Selected quad for the lemma: reason_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
reason_n	angle_n	equal_a	line_n	4,117	5	11.1250	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A07035	The art of fortification, or architecture militaire as vvell offensiue as defensiue, compiled & set forth, by Samuell Marolois revievved, augmented and corrected by Albert Girard mathematician: & translated out of French into English by Henry Hexam; Opera mathematica. Part 3.	Marolois, Samuel.; Girard, Albert, 1595-1632.; Hexham, Henry, 1585?-1650?	1631 (1631)	STC 17451; ESTC S101439	115,154	146

his_o equal_a cdf_n &_o the_o face_n d●_n and_o then_o in_o the_o triangle_n dcf_n the_o line_n df_n fc_n fb_n fp_n and_o final_o bf_a 44_o 93555_o which_o make_v 〈◊〉_d for_o which_o reason_n one_o must_v reduce_v the_o part_n into_o rodd_n but_o this_o reason_n may_v be_v set_v down_o more_o easy_a in_o take_v but_o 7_o 48926_o which_o make_v 1000000_o part_n now_o if_o you_o have_v rather_o work_v it_o by_o multiplication_n without_o division_n than_o you_o must_v take_v but_o 100000_o part_n which_o make_v 13352_o rod_n and_o so_o you_o shall_v find_v dc_o 25_o 38302_o cb_n 9_o 34672_o basilius_n 13_o 35246_o dp_n 81_o 01484_o bh_n 33_o 38115_o the_o angle_n flank_v 79._o 33._o the_o 14._o figure_n the_o 1._o of_o the_o two_o in_o this_o fortress_n hexagonall_a let_v eh_n be_v 60_o rod_n and_o ei_o bc_n kh_n equal_a the_o flank_n kc_n 8_o rod_n it_o behoove_v we_o to_o find_v out_o the_o rest_n by_o the_o help_n of_o the_o rule_n of_o algeber_n you_o shall_v find_v that_o the_o angle_n eif_n will_v be_v 69_o degree_n 4_o 1_o 2_o minute_n then_o in_o the_o triangle_n ibc_n the_o angle_n i_o &_o the_o side_n ib_n 8_o rod_n be_v give_v you_o shall_v find_v bc_n 20_o 9229_o for_o the_o face_n ei_o also_o i_fw-mi will_v be_v 22_o 4002_o then_o aec_fw-la will_v make_v 43_o 3231_o the_o rest_n be_v easy_a without_o the_o second_o flank_n the_o 15._o figure_n this_o figure_n have_v for_o the_o flank_n 10_o rod_n the_o rest_n be_v like_a to_o the_o former_a they_o shall_v be_v put_v in_o order_n in_o the_o table_n follow_v the_o 16._o figure_n &_o 5._o plate_n the_o flank_n of_o this_o heptagone_n that_o be_v a_o fortress_n with_o seven_o angle_n or_o bulwark_n make_v 10_o rod_n the_o distance_n of_o the_o point_v 72_o rod_n and_o the_o angle_n flank_v 80_o degree_n how_o many_o then_o will_v the_o rest_n make_v when_o as_o the_o second_o flank_n be_v 10_o rod_n see_v that_o the_o angle_n of_o the_o bulwark_n make_v 80_o degree_n and_o the_o angle_n of_o the_o 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6436_o and_o he_o 19_o 6836_o let_v we_o make_v the_o second_o flank_n 7_o 3164_o and_o then_o the_o curtain_n will_v be_v 26_o df_n 23_o the_o face_n open_v 25_o 2901._o mp_n 46_o 9337._o marolois_n have_v give_v here_o 80_o degree_n but_o take_v no_o more_o than_o 79_o 25_o 43_o for_o the_o angle_n flank_v which_o he_o do_v without_o premeditation_n the_o 18._o figure_n &_o 5._o plate_n in_o this_o present_a heptagone_n the_o angle_n flank_v be_v 79_o degree_n the_o gorge_n 12_o rod_n &_o the_o curtain_n 32_o rod_n how_o much_o will_v the_o other_o line_n &_o angle_n make_v of_o this_o fortress_n to_o resolve_v this_o question_n you_o must_v suppute_v from_o the_o triangle_n cbg_n the_o line_n cg_n gb_n and_o you_o shall_v have_v gb_n the_o second_o flank_n now_o if_o to_o bg_n you_o add_v ab_fw-la you_o shall_v have_v agnostus_n than_o the_o triangle_n dag_n will_v be_v know_v afterward_o the_o triangle_n dae_n and_o final_o the_o triangle_n dcf_n and_o dhk_v alb._n girard_n note_v that_o in_o this_o question_n marolois_n have_v set_v down_o the_o angle_n flank_v 79_o and_o 2_o seventh_n yet_o do_v not_o follow_v this_o number_n but_o 79_o degree_n the_o same_o error_n also_o be_v commit_v in_o his_o supputation_n of_o the_o former_a and_o in_o the_o 14_o figure_n one_o can_v guess_v well_o his_o supposition_n nevertheless_o that_o need_v not_o to_o stay_v the_o reader_n for_o i_o have_v set_v down_o the_o question_n as_o it_o seem_v he_o will_v have_v propound_v it_o but_o the_o worst_a be_v he_o make_v 2_o figure_n and_o one_o can_v understand_v well_o of_o which_o of_o either_o of_o they_o he_o will_v speak_v but_o we_o will_v speak_v more_o thereof_o hereafter_o howsoever_o those_o which_o be_v most_o intricate_a aswell_o by_o reason_n of_o the_o fault_n escape_v in_o the_o impression_n of_o the_o former_a edition_n as_o by_o the_o error_n of_o his_o disciple_n which_o do_v calculate_v they_o shall_v be_v partly_o omit_v and_o shall_v give_v no_o impediment_n 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bf_n will_v be_v find_v to_o be_v 20_o 1982_o and_o df_n 22_o 6014_o whereof_o the_o double_a take_v from_o dp_n 82_o there_o will_v remain_v for_o the_o curtain_n bh_n 36_o 7972_o and_o consequent_o for_o the_o second_o flank_n gh_o 14_o 6350._o now_o know_v the_o triangle_n dea_n you_o shall_v find_v the_o capital_a also_o the_o gorge_n ab_fw-la 12_o 8744._o if_o in_o the_o triangle_n dkh_n you_o seek_v dh_n than_o you_o shall_v find_v it_o to_o be_v 62_o 7388._o now_o the_o reason_n of_o the_o demy-diameter_n or_o middle_a line_n in_o the_o side_n of_o this_o heptagone_n be_v as_o 1000000000_o to_o 867767478._o the_o 20._o figure_n this_o present_a octogone_n that_o be_v a_o fortress_n of_o 8_o angle_n or_o bulwark_n have_v the_o distance_n dp_v divide_v into_o 7_o equal_a part_n whereof_o df_n &_o fb_n make_v each_o of_o they_o 2_o &_o bp_o fichant_n be_v 60_o rod_n the_o angle_n flank_v 82_o ½_o degree_n the_o line_n bf_n be_v 2_o and_o fp_n 5_o than_o the_o square_a of_o bp_o 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will_v be_v find_v 24_o 11937_o and_o the_o gorge_n ab_fw-la 13_o 05342_o if_o you_o calculate_v the_o triangle_n dea_n the_o 21._o figure_n in_o this_o octogone_n let_v the_o face_n be_v 24_o rod_n the_o flank_n 12_o &_o the_o curtain_n 36_o rod_n &_o the_o angle_n flank_v right_a it_o be_v require_v how_o many_o the_o other_o line_n &_o angle_n will_v be_v the_o angle_n adf_n 67_o ½_o &_o adc_n 45_o will_v make_v know_v cdf_n to_o be_v 22_o ½_o now_o dc_o be_v 24_o rod_n by_o consequence_n df_n fc_n dp_n fb_n will_v be_v know_v also_o see_v that_o

