then_o count_v the_o perpendicular_a 50_o â…_n on_o one_o leg_n and_o half_a the_o distance_n between_o the_o rafter-feet_n or_o half_a breadth_n of_o the_o frame_n at_o that_o place_n on_o the_o other_o leg_n and_o the_o parallel_n distance_n between_o shall_v be_v the_o true_a length_n of_o the_o rafter_n require_v then_o for_o the_o hip_n length_n first_o make_v be_o and_o bm_v equal_a to_o ae_n or_o be_v at_o one_o end_n and_o make_v cn_fw-la and_o dn_n equal_a to_o df_n or_o cf_n and_o draw_v the_o line_n mm_o and_o nn_n at_o both_o end_n also_o make_v ab_n and_o bb_n equal_a to_o ab_fw-la and_o ca_o and_o da_n equal_a to_o cd_o and_o draw_v the_o diagonal_a line_n as_o and_o b_n and_o co_n and_o do_v at_o each_o end_n extend_v and_o set_v down_o the_o measure_n of_o they_o these_o line_n be_v draw_v you_o have_v the_o half_a diagonal_a line_n as_o and_o b_n and_o co_n and_o do_v and_o may_v measure_v they_o by_o your_o scale_n to_o find_v their_o length_n or_o set_v the_o rule_n to_o the_o angle_n ae_z and_o bes_z find_v they_o by_o the_o 30_o scale_n also_o you_o have_v the_o bevel-end_n of_o the_o frame_n and_o the_o near_a distance_n over_o from_o side_n to_o side_n of_o the_o frame_n then_o for_o the_o hip_n length_n and_o angle_n thus_o set_v the_o 30_o scale_n square_a and_o count_v the_o perpendicular_a height_n lg_n on_o one_o leg_n 50_o â…_n and_o each_o half_a diagonal_a line_n etc._n etc._n one_o after_o another_o on_o the_o other_o leg_n and_o the_o parallel_n distance_n between_o shall_v be_v the_o true_a length_n of_o the_o several_a hip_n require_v example_n the_o parallel_n distance_n between_o 5_o foot-o-inche_n â…_n the_o common_a perpendicular_a count_v on_o one_o 30_o scale_n and_o 8_o foot_n 1_o inch_n the_o long_a diagonal_a line_n as_o count_v on_o the_o other_o 30_o scale_n shall_v give_v 9_o foot_n 6_o inch_n for_o the_o length_n of_o one_o hippolito_n ap_n lateral_o and_o the_o parallel_n distance_n between_o 5_o 0-â…_a the_o common_a 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30_o scale_n it_o shall_v give_v the_o true_a angle_n of_o the_o hip_n at_o the_o raising-piece_n and_o king-post_n which_o 8_o angle_n in_o these_o 4_o hip_n be_v express_v by_o the_o line_n and_o letter_n in_o the_o scheme_n thus_o pas_n and_o pb_n pdo_n and_o pco_n the_o 4_o angle_n at_o the_o raising-piece_n and_o spa_n and_o spb_n opc_n and_o opd_v the_o 4_o angle_n at_o the_o king-post_n who_o length_n you_o may_v prove_v by_o mr._n pope_n excellent_a way_n making_z sp_z and_o open_v equal_a to_o lg_n on_o the_o extend_a diagonal_a line_n and_o draw_v the_o line_n ap_n bp_o cp_n dp_n for_o the_o 4_o hip-rafter_n length_n require_v or_o for_o more_o proof_n thus_o also_o by_o the_o rule_n as_o before_o in_o square_a roof_n set_a the_o 30_o scale_n square_a and_o take_v the_o parallel_n extent_n from_o 7_o foot_n 2¾_a the_o rafter_n length_n at_o the_o broad_a end_n on_o one_o 30_o scale_n to_o 6_o foot_n 2_o inch_n more_o than_o ½_o the_o bevel-end_n by_o 1_o foot_n the_o half_a quantity_n of_o bevel_v and_o it_o shall_v give_v 9_o foot_n 6_o inch_n for_o the_o hip_n ap_n and_o from_o 72¾_a to_o 4-2_a one_o foot_n less_o than_o the_o half_a bevel-end_n to_o 8_o foot_n ¼_n the_o hip_n bp_o also_o the_o par._n extent_n from_o 6_o foot_n 5½_a the_o rafter_n length_n at_o the_o lesser_a end_n to_z 5_o foot_n 0¾_a one_o foot_n more_o than_o 4-0-¾_a the_o half_n little_a bevel-end_n shall_v give_v 8_o 1½_a for_o the_o hip_n pc_n and_o the_o extent_n