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 of_o the_o line_n of_o the_o length_n of_o the_o oval_n and_o with_o the_o other_o point_n of_o the_o compass_n make_v a_o circle_n then_o remove_v the_o point_n of_o the_o compass_n and_o set_v it_o upon_o the_o left_a hand_n point_n of_o the_o aforesaid_a line_n and_o make_v another_o circle_n and_o the_o two_o extreme_n of_o these_o two_o circle_n represent_v the_o length_n of_o the_o oval_a then_o remove_v your_o compass_n and_o set_v one_o point_n of_o it_o upon_o the_o low_a cut_v of_o these_o two_o circle_n and_o set_v it_o at_o that_o distance_n that_o have_v a_o point_n upon_o this_o low_a cut_v of_o these_o two_o circle_n it_o may_v conjoin_v with_o a_o true_a circularie_a line_n the_o two_o upper_a extreme_n of_o the_o two_o circle_n that_o be_v do_v remove_v your_o compass_n without_o any_o alteration_n of_o the_o distance_n and_o set_v one_o of_o the_o point_n of_o it_o upon_o the_o upper_a cut_n of_o the_o two_o circle_n and_o with_o the_o other_o point_n of_o it_o make_v a_o true_a circulary_a line_n to_o join_v the_o two_o low_a extreme_n of_o the_o two_o circle_n and_o this_o be_v trace_v with_o a_o pen_n will_v produce_v a_o perfect_a oval_n one_o three_o part_n long_o than_o it_o be_v broad_a some_o make_v it_o half_o as_o long_o again_o as_o it_o be_v broad_a but_o the_o side_n be_v too_o flat_a and_o the_o oval_n be_v by_o it_o more_o deform_v and_o not_o so_o seemly_a and_o complete_a as_o this_o be_v demonstrate_v in_o figure_n 16._o now_o to_o make_v the_o most_o perfect_a rhomb_n that_o can_v be_v make_v rhomb_n of_o the_o set_n out_o of_o the_o rhomb_n you_o be_v but_o to_o draw_v four_o line_n dioganall_a wise_a from_o the_o upper_a and_o low_a point_n of_o the_o perpendicularie_n diametrical_a line_n that_o cut_v the_o two_o broad_a side_n of_o the_o oval_n to_o the_o two_o point_n of_o the_o diametrical_a line_n of_o the_o length_n of_o the_o oval_n and_o it_o will_v produce_v a_o perfect_a rhomb_n as_o in_o figure_n 18._o and_o to_o make_v the_o two_o equilateral_a triangle_n and_o the_o four_o scalene_a triangle_n out_o of_o the_o small_a rhomb_n triangle_n of_o the_o set_n out_o of_o the_o 6._o triangle_n you_o be_v but_o to_o draw_v four_o line_n diagonall-wise_a from_o the_o two_o end_n of_o the_o first_o line_n that_o do_v represent_v the_o eight_o part_n of_o the_o oval_a to_o the_o two_o upper_a and_o low_a cut_n of_o the_o two_o circle_n and_o these_o line_n with_o the_o two_o diametrical_a line_n of_o the_o length_n and_o breadth_n of_o the_o oval_a already_o draw_v will_v produce_v these_o six_o foresaid_a triangle_n and_o to_o set_v out_o the_o paralograme_n paralograme_n of_o the_o set_n out_o of_o the_o paralograme_n you_o be_v to_o divide_v the_o breadth_n of_o the_o oval_a into_o four_o equal_a part_n and_o the_o length_n of_o it_o into_o nine_o equal_a part_n and_o at_o every_o division_n to_o make_v a_o point_n and_o to_o draw_v four_o strait_a line_n upon_o the_o first_o point_n of_o these_o two_o division_n and_o they_o will_v produce_v this_o paralograme_n that_o be_v a_o nine_o part_n long_o than_o twice_o his_o breadth_n chap._n ix_o of_o the_o superficies_n contain_v in_o the_o 3._o plate_n the_o first_o circle_n of_o this_o plate_n contain_v a_o equilateral_a triangle_n square_a of_o the_o set_n out_o of_o the_o equilateral_a triangle_n out_o of_o a_o square_a a_o square_a and_o a_o pentagon_n and_o because_o all_o the_o equilateral_a triangle_n that_o be_v set_v out_o by_o a_o square_a have_v their_o two_o side_n long_o than_o their_o square_n by_o anâ—_n part_n i_o have_v draw_v two_o demy-circle_n to_o show_v where_o the_o upper_a angle_n of_o the_o triangle_n shall_v come_v to_o be_v perfect_a and_o equal_a of_o all_o her_o