angle_n g_o k_o a_o will_v be_v equal_a to_o the_o angle_n d_o a_o k_o and_o so_o the_o figure_n will_v be_v draw_v according_a to_o the_o say_a proportion_n the_o 10._o plate_n &_o the_o 51_o 52_o 53_o 54_o 55_o 56_o 57_o 58_o 59_o figure_n second_o if_o the_o line_n from_o angle_n to_o angle_v be_v give_v ab_fw-la for_o example_n 80_o rod_n you_o shall_v then_o proportion_v out_o the_o face_n to_o the_o curtain_n sesquialtera_fw-la as_o 2_o to_o 3._o then_o in_o conformity_n of_o the_o table_n of_o angle_n set_v down_o before_o will_v be_v make_v the_o angle_n cab_n according_a to_o its_o form_n to_o wit_n in_o a_o square_a of_o 15_o degree_n in_o a_o pentagone_n of_o 89_o ½_o degree_n in_o the_o exagone_fw-mi 22_o ½_o degree_n and_o so_o consequent_o of_o the_o rest_n then_o for_o the_o form_n of_o the_o abovesaid_a proportion_n of_o the_o face_n to_o the_o curtain_n which_o be_v as_o we_o have_v say_v as_o 2_o to_o 3._o aswell_o in_o a_o square_a fortress_n as_o in_o a_o pentagone_n and_o other_o follow_v you_o shall_v take_v upon_o a_o scale_n of_o a_o reasonable_a greatness_n 2._o which_o shall_v be_v set_v from_o a_o to_o c_o and_o 3_o from_o a_o to_o d_o suppose_v that_o it_o be_v at_o c_o &_o d_o then_o from_o the_o point_n d_o will_v be_v make_v the_o arch_n g_o of_o the_o distance_n of_o 2_o to_o wit_n ca_n &_o from_o c_z of_o the_o distance_n of_o d_o a_o 3_o will_v be_v make_v the_o arch_n which_o cut_n between_o each_o other_o at_o g_o by_o which_o a_o right_a line_n be_v draw_v to_o a_o where_o the_o same_o cut_v the_o line_n fb_n as_o here_o at_o f_o you_o have_v the_o face_n of_o the_o bulwark_n and_o by_o that_o the_o curtain_n will_v be_v know_v to_o know_v also_o the_o line_n of_o the_o gorge_n and_o the_o flank_n and_o all_o the_o other_o part_n you_o must_v make_v the_o angle_n form_v the_o gorge_n and_o the_o flank_n of_o 40_o degree_n as_o be_v here_o the_o angle_n kie_fw-mi cut_v the_o diameter_n p_o a_o at_z i_o from_o which_o 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a_o fortress_n have_v 5_o term_n to_o wit_n the_o name_n of_o the_o figure_n the_o angle_n of_o the_o bastion_n m_o the_o angle_n form_v the_o flank_n gbh_n and_o the_o reason_n of_o the_o face_n dg_n to_o the_o curtain_n hl_n conceive_v therein_o that_o ab_fw-la be_v give_v this_o figure_n be_v in_o stead_n of_o the_o 2_o 50_o be_v worth_a nothing_o i_o be_o so_o constrain_v to_o make_v a_o new_a figure_n and_o a_o new_a explanation_n let_v then_o nba_n be_v equal_a to_o the_o demy-angle_n of_o the_o polygone_a and_o let_v the_o angle_n form_v the_o flank_n be_v hbg_n lak_n so_o you_o shall_v have_v the_o intersection_n or_o cut_v between_o c_o then_z dbf_a the_o half_a equal_a to_o the_o demy-angle_n flank_v and_o find_v the_o point_n f_o so_o that_o ab_fw-la to_o bf_n be_v the_o reason_n of_o the_o curtain_n to_o the_o face_n and_o have_v draw_v cf_n cut_v nb_n at_o d_o and_o make_v dg_n a_o parallel_n with_o fb_n meet_v with_o bc_n at_o g_o you_o have_v final_o gh_v the_o perpendicular_a with_o basilius_n and_o do_v the_o like_a also_o on_o the_o other_o side_n you_o shall_v have_v all_o the_o part_n require_v the_o demonstration_n be_v manifest_a see_v that_o c_z be_v the_o common_a point_n of_o the_o like_a figure_n fba_n dgk_n then_o as_o fb_n to_o basilius_n so_o dg_v the_o face_n wilbe_n to_o gk_n or_o hl._n the_o 11._o plate_n and_o 60_o 61_o 62_o 63_o 64_o 65_o 66_o ●7_n ●●_o figure_n if_o there_o be_v a_o question_n to_o proportion_n out_o the_o face_n to_o the_o curtain_n and_o the_o gorge_n to_o the_o flank_n when_o the_o line_n of_o be_v give_v you_o shall_v do_v as_o the_o figure_n in_o appearance_n demonstrate_v to_o the_o eye_n namely_o you_o shall_v first_o proportion_v out_o the_o face_n to_o the_o curtain_n by_o the_o former_a rule_n and_o make_v it_o so_o that_o the_o curtain_n of_o a_o square_a fortress_n be_v 300_o foot_n and_o the_o face_n hf_n 250_o foot_n have_v reason_n as_o 5_o to_z 6_o according_a to_o which_o you_o shall_v find_v the_o point_n g_o which_o be_v the_o angle_n of_o the_o shoulder_n to_o find_v then_o afterward_o the_o gorge_n &_o the_o flank_n you_o shall_v make_v the_o covert_a line_n gh_n which_o must_v be_v divide_v into_o 4_o equal_a part_n in_o four_o because_o you_o will_v have_v the_o reason_n of_o the_o gorge_n in_o the_o flank_n as_o 4_o to_o 3_o whereof_o you_o must_v place_v three_o of_o they_o from_o h_n to_o i_o and_o then_o draw_v i_o g_o cut_v through_o the_o semy-diameter_n of_o the_o polygone_a which_o be_v here_o square_v into_o a_o and_o then_o the_o line_n ab_fw-la be_v draw_v it_o will_v be_v parallel_v with_o of_o and_o so_o you_o have_v the_o side_n of_o the_o polygone_a interior_a upon_o which_o the_o 2_o perpendicular_o be_v draw_v gc_n hd_n you_o have_v the_o 2_o flank_n and_o consequent_o the_o essential_a part_n of_o a_o square_a fortification_n know_v you_o must_v also_o observe_v the_o same_o in_o the_o other_o figure_n follow_v if_o the_o line_n of_o which_o 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foot_n bank_n thereof_o i._o m._n the_o parapett_n of_o the_o falsebray_n m._n n._n the_o scharfe_n p._n o._n n._n m._n the_o moat_n p._n o._n the_o counterscharfe_n p._n q._n the_o covert_a way_n r._n q._n the_o foot_n bank_n thereof_o t._n s._n r._n the_o parapett_n of_o the_o covert_a way_n the_o other_o name_n which_o have_v need_n of_o explanation_n shall_v be_v declare_v in_o their_o due_a place_n before_o we_o come_v to_o instruct_v you_o particular_o in_o the_o art_n of_o fortification_n we_o will_v brief_o treat_v of_o the_o calculation_n thereof_o in_o which_o supputation_n you_o shall_v have_v first_o set_v down_o the_o know_a term_n and_o under_o they_o the_o disposition_n of_o the_o character_n or_o letter_n beginning_n with_o a_o square_a fortress_n with_o four_o angle_n or_o bulwark_n and_o proceed_v on_o to_o a_o dodecagone_fw-mi a_o fortress_n with_o twelve_o angle_n or_o bulwark_n make_v upon_o every_o polygone_a 3_o or_o 4_o trial_n that_o afterward_o one_o may_v 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add_v to_o the_o angle_n flank_v interior_a 90_o degree_n than_o the_o sum_n will_v be_v the_o angle_n of_o the_o shoulder_n to_o find_v out_o the_o angle_n of_o the_o polygone_a from_o the_o number_n of_o its_o subtract_v namely_o 2_o the_o remaynder_n must_v be_v multiply_v by_o 2._o and_o the_o product_n will_v be_v the_o number_n of_o the_o right_a angle_n which_o such_o a_o polygone_a contain_v as_o you_o may_v see_v by_o this_o exemple_n follow_v or_o thus_o and_o by_o the_o same_o rule_n you_o shall_v find_v the_o angle_n of_o the_o subsequent_a polygones_n begin_v from_o a_o square_a fortress_n to_o a_o dodecagone_fw-mi 4._o 5._o 6._o 7._o 8._o 9_o 10._o 11._o 12_o 90._o 72._o 60._o 51_o 3_o 7._o 45._o 40._o 36._o 32_o 8_o ●●_o 30_o the_o angl_n of_o the_o centre_n 90._o