from_o 6-5-4_a 8_o the_o rafter_n length_n to_o 30¾_a 1_o foot_n less_o than_o the_o half_a bevel-end_n give_v 71½_a the_o hip_n dp_n as_o before_o note_v that_o by_o the_o work_a these_o 3_o way_n you_o may_v be_v sure_a to_o prevent_v any_o mistake_n that_o may_v happen_v in_o work_v one_o way_n only_o last_o for_o the_o angle_n on_o the_o outside_n or_o back_n of_o the_o hip_n take_v every_o several_a hip_n length_n lateral_o between_o your_o compass_n and_o set_v one_o point_n in_o the_o length_n of_o the_o bevel-end_n or_o rather_o in_o a_o mean_a between_o ab_fw-la the_o bevel-end_n and_o mm_o at_o the_o great_a end_n or_o cd_o and_o nn_n at_o the_o lesser_a end_n and_o open_a or_o shut_v the_o 30_o scale_n till_o the_o other_o point_n fall_v on_o the_o hip_n length_n that_o you_o work_v for_o then_o for_o the_o long_a hip_n count_v more_o and_o for_o the_o short_a hip_n less_o by_o half_a the_o inch_n bevel_v than_o the_o whole_a breadth_n over_o at_o the_o end_n and_o take_v the_o parallel_n near_a distance_n from_o thence_o to_o the_o 30_o scale_n for_o a_o near_a distance_n which_o near_a distance_n you_o must_v keep_v then_o take_v the_o whole_a diagonal_a line_n perpendicular_a to_o the_o hip_n wrought_v for_o viz._n the_o short_a diagonal_a line_n for_o the_o long_a hip_n at_o each_o end_n between_o your_o compass_n and_o make_v it_o a_o parallel_n in_o the_o near_a distance_n last_o find_v and_o then_o the_o 30_o scale_n be_v set_v to_o the_o angle_n of_o the_o back_n of_o the_o hip_n require_v example_n in_o this_o figure_n for_o the_o long_a hip._n take_v 9_o foot_n 6_o inch_n the_o hip_n length_n ap_n between_o your_o compass_n and_o set_v one_o point_n in_o 10_o foot_n one_o inch_n a_o mean_a between_o ab_fw-la 10_o foot_n 4_o and_o mm_o 9-10_a ½_o and_o open_a or_o shut_v the_o rule_n till_o the_o other_o point_n stand_v in_o 9_o foot_n 6_o the_o hip_n length_n first_o take_v then_o the_o near_a distance_n from_o 9_o foot_n 4_o inch_n one_o foot_n less_o than_o 10_o f._n 4_o inch._n the_o bevel-end_n distance_n over_o to_o the_o other_o 30_o scale_n shall_v give_v 7_o 10_o a_o near_a distance_n from_o the_o point_n b_o to_o ap_n the_o hippolito_n raise_v over_o as_o the_o diagonal_a then_o the_o lateral_a extent_n bb_v of_o the_o whole_a diagonal_a line_n 12_o f._n 7_o inch._n be_v make_v a_o parallel_n in_o 7_o 10_o the_o near_a distance_n last_o find_v shall_v set_v the_o 30_o scale_n to_o the_o angle_n at_o r_o the_o back_n of_o the_o hip_n require_v measure_v by_o take_v parallel_n 30_o and_o measure_v at_o lateral_o on_o the_o chord_n it_o be_v about_o 108_o degree_n second_o take_v 8_o f._n 4_o inches_n the_o other_o hip_n length_n between_o your_o compass_n and_o set_v 1_o point_n in_o 10_o f._n 1_o inch_n as_o before_o open_a or_o shut_v the_o rule_n till_o the_o other_o point_n fall_v in_o 8_o 4_o the_o hip_n length_n then_o the_o near_a distance_n from_o 11_o foot_n 4_o inch_n 1_o foot_n more_o than_o 10_o foot_n 4_o inch_n the_o distance_n of_o the_o bevel-end_n to_o the_o common_a line_n of_o the_o other_o 30_o scale_n shall_v be_v when_o measure_v lateral_o from_o the_o centre_n 9_o foot_n fere_n for_o a_o near_a distance_n then_o 15_o foot_n 8_o inch_n the_o long_a diagonal_a line_n be_v make_v a_o parallel_n in_o 9_o foot_n fere_n the_o near_a distance_n last_o find_v set_v the_o 30_o scale_n to_o 121_o degree_n the_o angle_n require_v the_o back_n of_o the_o short_a hip_n require_v the_o same_o work_n serve_v for_o the_o other_o