have_v a_o way_n of_o some_o eight_o foot_n broad_a be_v present_o form_v that_o run_v between_o the_o battlement_n of_o the_o wall_n and_o the_o rampire_n that_o be_v call_v the_o way_n of_o the_o round_n and_o in_o french_a le_fw-fr chemin_fw-fr des_fw-fr rondes_fw-la and_o this_o way_n go_v from_o one_o bastion_n to_o another_o round_o about_o the_o garrison_n a_o very_a commodious_a and_o necessary_a mean_n for_o the_o round_n to_o go_v safe_a in_o the_o night_n and_o to_o discover_v by_o look_v out_o of_o the_o port_n hole_n of_o the_o battlement_n if_o any_o appear_v near_o to_o their_o counterscarp_n or_o corridor_n that_o be_v on_o the_o other_o side_n of_o their_o dike_n and_o the_o rampire_n beside_o the_o brest-work_n upon_o it_o be_v common_o six_o foot_n high_o than_o the_o battlement_n of_o the_o wall_n and_o the_o brest-worke_n six_o foot_n more_o so_o that_o the_o splinter_n of_o the_o wall_n during_o a_o battery_n can_v offend_v the_o soldier_n that_o defend_v the_o brest-worke_n of_o the_o rampire_n four_o the_o rampire_n be_v common_o thirty_o foot_n high_a beside_o the_o height_n of_o the_o brest-worke_n that_o be_v six_o foot_n high_a within_o side_n for_o it_o have_v a_o foot-step_n of_o eighteen_o inch_n high_a and_o two_o foot_n broad_a and_o the_o whole_a height_n of_o the_o rampire_n with_o his_o brest-worke_n be_v within_o side_n thirty_o six_o foot_n and_o without_o thirty_o three_o foot_n high_a because_o the_o top_n of_o the_o brest-work_n be_v carry_v three_o foot_n slope_n because_o it_o be_v always_o twenty_o foot_n broad_a at_o top_n to_o be_v of_o cannon-proof_n and_o the_o rampire_n beside_o the_o thickness_n of_o the_o brest-worke_n and_o the_o slope_n of_o the_o two_o side_n defalk_v be_v always_o forty_o foot_n broad_a on_o both_o side_n on_o which_o they_o plant_v elm_n or_o sicamore_n tree_n that_o in_o few_o year_n make_v very_o pleasant_a and_o shady_a walk_n that_o serve_v in_o time_n of_o siege_n for_o a_o way_n for_o horse_n man_n foot_n and_o carriage_n to_o come_v to_o defend_v and_o relieve_v the_o rampire_n five_o the_o distance_n from_o the_o centre_n of_o one_o bastion_n to_o another_o be_v common_o from_o two_o hundred_o fifty_o yard_n to_o three_o hundred_o yard_n six_o the_o face_n of_o their_o bastion_n be_v ordinary_o from_o one_o hundred_o yard_n to_o one_o hundred_o twenty_o yard_n beside_o the_o turn_n of_o their_o orillon_n that_o be_v about_o twenty_o yard_n seven_o the_o whole_a gorge_n of_o their_o bastion_n be_v from_o one_o hundred_o yard_n to_o a_o hundred_o and_o twenty_o yard_n from_o out_o to_o out_o 8_o the_o breast_n of_o their_o bastion_n be_v from_o 120._o yard_n to_o 130._o yard_n 9_o and_o from_o the_o centre_n of_o the_o bastion_n to_o the_o point_n or_o the_o utmost_a angle_n of_o the_o bastion_n from_o 80_o yard_n to_o 100_o yard_n 10_o the_o flank_n of_o the_o bastion_n from_o 42_o yard_n to_o 50_o yard_n that_o be_v divide_v into_o three_o equal_a part_n if_o they_o make_v orillon_n one_o part_n be_v allow_v for_o the_o flank_n and_o the_o other_o two_o for_o the_o turn_n and_o the_o frame_n of_o the_o orillon_n 11_o their_o curtain_n be_v always_o between_o 160_o yard_n to_o 200_o yard_n 12_o the_o line_n of_o defence_n to_o be_v good_a for_o the_o defence_n of_o the_o musket_n shot_n be_v to_o be_v from_o 220_o yard_n to_o 250_o yard_n at_o the_o most_o 13_o the_o slope_n of_o their_o brest-work_n without_o side_n be_v one_o foot_n for_o every_o yard_n if_o the_o earth_n be_v good_a turn_n