108._o 120._o 128_o 4_o 7._o 135._o 140._o 144._o 147_o ●_o 11._o 150_o the_o angle_n of_o the_o polig_n 45._o 54._o 60._o 64_o 2_o 7._o 67_o 1_o 2._o 70._o 72._o 73_o 7_o 11._o 75_o the_o half_a 15._o 15._o 15._o 15._o 15._o 15._o 15._o 15._o 15._o sum._n 60._o 69._o 75._o 79_o 2_o 7._o 82_o 1_o 2._o 85._o 87._o 88_o 7_o 11._o 90_o the_o angle_n flank_v remain_v 30._o 39_o 45._o 49_o 2_o 7._o 52_o 1_o 2._o 55._o 57_o 58_o 7_o 11._o 60_o the_o double_a of_o the_o ang_fw-mi 180._o 180._o 180._o 180._o 180._o 180._o 180._o 180_o 180_o the_o flank_n interior_a 150._o 141._o 135._o 130_o 5_o 7._o 127_o 1_o 2._o 125._o 123._o 121_o 4_o 11._o 120_o the_o flank_n exterior_a 15._o 19_o 1_o 2._o 22_o 1_o 2._o 24_o 9_o 14._o 26_o 1_o 4._o 27_o 1_o 2._o 28_o 1_o 2._o 29_o 7_o 22._o 30_o the_o flank_n interior_a 90._o 90._o 90._o 90._o 90._o 90._o 90._o 90._o 90_o 105._o 109_o 1_o 2._o 112_o 1_o 2._o 114_o 9_o 14._o 116_o 1_o 4._o 117_o 1_o 2._o 118_o 1_o 2._o 119_o 1_o 22._o 120_o the_o ang_fw-mi of_o the_o shall_v and_o see_v the_o angle_n flank_v of_o a_o dodecagone_n be_v right_a and_o able_a to_o resist_v a_o battery_n which_o be_v also_o make_v always_o with_o right_a angle_n to_o shake_v the_o more_o the_o face_n of_o the_o bulwark_n one_o must_v fortify_v the_o polygones_n which_o be_v above_o the_o dodecagone_fw-mi with_o a_o right_a angle_n that_o the_o line_n of_o defence_n may_v come_v the_o more_o into_o the_o curtain_n that_o one_o may_v give_v the_o more_o fire_n upon_o it_o but_o the_o polygones_n which_o be_v under_o a_o dodecagone_fw-mi must_v be_v fortify_v according_a to_o the_o precedent_a table_n and_o the_o calculation_n thereof_o shall_v be_v make_v hereafter_o some_o time_n we_o augment_v aswell_o the_o angle_n of_o the_o bulwark_n as_o the_o octogone_n a_o fortress_n with_o 8_o angle_n or_o bulwark_n with_o a_o right_a angle_n and_o those_o above_o be_v always_o right_a &_o those_o under_o diminish_v to_o the_o square_a fortress_n which_o have_v the_o 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7._o 22_o ●_o 2._o the_o angle_n flank_v interior_a 90._o 90._o 90._o 90._o 90._o the_o angle_n which_o be_v the_o flank_n always_o 105._o 108._o 110._o 111_o 3_o 7._o 112_o 1_o 2._o the_o angle_n of_o the_o shoulder_n in_o the_o same_o manner_n also_o may_v be_v make_v right_o the_o angle_n flank_v of_o the_o decagone_fw-mi a_o fortress_n of_o ten_o bulwark_n where_o you_o must_v note_v also_o that_o before_o we_o proceed_v further_o we_o will_v make_v use_n in_o the_o supputation_n follow_v of_o the_o ten_o or_o decinall_a number_n which_o though_o it_o give_v some_o imperfection_n yet_o see_v the_o thing_n which_o we_o omit_v therein_o be_v of_o no_o great_a consequence_n it_o be_v ridiculous_a to_o make_v any_o further_a search_n thereof_o consider_v likewise_o that_o the_o table_n of_o sines_n tangent_n and_o secant_v be_v one_o and_o the_o same_o i_o think_v it_o good_a therefore_o to_o make_v use_n of_o these_o follow_v the_o i._o question_n the_o 1._o figure_n &_o 1_o plate_n 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know_v term_n in_o a_o triangle_n but_o that_o one_o may_v know_v the_o other_o three_o or_o one_o of_o they_o only_o which_o one_o desire_n as_o the_o reader_n may_v know_v it_o the_o manner_n &_o order_n thereof_o be_v much_o more_o facile_a &_o easy_a to_o conceive_v than_o the_o read_n of_o our_o author_n in_o his_o former_a edition_n be_v obscure_a troublesome_a and_o hard_o to_o be_v attain_a unto_o for_o this_o reason_n the_o learner_n of_o this_o science_n be_v require_v to_o be_v forewarn_v before_o they_o come_v to_o the_o read_n of_o this_o book_n &_o be_v advertise_v hereby_o that_o when_o it_o be_v say_v a_o triangle_n have_v three_o term_n he_o must_v understand_v know_v or_o give_v and_o that_o when_o i_o say_v that_o a_o triangle_n rightangle_n have_v three_o term_n that_o then_o i_o be_o bind_v but_o to_o show_v two_o of_o they_o see_v that_o this_o word_n rightangle_n presuppose_v that_o the_o triangle_n have_v a_o right_a angle_n to_o weet_v of_o 90_o degree_n 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for_o one_o entire_a flank_n zbc_n &_o see_v that_o the_o angle_n of_o the_o shoulder_n bcd_n be_v exterior_a in_o the_o triangle_n cdi_o it_o will_v be_v equal_a to_o the_o two_o interiour_o d._n 90_o and_o 1._o 13_o 24_o and_o therefore_o the_o shoulder_n c_o will_v be_v 103._o 24._o to_o find_v out_o the_o length_n in_o a_o triangle_n ambligone_o bni_n the_o side_n by_o be_v 600_o foot_n &_o the_o angle_n be_v find_v therefore_o the_o other_o term_n shall_v be_v find_v to_o wit_n bn_n the_o capital_a line_n 196_o 64_o &_o ni_fw-fr 444_o 62_o but_o nd_o be_v the_o

three_o of_o diego_n for_o nd_o be_v 7_o diego_n be_v 21_o by_o the_o hypothesis_n therefore_o the_o four_o part_n of_o the_o find_v ni_fw-fr will_v be_v nd_o 111_o 15_o the_o gorge_n &_o the_o rest_n will_v be_v 333_o 46_o for_o diego_n the_o curtain_n now_o diego_n to_o dc_o be_v as_o 21_o to_o 5_o therefore_o one_o may_v say_v if_o 21_o give_v i_o 5_o how_o many_o devil_n 333_o 46_o give_v i_o facit_fw-la for_o cd_o 79_o 40._o &_o so_o ci_o will_v be_v 343_o 84_o which_o take_v from_o by_o 600_o there_o will_v remain_v bc_n 257_o 16_o for_o the_o face_n final_o the_o triangle_n rightangle_n vdh_n have_v three_o term_n dh_n 600_o and_o the_o angle_n h_n equal_a to_o hdi._n 13_o 24_o because_o of_o the_o parallel_v diego_n uh_o than_o you_o shall_v know_v vd_v 139_o 05_o &_o uh_o 583_o 67_o from_o which_o side_n subtract_v up_o equal_a to_o demy_fw-fr diego_n 166_o 73_o there_o will_v remain_v ph_n 416_o 94_o whereof_o the_o double_a will_v be_v for_o bh_n 833_o 88_o and_o pa_n will_v be_v 416_o 