end_n be_v near_o the_o same_o angles_n which_o you_o may_v prove_v by_o m._n will._n pope_n excellent_a way_n thus_o find_v the_o middle_n between_o saint_n and_o a_o or_o saint_n and_o b_o at_o q_n than_o the_o near_a distance_n from_o q_o to_o bp_o or_o ap_n near_o lie_v to_o r_o and_o draw_v the_o line_n re_fw-mi rm_n for_o the_o angle_n at_o r_o the_o back_n of_o the_o hip_n require_v moreover_o if_o you_o raise_v 4_o perpendicular_o cut_v the_o point_n o_o and_o s_o the_o two_o place_n of_o the_o king-post_n be_v perpendicular_a to_o the_o raising-piece_n ac_fw-la end_n bd_o as_o the_o
4_o prick-line_n ♉_o saint_n â™_o s_o ♌_o o_o and_o ♊_o o_fw-fr do_v show_v and_o lay_v the_o length_n of_o each_o hip_n from_o his_o proper_a corner_n abcd_v as_o ap_n from_o a_o to_o ♈_o and_o â™_o bp_o from_o b_o to_o ♈_o and_o ♉_o dp_v from_o d_o to_z ♊_o and_o ♋_o cp_n from_o c_z to_z ♌_o and_o ♋_o then_z draw_z line_n from_o point_n to_o point_n as_o in_o the_o figure_n then_o â™_o ♌_o and_o ♉_o ♊_o be_v the_o two_o ridge_n when_o turn_v right_o over_o os_fw-la and_o c_o ♋_o d_o be_v the_o least_o hip_n and_o a_o ♈_o b_o be_v the_o great_a hip_n as_o mr._n pope_n have_v well_o show_v thus_o much_o for_o hipt_v roof_n use_v xvii_o to_o find_v the_o length_n and_o angle_n of_o every_o principal_a particular_a rafter_n in_o frames_n broad_a at_o one_o end_n than_o the_o other_o the_o perpendicular_a as_o before_o be_v hint_v be_v to_o be_v the_o same_o all_o over_o the_o roof_n therefore_o open_a the_o rule_n square_n and_o take_v from_o the_o perpendicular_a on_o one_o leg_n to_o the_o half_a breadth_n of_o the_o frame_n on_o the_o other_o leg_n measure_v it_o from_o the_o centre_n and_o that_o be_v the_o length_n require_v for_o the_o angle_n lay_v a_o rule_n to_o the_o compass-point_n and_o set_v a_o bevel_n as_o before_o be_v show_v and_o you_o have_v the_o angle_n at_o the_o raising-piece_n and_o ridge_n of_o the_o house_n to_o cut_v the_o rafter_n foot_n by_o the_o same_o rule_n serve_v to_o draw_v out_o a_o pair_n of_o well-stair_n to_o give_v hypothenusaes_n or_o string_n at_o any_o particular_a height_n and_o breadth_n for_o the_o 30_o scale_n set_v square_a and_o the_o perpendicular_a height_n count_v on_o one_o leg_n and_o the_o breadth_n on_o the_o other_o leg_n the_o measure_n between_o be_v always_o the_o hypothenusa_n or_o string_n in_o fly_a stair_n as_o may_v plain_o appear_v use_v xviii_o to_o find_v the_o length_n and_o angle_n of_o collar-beam_n in_o any_o roof_n take_v the_o whole_a breadth_n of_o the_o frame_n between_o your_o compass_n and_o set_v one_o point_n in_o the_o length_n of_o the_o rafter_n on_o one_o leg_n and_o the_o other_o point_n in_o the_o same_o place_n on_o the_o either_o leg_n then_o the_o two_o leg_n represent_v the_o two_o principal_a rafter_n and_o a_o rule_n lay_v to_o the_o compass-point_n represent_v the_o raising-piece_n then_o at_o any_o height_n that_o you_o please_v above_o the_o raising-piece_n apply_v a_o rule_n parallel_n to_o it_o and_o the_o measure_n between_o lay_v from_o the_o centre_n give_v the_o length_n remember_v to_o add_v wood_n for_o the_o tenon_n and_o a_o bevel_n lay_v to_o the_o 30_o scale_n and_o rule_n give_v the_o true_a angle_n to_o cut_v it_o by_o where_o the_o timber_n be_v square_a use_v xix_o to_o find_v the_o length_n and_o angle_n of_o rafter_n and_o purloyn_n in_o bevel_n frames_n the_o length_n of_o the_o rafter_n be_v show_v before_o to_o find_v it_o by_o the_o half_a breadth_n of_o the_o