now_o let_v the_o reader_n judge_n whether_o these_o costly_a fortification_n be_v for_o our_o turn_n and_o within_o side_n a_o foot_n in_o two_o yard_n but_o if_o it_o be_v a_o sandy_a ground_n or_o a_o run_a clay_n it_o require_v a_o foot_n and_o half_a without_o side_n and_o a_o foot_n within_o for_o a_o yard_n high_a 14_o the_o foundation_n of_o these_o wall_n begin_v from_o the_o bottom_n of_o the_o dike_a and_o be_v carry_v up_o to_o the_o upper_a 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 at_o the_o least_o except_o it_o be_v the_o circulary_a form_n that_o have_v but_o one_o circulary_a line_n that_o begin_v at_o one_o point_n and_o end_n at_o the_o same_o the_o point_n be_v a_o small_a touch_n of_o the_o pen_n point_n of_o the_o point_n that_o can_v be_v divide_v because_o it_o have_v no_o part_n but_o be_v the_o beginning_n and_o end_n of_o all_o line_n and_o the_o centre_n of_o all_o form_n see_v figure_n 1._o a_o line_n be_v the_o continuance_n of_o a_o point_n that_o be_v incapable_a of_o division_n line_n of_o line_n but_o in_o the_o length_n of_o it_o it_o be_v distinguish_v by_o divers_a term_n as_o by_o the_o strait_a line_n in_o figure_n 2._o by_o the_o circulary_a line_n in_o figure_n 3._o by_o the_o perpendiculary_a line_n in_o figure_n 7._o by_o the_o parallel_n line_n in_o figure_n 8._o by_o the_o diagonall_a line_n in_o figure_n 11._o and_o by_o the_o diametrical_a line_n in_o figure_n 12._o there_o be_v divers_a other_o distinction_n of_o line_n but_o 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
allow_v they_o to_o be_v of_o 150._o square_a yard_n that_o be_v a_o piece_n of_o ground_n of_o twelve_o yard_n and_o half_a square_n of_o all_o side_n by_o which_o scantling_n the_o 41666._o yard_n be_v divide_v there_o will_v appear_v to_o be_v in_o this_o continent_n 277._o dwell_a house_n and_o as_o many_o inhabitant_n that_o may_v billet_n four_o hundred_o foot_n and_o two_o troop_n of_o horse_n and_o this_o number_n be_v sufficient_a to_o maintain_v this_o fort_n against_o a_o army_n of_o five_o thousand_o man_n three_o month_n so_o it_o be_v provide_v with_o sixteen_o piece_n of_o ordnance_n ball_n ammunition_n arm_n victual_n and_o all_o other_o necessary_n fit_a for_o a_o siege_n object_n some_o will_v wonder_v why_o so_o small_a a_o fort_n shall_v have_v sixteen_o piece_n of_o ordnance_n when_o many_o of_o our_o garrison_n that_o be_v of_o two_o or_o three_o mile_n circumference_n have_v not_o so_o many_o ans_fw-fr i_o answer_v the_o more_o be_v the_o pity_n they_o be_v no_o 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foot_n square_a and_o they_o have_v a_o slant_v descent_n from_o the_o gorge_n of_o the_o bastion_n with_o a_o door_n to_o come_v down_o to_o they_o but_o because_o they_o have_v no_o other_o light_n then_o the_o porthole_n nor_o no_o other_o evacuation_n for_o the_o smoke_n than_o the_o door_n when_o the_o cannoneer_n fire_v their_o piece_n they_o be_v so_o encumber_v with_o the_o smoke_n in_o the_o vault_n that_o they_o can_v not_o sudden_o charge_v their_o piece_n again_o but_o be_v enforce_v to_o sâ—ay_v till_o the_o evacuation_n of_o the_o smoke_n be_v pass_v whereupon_o the_o modern_a enginier_n to_o prevent_v these_o defect_n have_v invent_v the_o upper_a and_o low_a slank_n where_o they_o always_o place_v two_o piece_n of_o ordnance_n in_o every_o one_o of_o they_o so_o that_o according_a to_o the_o foreign_a method_n every_o bastion_n be_v to_o have_v ten_o piece_n of_o ordnance_n at_o the_o least_o two_o demy-cannon_n in_o every_o flank_n and_o two_o long_a culverin_n to_o 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