94_o for_o it_o be_v equal_a to_o ph_n and_o thus_o must_v you_o do_v with_o the_o rest_n follow_v the_o 3._o figure_n 1._o plate_n let_v there_o be_v a_o square_a fortress_n whereof_o the_o curtain_n diego_n be_v 4_o part_n the_o flank_n cd_o shall_v have_v 1_o of_o they_o and_o also_o the_o gorge_n 1_o the_o defence_n run_v from_o the_o angle_n of_o the_o flank_n form_v the_o face_n &_o the_o length_n by_o be_v of_o 600_o foot_n one_o require_v the_o greatness_n of_o the_o other_o line_n of_o the_o same_o fortress_n to_o find_v out_o the_o angles_n the_o triangle_n rightangle_n cdi_o have_v three_o term_n as_o the_o reason_n of_o the_o side_n cd_o 1_o to_z di_z 4_o therefore_o the_o flanck_a interior_a cdi_o will_v be_v 14_o degree_n &_o 2_o minute_n whereunto_o add_v d_o 90_o degree_n than_o the_o shoulder_n bcd_n will_v be_v 104_o 2_o now_o if_o you_o double_v the_o angle_n cid_n it_o will_v be_v 28_o 4_o cmd_v and_o its_o adjunct_n cmf_n 151_o 56_o for_o the_o angle_n 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such_o a_o fortress_n will_v be_v for_o the_o angle_n see_v that_o the_o line_n flank_v be_v 60_o then_o in_o the_o triangle_n bni_n the_o angle_n b_o will_v be_v 30_o &_o n_o 135_o the_o adjunct_n of_o any_o 45_o whereof_o the_o angle_n remain_v be_v will_v be_v 15_o degree_n therefore_o the_o triangle_n bdi_n will_v be_v isosceles_a that_o be_v two_o side_n a_o like_a such_o that_o dbi_n be_v aswell_o 15_o as_o bid_v the_o shoulder_n c_o will_v be_v 105_o and_o in_o the_o triangle_n dmi_n the_o angle_n upon_o the_o basis_n be_v each_o of_o they_o 15_o degree_n the_o remaynder_n m_o will_v be_v then_o 150_o for_o the_o flank_a exterior_a bmh._n for_o the_o side_n the_o triangle_n isoceles_a bdi_n have_v three_o term_n the_o defence_n by_o 600_o foot_n and_o the_o point_a angle_n every_o of_o they_o 15_o degree_n therefore_o diego_n the_o curtain_n will_v be_v 310_o 584._o also_o the_o triangle_n cdi_o have_v three_o term_n the_o angle_n d_o right_n i_o 15._o and_o the_o curtain_n diego_n than_o cd_o the_o flank_n 83_o 217_o &_o ci_o 321_o 535_o which_o be_v take_v from_o by_o 600_o there_o will_v remain_v bc_n the_o face_n 278_o 465._o moreover_o in_o the_o triangle_n ibn_n the_o angle_n b_o 30_o i_o 15_o &_o the_o defence_n 600_o so_o than_o bn_n the_o capital_a line_n will_v be_v 219_o 623_o &_o ni_fw-fr 424._o 268._o from_o whence_o diego_n be_v take_v there_o will_v remain_v the_o gorge_n nd_o 113._o 684._o moreover_o the_o triangle_n bill_n have_v three_o term_n l_o right_o b_o equal_a to_o cid_n 15._o and_o by_o 600_o then_o i_o will_v will_v be_v 155_o 292_o &_o bl_v 579_o 558_o and_o take_v from_o it_o pl_z 155._o 292._o which_o be_v the_o half_a of_o the_o curtain_n there_o will_v remain_v bp_o 424_o 266_o for_o ap_n also_o it_o be_v double_a be_v bh_n 848_o 532_o one_o may_v also_o easy_o know_v basilius_n 600._o the_o 5._o figure_n &_o 2._o plate_n albert_n girard_n heitherto_o of_o the_o defence_n draw_v from_o the_o angle_n of_o the_o flank_n but_o in_o those_o follow_a there_o be_v a_o second_o flank_n and_o the_o distance_n of_o the_o angle_n of_o the_o flank_n even_o to_o the_o angle_n flank_v which_o be_v call_v fichant_n the_o 5_o figure_n be_v the_o design_n or_o draught_n of_o a_o square_a fortress_n whereof_o the_o line_n of_o defence_n fichant_n dh_n make_v 600_o foot_n the_o angle_n flank_v 60_o degree_n the_o line_n hb_n be_v divide_v into_o 7_o equal_a part_n whereof_o the_o one_o which_o be_v between_o the_o character_n 1._o 2._o be_v subdivide_v into_o 5_o equal_a part_n and_o from_o the_o centre_n b_o be_v make_v the_o arch_n 4_o n_o cut_v the_o capital_a line_n at_o n_o from_o which_o point_n be_v draw_v ndz_n parallel_n to_o bh_n and_o from_o the_o point_n v_n which_o be_v in_o the_o character_n 2_o the_o perpendicular_a vd_n the_o question_n be_v how_o many_o these_o line_n and_o the_o angle_n of_o such_o a_o fortress_n will_v make_v for_o the_o line_n in_o part_n indetermine_v forasmuch_o as_o bh_n contain_v 35_o such_o part_n as_o bn_n 9_o bv_o 10_o uh_o 25_o you_o shall_v have_v in_o the_o triangle_n rightangle_n btn_n term_n enough_o to_o know_v bt_n or_o tn_z 6_o 364_o and_o tu_fw-la or_o nd_o 3_o 636_o also_o the_o triangle_n rightangle_n duh_o have_v three_o term_n vd_v equal_a to_o tn_n 6364_o &_o uh_o 25_o than_o dii_n will_v be_v 25_o 797_o part_n or_o the_o same_o dh_fw-mi be_v 600_o foot_n we_o must_v then_o calculate_v according_a to_o this_o reason_n the_o line_n above_o mention_v to_o bring_v they_o into_o foot_n say_v as_o follow_v for_o the_o line_n in_o part_n determine_v to_o wit_n bring_v into_o foot_n the_o 25_o 797_o part_n make_v 600_o foot_n how_o many_o then_o will_v bh_n 35._o come_v to_o there_o will_v be_v for_o bh_n 814_o 04_o foot_n likewise_o vl_a or_o dz_n the_o curtain_n may_v be_v make_v so_o bh_n 7_o part_n make_v 814_o 04_o foot_n how_o many_o vl_a 3_o it_o will_v come_v for_o the_o curtain_n to_o 348_o 87_o &_o for_o bn_n the_o capital_a line_n you_o shall_v say_v if_o 35_o give_v 814_o 04_o how_o many_o 9_o you_o shall_v have_v for_o bn_n 209.32_o and_o so_o of_o the_o other_o for_o tn_n or_o bt_v or_o vd_n will_v be_v 148._o 02_o &_o tu_fw-la or_o the_o gorge_n nd_o 84_o 57_o and_o see_v that_o bh_n be_v know_v also_o vl_a the_o half_a of_o the_o remainder_n will_v be_v bv_o 232_o 58._o in_o the_o triangle_n rightangle_n buc_n the_o angle_n b_o will_v be_v 15_o degree_n for_o pbo_n be_v 45_o &_o cbo_n 30_o and_o bv_n be_v know_v therefore_o bc_n