square_a or_o bevel-end_n and_o the_o perpendicular_a answerable_a to_o that_o roof_n as_o afterward_o in_o the_o example_n and_o the_o angle_n of_o the_o foot_n and_o outside_n or_o back_n of_o the_o bevel-end_n rafter_n and_o the_o upright_a of_o the_o gable_n end_n must_v be_v to_o a_o angle_n less_o and_o more_o than_o 90_o degree_n by_o the_o angle_n at_o the_o corner_n of_o the_o frame_n where_o that_o rafter_n be_v to_o stand_v be_v more_o at_o the_o sharp_a angle_n and_o less_o at_o the_o blunt_a angle_n as_o in_o figure_n iv_o you_o may_v see_v the_o true_a quantity_n of_o which_o angle_n be_v thus_o find_v by_o the_o rule_n take_v the_o length_n of_o the_o rafter_n for_o the_o bevel-end_n in_o foot_n and_o inch_n and_o make_v it_o a_o parallel_n in_o 15_o then_o half_a the_o quantity_n of_o foot_n and_o inch_n bevel_v take_v from_o the_o same_o scale_n and_o carry_v parallelly_n till_o it_o stay_v in_o like_a part_n shall_v show_v right_a against_o it_o in_o the_o tangent_n the_o degree_n and_o minute_n require_v and_o this_o be_v the_o angle_n that_o the_o blunt_a corner_n be_v to_o be_v lay_v in_o legement_n more_o than_o a_o square_n and_o the_o sharp_a angle_n less_o than_o a_o square_n or_o 90_o degree_n both_o out_o of_o level_n and_o out_o of_o square_a also_o when_o you_o tumble_v or_o stripe_n in_o the_o tenon_n of_o the_o purloin_v the_o thing_n desire_v example_n let_v ab_fw-la represent_v the_o bevel-end_n of_o a_o frame_n be_v out_o of_o square_n from_o the_o line_n ac_fw-la 6_o foot_n as_o the_o line_n cb_n show_v then_o if_o ac_fw-la be_v 20_o foot_n ab_fw-la will_v be_v 20_o foot_n 10_o inch_n then_o draw_v el_n the_o middle_a line_n of_o the_o frame_n and_o gm_n and_o hk_n the_o line_n at_o three_o quarter_n of_o the_o breadth_n of_o the_o frame_n then_o take_v ed_z the_o half_a bevel_n and_o lay_v it_o from_o g_o to_o f_o and_o from_o h_n to_o i_o and_o draw_v the_o line_n of_o and_o by_o for_o the_o outside_n line_n of_o the_o two_o bevel-end_n rafter_n end_v the_o two_o other_o line_n parallel_v to_o they_o according_a to_o the_o breadth_n or_o scantling_n of_o your_o bevel-end_n rafter_n as_o here_o in_o the_o figure_n 8_o inch_n broad_a thus_o the_o line_n of_o and_o by_o represent_v the_o two_o end_n rafter_n lay_v in_o legement_n to_o fit_v in_o the_o purloin_v as_o for_o their_o lie_v out_o of_o square_n from_o the_o raising-piece_n and_o to_o the_o same_o angle_n they_o be_v to_o be_v lay_v out_o at_o level_n that_o the_o cut_n of_o the_o purloin_v end_n may_v fit_v the_o rafter_n side_n when_o erect_v in_o their_o place_n according_a as_o the_o end_v f_o and_o i_o of_o the_o bevel-end_n rafter_n be_v according_a to_o the_o angel_n afg_n and_o afm_n the_o one_o be_v 11_o degree_n and_o 20_o minute_n under_o and_o the_o other_z 11_o 20_o above_o 90_o degree_n also_o note_v that_o if_o pk_a and_o om_fw-la do_v represent_v a_o pair_n of_o square_a rafter_n at_o any_o intend_a distance_n from_o a_o then_o it_o be_v and_o rq_n will_v give_v the_o true_a length_n of_o the_o purloin_v fit_a for_o those_o place_n rq_n be_v the_o short_a purloin_v and_o it_o be_v the_o long_a purloin_v to_o find_v by_o the_o rule_n only_o how_o long_o the_o purloin_v must_v be_v on_o the_o outside_n more_o or_o less_o than_o the_o distance_n on_o the_o raising-piece_n where_o you_o intend_v the_o two_o square_a rafter_n foot_n shall_v stand_v do_v thus_o set_v the_o 30_o scale_n to_o the_o same_o angle_n that_o the_o bevel-end_n rafter_n lie_v out_o