfluous_a to_o make_v they_o large_a to_o conclude_v the_o irregularity_n of_o the_o triangle_n cause_v the_o dimension_n of_o her_o bastion_n to_o be_v these_o the_o flank_n be_v forty_o yard_n with_o the_o thickness_n of_o the_o 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to_o set_v out_o all_o the_o 18_o flanker_n one_o after_o another_o as_o you_o be_v direct_v in_o the_o last_o chapter_n chap._n xxviii_o of_o the_o fortification_n of_o the_o square_n fortify_v by_o small_a flanker_n demonstrate_v by_o a_o demi-square_a in_o plate_n 21._o this_o square_n fortify_v by_o twenty_o four_o small_a flank_n if_o of_o 6000_o yard_n circumference_n that_o be_v three_o english_a mile_n and_o two_o five_o part_n of_o a_o mile_n it_o be_v very_o convenient_a to_o fortify_v a_o great_a corporation_n town_n that_o may_v be_v redece_v to_o that_o form_n his_o continent_n be_v great_a because_o of_o the_o solidity_n of_o his_o body_n for_o it_o contain_v 2250000_o square_a yard_n the_o angle_n of_o his_o centre_n be_v of_o 90_o degree_n and_o it_o be_v to_o be_v set_v out_o in_o the_o field_n after_o the_o very_a same_o manner_n of_o the_o end-side_n of_o the_o paralograme_n describe_v in_o the_o last_o chapter_n and_o his_o flanker_n be_v to_o be_v set_v out_o 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