fb_n be_v find_v or_o ea_fw-la you_o have_v also_o the_o triangle_n dea_n by_o which_o you_o shall_v find_v da_fw-la ab_fw-la &_o shall_v find_v dg_n in_o the_o triangle_n cbg_n for_o have_v cg_n you_o add_v unto_o it_o dc_o final_o dh_fw-mi in_o the_o triangle_n dkh_n dp_n 80_o 34624_o dg_n 55_o 35756_o dh_n 61_o 91032_o gh_n 7_o 02948._o ab_fw-la 13_o 39837._o ai_fw-fr 62_o 39837._o the_o 22._o figure_n in_o this_o present_a octogone_n the_o flank_n be_v 11_o rod_n the_o angle_n flank_v 82_o ½_o degree_n the_o line_n dp_v 76_o rod_n &_o the_o second_o flank_n to_o be_v choosen_n how_o many_o then_o will_v be_v the_o other_o part_n of_o this_o fortress_n the_o angle_n of_o the_o octogone_n be_v know_v and_o the_o angle_n flank_v also_o their_o half_a adf_n adc_n therefore_o the_o rest_n and_o the_o triangle_n cbg_n will_v be_v know_v and_o bg_n will_v be_v find_v 22_o 3058_o suppose_v the_o second_o flank_n gh_o be_v 9_o 6942_o than_o bh_n 32_o rod_n likewise_o as_o much_o for_o fk_v but_o dp_n be_v 76_o therefore_o df_n make_v 22_o and_o after_o you_o have_v calculate_v the_o triangle_n dcf_n dae_n you_o shall_v find_v dc_o 24_o 52978_o dg_v 49_o 40045_o ab_fw-la 12_o 94981_o and_o final_o in_o the_o triangle_n dkh_n the_o fichant_n dh_n 58_o 25283._o the_o 23._o figure_n let_v the_o angle_n flank_v be_v 82_o ½_o of_o this_o present_a octogone_n the_o line_n of_o defence_n dg_n 50_o rod_n dp_v 76_o and_o cab_n the_o half_a of_o the_o angle_n flank_v after_o you_o have_v carry_v ac_fw-la which_o be_v forget_v if_o you_o count_v the_o triangle_n dag_n acg_n you_o shall_v find_v the_o capital_a the_o face_n and_o the_o gorge_n and_o bg_n than_o the_o triangle_n cbg_n dcf_n afterward_o you_o shall_v have_v cb_n cf_n df_n and_o therefore_o fk_a or_o the_o curtain_n and_o the_o second_o flank_n than_o dh_n 58_o 3498._o the_o capital_a 23_o 93655_o the_o gorge_n 12_o 84323_o the_o flank_n 11_o 26362_o the_o second_o flank_n 9_o 15324_o the_o face_n 24_o 53340_o the_o 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one_o part_n its_o root_n substract_v will_v be_v 11_o 14172_o for_o one_o part_n whereof_o the_o double_a will_v be_v for_o df_n or_o fb_n and_o the_o triple_a for_o fk_v or_o the_o curtain_n 33_o 42516_o bh_n and_o after_o you_o have_v calculate_v the_o triangle_n dfc_a cbg_n dea_n you_o shall_v find_v the_o other_o line_n dc_o 25_o 1219_o dg_n 48_o 25855_o gh_v 12_o 9026_o bc_n will_v be_v 10_o 68335_o the_o seven_o part_n of_o the_o number_n above_o say_v will_v be_v for_o dp_n 77_o 99204._o the_o 26._o figure_n in_o this_o present_a enneagone_n let_v the_o angle_v flank_v be_v 85_o degree_n the_o defence_n 50_o rod_n the_o other_o 60_o &_o the_o gorge_n in_o the_o flank_n as_o 4_o to_o 3._o the_o rest_n be_v require_v if_o you_o take_v away_o the_o angle_n adc_n 42_o ½_o from_o adf_n 70_o the_o angle_v gdr_n will_v remain_v and_o dg_n make_v 50_o rod_n than_o the_o triangle_n dgr_n will_v be_v know_v to_o wit_n dr_n &_o rg_n or_o kh_n 23._o 0875_o which_o will_v make_v you_o know_v the_o triangle_n dkh_n for_o dh_n be_v 60_o then_o rk_v or_o gh_n will_v be_v 11_o 0301._o moreover_o in_o the_o triangle_n dea_n the_o side_n ea_fw-la be_v equal_a to_o kh_n by_o de_fw-fr &_o dr_n you_o shall_v have_v agnostus_n 35_o 94735_o then_o in_o carry_v ac_fw-la you_o shall_v seek_v the_o angle_n a_o from_o the_o triangle_n abc_n set_v down_o ab_fw-la 4_o and_o bc_n 3_o part_n according_a to_o the_o hypatese_n you_o shall_v find_v then_o the_o angle_n a_o of_o 36_o degree_n 52_o min._n &_o 12._o second_n then_o let_v we_o go_v to_o the_o triangle_n acg_n have_v agnostus_n know_v the_o angle_n a_o &_o g_o equal_a to_o cdf_n 27_o ½_o degree_n to_o find_v cg_n afterward_o the_o face_n and_o consequent_o df_n fk_n of_o or_o ab_fw-la the_o gorge_n which_o will_v be_v 14_o 72808_o therefore_o if_o 4_o give_v 3_o how_o many_o than_o ab_fw-la you_o shall_v have_v bc_n for_o the_o flank_n 11_o 04606_o &_o dp_v 78_o 51183_o da_n 24_o 56925._o the_o 27._o figure_n the_o distance_n 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be_v require_v how_o many_o the_o other_o part_n thereof_o will_v make_v the_o reason_n sesquitertia_fw-la be_v as_o 4_o to_z 3_o for_o ab_fw-la to_o bc_n than_o the_o imaginall_a angle_n bac_n will_v be_v 36_o degree_n 52_o minute_n 12_o second_n moreover_o the_o angle_n adc_n be_v 43_o ½_o then_o cdf_n will_v be_v 28_o ½_o which_o be_v a_o angle_n of_o the_o triangle_n gdr_n and_o which_o may_v be_v know_v see_v that_o dg_n be_v 50_o rod_n therefore_o graccus_n 23_o 858_o or_o its_o equal_a kh_n and_o for_o as_o much_o as_o dh_n be_v 60_o than_o dk_n will_v be_v know_v and_o also_o dr_n withal_o and_o so_o you_o shall_v find_v rk_n for_o the_o second_o flank_n likewise_o the_o triangle_n dea_n will_v be_v know_v then_o ed_z and_o dr_n will_v make_v know_v er_fw-mi or_o agnostus_n &_o consequent_o you_o shall_v have_v the_o triangle_n acg_n for_o the_o angle_n a_o be_v find_v above_o and_o the_o angle_n g_z be_v 28_o ½_o degree_n than_o the_o face_n will_v be_v 26_o 11334_o and_o have_v find_v out_o df_n you_o shall_v find_v of_o or_o the_o gorge_n 15_o 197_o also_o fk_v for_o the_o curtain_n 32_o 10821_o dp_v 78_o 00619._o the_o 29._o figure_n in_o this_o decagone_fw-mi the_o angle_n flank_v make_v 87_o degree_n the_o defence_n fichant_n 60_o rod_n the_o flank_n 12_o rod_n and_o the_o gorge_n 16_o rod_n it_o be_v require_v how_o much_o the_o other_o part_n make_v the_o quadrangle_n abcd_v have_v five_o term_n give_v you_o shall_v find_v the_o other_o part_n also_o the_o triangle_n dfc_n whereby_o you_o have_v fb_n or_o kh_n &_o consequent_o

the_o triangle_n dkh_n and_o have_v fd_v &_o dk_n their_o sum_n and_o difference_n will_v be_v for_o dp_n 78_o 65_o and_o bh_n 30_o 32633_o the_o face_n shall_v be_v 27_o 49377_o dg_n 52_o 64265._o the_o 30._o figure_n in_o this_o decagone_fw-mi let_v the_o curtain_n be_v 36_o rod_n and_o the_o flank_n 12_o the_o face_n 24_o and_o the_o reason_n of_o the_o bulwark_n in_o the_o flank_a interior_a as_o 58_o to_o 19_o the_o half_n of_o 58_o be_v 29_o than_o the_o angle_n adg_n to_o agd_n or_o cdf_n will_v be_v as_o 29_o to_o 19_o therefore_o set_v down_o adf_n to_o cdf_n it_o will_v be_v as_o 48_o to_o 19_o but_o adf_n be_v 72_o degree_n than_o cdf_n will_v be_v 28_o ½_o degree_n &_o then_o the_o angle_n flank_v 87_o degree_n the_o rest_n be_v easy_a for_o the_o triangle_n dcf_n cbg_n will_v make_v know_v cf_n fd_n dg_n be_v 49_o 14888_o dh_fw-mi 61_o 68324_o dp_v 78_o 18336_o ad_fw-la 24_o 52336._o the_o 31._o figure_n let_v there_o be_v a_o decagone_fw-mi whereof_o the_o face_n be_v 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know_v the_o angle_n remain_v adc_n 43_o 55_o 48_o and_o the_o other_o or_o cdf_n 29_o 42_o 23_o therefore_o in_o the_o triangle_n cdf_n the_o angle_n d_o be_v than_o so_o and_o dc_o 24_o rod_n and_o then_o df_n dk_n dp_n will_v be_v know_v likewise_o fc_a fb_n bp_o &_o ad_fw-la de_fw-fr &_o of_o for_o the_o gorge_n here_o be_v nothing_o but_o that_o which_o be_v ill_o calculate_v by_o the_o autheur_n or_o rather_o by_o his_o disciple_n as_o from_o the_o beginning_n without_o all_o doubt_n seeke_v to_o help_v themselves_o with_o the_o figure_n put_v here_o under_o which_o be_v needless_a suppose_v that_o they_o have_v skill_n in_o trigonometrie_n i_o will_v only_o set_v down_o here_o the_o reason_n of_o the_o ray_v or_o semy-diameter_n in_o the_o side_n of_o the_o vndecagone_fw-mi inscribe_v in_o the_o circle_n be_v as_o 100000000000000_o to_o 563●46511368285_n so_o that_o you_o may_v take_v here_o the_o reason_n or_o proportion_n as_o precise_o as_o you_o will_n the_o 33._o figure_n 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68112_o as_o much_o make_v fk_a which_o take_v from_o 75_o the_o half_a of_o the_o rest_n will_v be_v for_o df_n you_o shall_v find_v then_o also_o fc_a afterward_o ab_fw-la be_v set_v down_o upon_o 4._o bc_n will_v be_v 3_o than_o the_o triangle_n bac_n will_v be_v 36_o 52_o 12_o which_o take_v from_o dab_n there_o will_v remain_v dac_n &_o so_o the_o triangle_n dac_n shall_v have_v 3_o term_n you_o shall_v then_o seek_v da_fw-la or_o ac_fw-la by_o which_o you_o shall_v come_v to_o have_v ab_fw-la 13_o 7969_o &_o bc_n 10_o 34743_o then_o in_o the_o triangle_n cbg_n you_o shall_v find_v bg_n afterward_o dr_n or_o cg_n be_v know_v you_o shall_v have_v dg_n 41_o 54579_o gh_v 11_o 57496_o and_o final_o dh_fw-mi the_o fichant_n 58_o 96636._o the_o 34._o figure_n moreover_o for_o this_o vndecagone_fw-mi let_v the_o angle_v flank_v be_v 88_o degree_n 38_o 11_o &_o the_o face_n to_o the_o curtain_n as_o 2_o to_o 3_o and_o the_o gorge_n in_o the_o flank_n as_o 8_o to_o 5._o &_o the_o fichant_n 60_o rod_n how_o many_o will_v all_o these_o line_n make_v you_o must_v first_o find_v out_o the_o quadrangle_n abcd_n whereof_o the_o angle_n be_v know_v &_o set_v down_o ab_fw-la 8_o part_n then_o bc_n will_v be_v 5_o &_o dc_o be_v find_v you_o shall_v find_v df_n fc_n also_o bh_n see_v that_o dc_o be_v to_o bh_n as_o 2_o to_o 3_o have_v dk_n and_o kh_n you_o shall_v have_v dh_n in_o part_n which_o make_v 60_o rod_n by_o which_o reason_n you_o have_v the_o line_n require_v ab_fw-la 14_o 6_o cb_n 9_o 127_o dg_v 42_o 374_o dc_o 23_o 7363_o bh_n 35_o 604_o &_o dp_v 76_o 99675._o upon_o the_o 35._o figure_n alb._n girard_n this_o question_n be_v defective_a in_o regard_n it_o have_v but_o four_o term_n know_v dp_v 70_o rod_n &_o cb_n 10_o the_o reason_n of_o dc_o to_o bg_n be_v as_o 10_o to_o 9_o and_o the_o name_n of_o the_o figure_n undecagonall_a which_o be_v but_o 4_o term_n he_o come_v to_o the_o 〈◊〉_d set_v down_o the_o face_n 20_o rod_n to_o see_v if_o there_o be_v not_o in_o it_o repugnancy_n as_o if_o the_o question_n be_v exceed_v so_o that_o he_o in_o this_o search_n commit_v the_o faulth_n which_o be_v call_v petitio_fw-la principij_fw-la which_o be_v speak_v not_o to_o defame_v the_o author_n but_o to_o show_v how_o this_o proceed_v from_o hence_o that_o in_o this_o time_n many_o thought_n that_o geometry_n have_v attain_v to_o her_o high_a degree_n though_o we_o have_v have_v but_o the_o a_o b_o c_o of_o it_o till_o that_o some_o undertake_n the_o restauration_n thereof_o have_v in_o part_n make_v it_o flourish_v again_o can_v nevertheless_o escape_v the_o blame_n of_o some_o envious_a &_o ignorant_a man_n in_o this_o divine_a science_n as_o happen_v often_o have_v let_v pass_v the_o most_o difficult_a of_o the_o analitica_fw-la which_o ought_v to_o adorn_v those_o that_o make_v profession_n thereof_o who_o contrariwise_o set_v down_o the_o limit_n of_o their_o pretension_n from_o the_o begin_n of_o their_o course_n by_o emulate_v one_o a_o other_o 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flank_v be_v right_o the_o defence_n flank_v &_o the_o fichant_n 45_o &_o 60_o rod_n the_o gorge_n in_o the_o flank_n be_v as_o 4_o to_o 3._o see_v that_o the_o angle_n dcf_n be_v 60_o degree_n dh_n 45_o will_v be_v the_o double_a of_o graccus_n 22_o ½_o rod_n dh_n hk_n be_v know_v dk_n will_v be_v 55_o 621_o dr_n 38_o 971_o than_o the_o second_o flank_n 16_o 65._o also_o de_fw-fr will_v be_v find_v by_o the_o triangle_n dae_n &_o take_v from_o dr_n there_o will_v remain_v agnostus_n the_o basis_n of_o the_o

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the_o face_n will_v be_v 240_o 78_o and_o vc_n 62_o 32_o which_o be_v take_v from_o vd_n there_o will_v remain_v cd_o the_o flank_n 85._o 7_o and_o therefore_o the_o triangle_n rightangle_n cdi_o shall_v have_v 3_o term_n the_o angle_n c_o 75_o &_o cd_o 85_o 7._o so_o diego_n will_v be_v 319_o 84_o ci_o 331_o 15_o likewise_o by_o flanck_v 571_o 93._o &_o i_o z_o 29._o 03_o for_o the_o second_o flank_v now_o open_v be_v know_v as_o be_v the_o half_a of_o bh_n now_o of_o the_o pentagons_n or_o fortress_n with_o five_o angle_n or_o bulwark_n 6._o figure_n &_o 2._o plate_n in_o the_o pentagone_n a_o fortress_n with_o five_o bulwark_n kfbdl_n let_v the_o line_n kl_n be_v 63_o rod_n &_o divide_v into_o 7_o equal_a part_n whereof_o the_o capital_a line_n ka_o be_v 2_o of_o they_o also_o le_z and_o from_o the_o character_n 2_o or_o g_z let_v gb_n be_v the_o perpendicular_a also_o the_o angle_n flank_v be_v 69_o degree_n according_a to_o the_o precedent_a table_n how_o many_o then_o will_v be_v the_o line_n and_o the_o angle_n of_o such_o a_o fortress_n alb._n girard_n before_o we_o come_v to_o the_o construction_n hereof_o take_v notice_n that_o ka_o in_o this_o figure_n be_v not_o answerable_a to_o the_o length_n of_o kg_n yet_o you_o must_v suppose_v they_o to_o be_v equal_a aswell_o as_o the_o cipher_n 1_o and_o the_o letter_n h_n be_v two_o differ_a point_n which_o will_v have_v be_v the_o better_a discern_v if_o the_o figure_n have_v be_v well_o make_v which_o may_v serve_v as_o a_o forewarning_a for_o some_o figure_n follow_a upon_o which_o one_o ought_v not_o to_o stand_v so_o much_o as_o upon_o the_o supposition_n or_o hypoteses_n of_o the_o proposition_n the_o triangle_n rightangle_n kha_n have_v three_o term_n ka_o 18_o rod_n the_o angle_n akh_n 54_o degree_n which_o be_v of_o kh_n will_v be_v 10_o 5802_o &_o ha_o or_o gb_n 14_o 5623._o now_o if_o you_o take_v away_o kh_n from_o kg_n there_o will_v remain_v for_o ab_fw-la the_o gorge_n 7._o 4198._o item_n the_o 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dg_n 32._o 69912_o to_o which_o add_v twice_o bq_n and_o you_o shall_v have_v bv_o 72_o 40711_o its_o half_a bk_n 36_o 20355_o and_o so_o the_o triangle_n rightangle_n bko_n have_v 3_o term_n the_o angle_n kbo_n will_v be_v 54_o degree_n you_o shall_v find_v boy_n 61_o 593099_o and_o quoth_fw-mi 49_o 82984._o final_o ub_z bq_n be_v know_v than_o the_o triangle_n right_a angle_n dqv_n have_v 3_o term_n give_v vq_n &_o qd_n you_o shall_v find_v the_o fichant_n dv_o easy_o alb._n girard_n if_o one_o will_v consider_v the_o difference_n which_o be_v between_o this_o operation_n and_o that_o of_o our_o author_n publish_v in_o his_o former_a edition_n he_o shall_v find_v that_o this_o question_n be_v defective_a for_o i_o have_v add_v thereunto_o that_o the_o second_o flank_n agnostus_n be_v rod_n see_v that_o in_o all_o such_o question_n where_o there_o be_v a_o second_o flank_n he_o ought_v to_o have_v set_v down_o 5_o know_v term_n &_o neither_o more_o nor_o less_o without_o the_o one_o 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as_o much_o as_o be_v lawful_a and_o possible_a to_o show_v his_o intention_n which_o i_o be_o bind_v to_o do_v moreover_o if_o peradventure_o same_o one_o shall_v find_v this_o manner_n of_o operation_n strange_a which_o i_o 〈…〉_z heitherto_o it_o be_v requisite_a for_o he_o to_o know_v also_o be_v it_o speak_v under_o ●●●●ection_n that_o he_o understand_v not_o much_o if_o he_o do_v not_o practice_v it_o so_o himself_o in_o other_o subject_n yea_o though_o i_o have_v not_o set_v down_o one_o only_a number_n which_o i_o shall_v have_v do_v already_o heretofore_o if_o i_o have_v not_o have_v regard_n to_o the_o obscurity_n which_o student_n may_v pretend_v to_o find_v which_o have_v clear_v heitherto_o you_o may_v go_v on_o with_o the_o more_o assurance_n in_o the_o rest_n follow_a and_o must_v also_o know_v that_o the_o 8_o figure_n be_v leave_v without_o explication_n by_o the_o author_n the_o 8._o 9_o 10_o figure_n &_o 3_o plate_n in_o this_o figure_n pentagonal_a the_o angle_n elo_n be_v divide_v into_o 2_o equal_a part_n by_o ld_n the_o flank_n fb_n 9_o rod_n the_o angle_n flank_v 69_o degree_n and_o the_o curtain_n 30_o rod_n how_o many_o then_o will_v the_o other_o line_n &_o angle_n be_v the_o angle_n of_o the_o polygone_n be_v part_v in_o the_o midst_n by_o ka_o and_o the_o angle_n akf_n be_v know_v also_o the_o rest_n shall_v be_v fkg_v or_o its_o equal_a fb_n whence_o follow_v that_o the_o triangle_n rightangle_n fbi_n shall_v have_v 3_o term_n fb_n and_o the_o angle_n therefore_o the_o other_o 3_o term_n will_v be_v know_v also_o the_o whole_a line_n db_n and_o the_o

gorge_n it_o be_v better_a to_o make_v the_o say_a cat_n upon_o the_o curtain_n yet_o so_o that_o they_o be_v no_o hindrance_n to_o the_o round_n go_v along_o the_o rampart_n for_o which_o reason_n they_o ought_v to_o be_v make_v more_o inward_a and_o so_o that_o they_o may_v lie_v as_o near_o the_o rampart_n as_o possible_o may_v be_v the_o better_a to_o discouver_fw-fr and_o command_v the_o field_n about_o they_o the_o 74._o 75._o figure_n the_o figure_n 74_o be_v the_o profile_n of_o the_o rampart_n falsebray_v its_o parapet_n moat_n covert_a way_n and_o its_o parapet_n whereof_o the_o length_n breadth_n height_n and_o depth_n be_v mark_v out_o in_o the_o say_a figure_n 74_o which_o be_v a_o section_n of_o the_o bulwarck_n as_o the_o figure_n 75_o the_o section_n of_o the_o midst_n of_o the_o rampart_n and_o because_o all_o the_o profile_n can_v commodious_o be_v draw_v out_o we_o have_v represent_v but_o a_o part_n of_o the_o moat_n the_o form_n of_o the_o raveline_n its_o height_n 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figure_n show_v on_o the_o outside_n you_o shall_v make_v a_o parallel_n of_o 10_o rod_n for_o the_o breadth_n of_o the_o moat_n and_o upon_o the_o brink_n of_o it_o to_o the_o tenaille_n you_o shall_v lay_v out_o the_o ravelin_n as_o we_o have_v say_v in_o the_o former_a figure_n 73_o in_o the_o angle_n of_o the_o bulwark_n you_o shall_v make_v also_o ravelin_n such_o as_o you_o see_v here_o mark_v with_o ay_o k_o l_o m_o which_o shall_v be_v defend_v by_o the_o ravelin_n e_o f_o g_o h_o &_o the_o better_a to_o defend_v the_o say_v ravelin_n which_o be_v in_o the_o angle_n of_o the_o bulwark_n you_o shall_v make_v the_o other_o work_v n_o o_o p_o &_o q_o which_o be_v call_v horne-work_n which_o be_v make_v in_o such_o a_o manner_n that_o the_o moat_n of_o these_o hornework_n come_v to_o answer_v to_o the_o falsebray_v which_o be_v in_o the_o flank_n of_o the_o bulwark_n so_o that_o their_o breadth_n saint_n t_o make_v about_o 32_o rod_n of_o which_o distance_n the_o two_o 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moat_n the_o true_a dimension_n thereof_o be_v clear_o express_v in_o the_o say_v profile_v by_o the_o help_n of_o the_o scale_n join_v to_o it_o as_o also_o by_o the_o mean_n of_o the_o alphabetical_a letter_n show_v the_o foot_n to_o be_v 12_o ynche_n whereof_o the_o length_n be_v 〈…〉_z in_o the_o 25_o plate_n of_o our_o geometry_n the_o benefit_n of_o such_o work_n be_v well_o know_v when_o they_o be_v make_v in_o place_n of_o advantage_n for_o deffence_n or_o where_o no_o man_n victual_n and_o ammunition_n of_o war_n be_v want_v as_o also_o where_o the_o ground_n be_v of_o a_o reasonable_a largeness_n to_o wit_n at_o the_o least_o 32_o rod_n that_o the_o face_n of_o the_o say_a horneworke_n may_v be_v about_o 12_o rod_n which_o be_v the_o least_o length_n that_o one_o can_v give_v to_o such_o work_n against_o the_o attempt_n which_o may_v be_v give_v by_o a_o army_n &_o as_o a_o great_a assault_n can_v be_v resist_v with_o a_o few_o man_n but_o up_o a_o small_a room_n so_o be_v it_o manifest_a that_o the_o greatness_n &_o largeness_n of_o such_o a_o place_n must_v be_v answerable_a therefore_o in_o my_o opinion_n it_o be_v a_o thing_n repugnant_a to_o the_o rule_n of_o fortification_n to_o make_v such_o horneworke_n in_o the_o angle_n of_o bulwark_n where_o they_o be_v so_o straighten_v for_o demy-bulwarke_n can_v by_o reason_n of_o their_o smallnesse_n be_v well_o maintain_v and_o defend_v and_o on_o the_o other_o side_n their_o two_o wing_n can_v be_v defend_v from_o the_o main_a fortress_n but_o with_o great_a disadvantage_n as_o we_o intend_v to_o discuss_v thereof_o more_o at_o large_a hereafter_o in_o the_o plateforme_a of_o gulick_n how_o citadel_n or_o castle_n may_v be_v join_v to_o town_n or_o town_n to_o castle_n the_o 16._o plate_n and_o the_o 81._o 82._o figure_n for_o to_o build_v a_o castle_n or_o a_o citadel_n to_o a_o town_n either_o to_o help_v to_o defend_v it_o the_o better_a or_o to_o curb_v it_o you_o must_v find_v out_o first_o the_o most_o advantageous_a place_n in_o case_n there_o be_v a_o river_n than_o you_o shall_v build_v your_o castle_n upon_o the_o side_n of_o it_o where_o it_o may_v best_o command_v with_o all_o the_o advantage_n that_o possible_o may_v be_v also_o take_v heed_n that_o it_o be_v build_v in_o such_o a_o place_n that_o the_o castle_n may_v receive_v no_o disadvantage_n thereby_o but_o that_o the_o rampart_n of_o the_o town_n where_o the_o letter_n a_o g_o be_v be_v the_o weak_a part_n to_o the_o end_n that_o by_o this_o mean_n you_o may_v frustrate_v the_o desire_n of_o the_o inhabitans_fw-la of_o the_o say_a town_n from_o oppose_v themselves_o against_o the_o say_a castle_n but_o

lay_v out_o the_o two_o side_n a_o b_o &_o c_o d_o aught_o to_o be_v lengthen_v so_o far_o that_o one_o may_v make_v the_o angle_n b_o k_o l_o &_o c_o r_o p_o of_o 75_o degree_n and_o that_o the_o line_n of_o defence_n o._n k._n termine_v in_o the_o curtain_n m._n q._n and_o to_o know_v the_o length_n of_o the_o face_n and_o flank_n the_o face_n shall_v be_v make_v double_a in_o the_o flank_n which_o be_v do_v if_o you_o place_v upon_o the_o perpendicular_a o_fw-fr n_o the_o half_a of_o the_o line_n of_o defense_n o_o k._n as_o from_o o_o to_o n._n and_o from_o the_o point_n k_o the_o privy_a right_a line_n n_o k_o be_v draw_v cut_v through_o the_o line_n c_o b_o at_o m_n you_o shall_v draw_v the_o line_n m_o l_o parallel_n to_o n_o o_o or_o perpendicular_a upon_o c_z b_o and_o so_o the_o flank_n l_o m_o wilbe_n the_o half_a of_o the_o face_n l_o k._n and_o thus_o we_o have_v finish_v the_o fortification_n of_o the_o abovesaid_a quadrilatere_a irregular_a which_o in_o my_o opinion_n 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the_o flank_n t_o can_v discover_v the_o angle_n flank_v f_o but_o if_o the_o expense_n be_v not_o too_o much_o which_o happen_v when_o one_o make_v the_o moat_n very_o deep_a one_o might_n to_o that_o end_n cut_v the_o part_n x_o y_fw-fr z_o and_o so_o this_o inconveniency_n will_v be_v remedy_v for_o i_o find_v that_o such_o bulwark_n be_v far_o better_o than_o those_o that_o be_v make_v upon_o a_o angle_n because_o the_o gorge_n be_v very_o large_a the_o flank_n the_o face_n and_o the_o curtain_n of_o a_o competent_a measure_n and_o according_a to_o our_o former_a rule_n to_o wit_n the_o line_n of_o defense_n fichant_n be_v 60_o rod_n or_o thereabouts_o the_o curtain_n 36_o the_o flank_n 12_o the_o face_n 24_o and_o the_o gorge_n well_o nigh_o 17_o rod_n which_o be_v much_o better_a and_o large_a then_o in_o the_o bulwark_n that_o be_v make_v upon_o some_o angle_n so_o that_o such_o bulwark_n ought_v to_o be_v prefer_v before_o the_o other_o be_v it_o not_o for_o some_o other_o 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which_o be_v the_o distance_n of_o each_o angle_n or_o the_o centre_n of_o the_o bulwark_n give_v for_o the_o capital_a line_n 28_o 97._o what_o will_v the_o distance_n of_o the_o centre_n of_o the_o bulwark_n give_v that_o which_o this_o rule_n will_v produce_v wilbe_n the_o capital_a line_n and_o in_o the_o like_a manner_n shall_v you_o find_v the_o line_n of_o the_o gorge_n the_o flank_n and_o the_o face_n of_o the_o bulwark_n h_o &_o g_z the_o bulwark_n i_o shall_v be_v make_v in_o the_o midst_n of_o the_o curtain_n m_o n._n or_o in_o

great_a mathematician_n consume_v his_o time_n whole_o in_o this_o point_n by_o maintain_v that_o all_o fortification_n aswell_o great_a as_o small_a ought_v to_o be_v make_v in_o a_o square-forme_n but_o see_v that_o the_o bulwark_n towards_o the_o 4_o angle_n by_o this_o mean_n become_v less_o forcible_a than_o the_o other_o as_o the_o figure_n a_o and_o b_o demonstrate_v in_o the_o 23_o plate_n i_o think_v this_o aught_o to_o be_v take_v into_o consideration_n whither_o it_o be_v not_o better_o to_o make_v a_o fortress_n whereof_o the_o bulwark_n and_o the_o line_n of_o defence_n be_v of_o a_o like_a force_n then_o to_o make_v they_o as_o above_o for_o it_o be_v impossible_a that_o one_o shall_v make_v a_o fortress_n strong_a in_o one_o place_n but_o you_o must_v diminish_v the_o strength_n thereof_o in_o a_o other_o place_n to_o wit_n as_o the_o common_a proverb_n be_v one_o ought_v to_o cut_v his_o coat_n according_a to_o his_o cloth_n but_o when_o the_o situation_n of_o the_o 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dp_n be_v 72_o &_o bh_n or_o fk_v 32_o than_o the_o half_a of_o the_o rest_n will_v be_v 20_o for_o df_n and_o also_o the_o angle_n abf_n be_v 60_o degree_n and_o adc_n 37_o 1_o 2_o than_o cdf_n will_v be_v 22_o 1_o 2_o degree_n whereby_o you_o shall_v know_v then_o the_o triangle_n rightangle_n cdf_n second_o fc_a find_v with_o cb_n give_v will_v make_v know_v fb_n but_o pf_n be_v 52_o than_o the_o triangle_n rightangle_n bfp_n have_v the_o side_n bf_n fp_n know_v &_o bp_o will_v be_v know_v in_o the_o triangle_n rightangle_n ade_n cbg_n their_o term_n will_v suffice_v to_o make_v the_o rest_n to_o be_v understand_v when_o you_o have_v da_fw-la its_o half_n will_v be_v for_o de_fw-fr which_o take_v from_o df_n there_o will_v remain_v ab_fw-la dg_n 21_o 64780_o fb_n 16_o 64800_o bp_o 54_o 60000_o ad_fw-la 19_o 22344_o de_fw-fr 9_o 61172_o ab_fw-la 10_o 38828_o gh_n 11_o 80804_o dg_n 43_o 50349_o the_o 13._o figure_n in_o this_o figure_n hexagonall_a the_o second_o flank_n be_v in_o the_o flank_n as_o 6_o to_z 7_o the_o flank_n have_v its_o gorge_n as_o 7_o to_z 10_o the_o gorge_n to_o the_o line_n of_o the_o polygone_a as_o 2_o to_o 9_o the_o question_n be_v how_o many_o they_o &_o the_o other_o dimension_n will_v be_v when_o as_o dh_n make_v 60_o rod_n if_o you_o make_v gh_a 6_o then_o ho_o will_v be_v 7_o &_o he_o 10_o the_o curtain_n will_v be_v 25_o then_o you_o shall_v calculate_v the_o triangle_n cbg_n dag_n by_o which_o you_o have_v the_o angle_n g_o or_o