of_o square_a when_o they_o lie_v in_o legement_n to_o frame_v which_o here_o be_v to_o 11_o degree_n and_o 20_o minute_n then_o count_v from_o the_o centre_n the_o quantity_n of_o foot_n and_o inch_n you_o intend_v to_o make_v the_o mortise-hole_n from_o the_o rafter_n foot_n in_o the_o rafter_n for_o the_o tenon_n of_o your_o purloin_v and_o take_v from_o thence_o to_o the_o near_a distance_n to_o the_o other_o line_n and_o that_o shall_v show_v the_o quantity_n that_o the_o one_o purloin_v be_v to_o be_v long_o and_o the_o other_o short_a than_o the_o distance_n between_o the_o rafter-feet_n on_o the_o raise_v piece_n example_n thus_o set_v the_o 30_o scale_n to_o the_o angle_n gaf_n then_o count_v as_o the_o place_n for_o the_o mortise-hole_n from_o a_o on_o the_o rafter_n and_o take_v the_o near_a distance_n from_o thence_o to_o the_o other_o 30_o scale_n that_o shall_v give_v sv_o the_o quantity_n how_o much_o it_o be_v be_v short_a than_o ao_o and_o how_o much_o rq_n be_v long_o than_o pb_n note_v that_o if_o you_o count_v 3_o quarter_n of_o 20_o foot_n 10_o inch_n be_v the_o length_n of_o the_o bevel-end_n that_o then_o you_o will_v make_v the_o rafter_n too_o long_o by_o 4_o inch_n and_o a_o half_a as_o in_o the_o figure_n you_o may_v see_v therefore_o the_o sure_a way_n to_o find_v the_o rafter_n length_n be_v to_o set_v the_o scale_n to_o a_o square_a and_o then_o to_o take_v the_o parallel_n extent_n from_o the_o common_a perpendicular_a to_o the_o ½_o breadth_n between_o the_o place_n for_o the_o rafter-feet_n on_o the_o raising-piece_n so_o that_o the_o length_n of_o the_o rafter_n for_o the_o bevel-end_n at_o true_a pitch_n be_v of_o 15_o foot_n 3_o inch_n and_o not_o awe_o 157½_a which_o be_v just_a 3_o quarter_n of_o ab_fw-la the_o bevel-end_n over_o which_o the_o two_o rafter_n be_v to_o stand_v for_o then_o the_o top_n of_o the_o bevel-end_n rafter_n will_v be_v too_o high_a for_o the_o top_n of_o the_o square_a rafter_n be_v cut_v just_a 15_o foot_n three_o quarter_n of_o 20_o foot_n the_o breadth_n of_o the_o frame_n use_v xx._n the_o use_n of_o the_o scale_n to_o lay_v down_o or_o measure_v out_o on_o paper_n
theorem_fw-la we_o have_v suppose_v the_o figure_n of_o all_o the_o material_n different_a from_o those_o in_o the_o second_o so_o likewise_o we_o must_v now_o change_v the_o posture_n as_o will_v appear_v in_o the_o theorem_fw-la follow_v theorem_fw-la 4._o if_o the_o material_n figure_v as_o before_o wedg-wise_a shall_v not_o be_v dispose_v levelly_a but_o in_o form_n of_o some_o arch_n or_o proportion_n of_o a_o circle_n point_v all_o to_o the_o same_o centre_n in_o this_o case_n neither_o the_o piece_n of_o the_o say_a arch_n can_v sink_v downward_o through_o want_n of_o room_n to_o descend_v perpendicular_o nor_o the_o supporter_n or_o butment_n as_o they_o be_v term_v of_o the_o say_a arch_n can_v suffer_v so_o much_o violence_n as_o in_o the_o precedent_n flat_a posture_n for_o the_o roundness_n will_v always_o make_v the_o incumbent_a weight_n rather_o to_o rest_v upon_o the_o supporter_n than_o to_o shove_v they_o whence_o may_v be_v draw_v a_o evident_a corollary_n that_o the_o safe_a of_o all_o arch_n be_v the_o semicircular_a and_o of_o all_o vault_n the_o hemisphere_n though_o not_o absolute_o exempt_v from_o some_o natural_a weakness_n as_o barn._n baldi_n abbot_n of_o guastalla_n in_o his_o comment_n upon_o aristotle_n mechanic_n do_v very_o well_o prove_v whence_o i_o note_v that_o when_o any_o thing_n be_v mathematical_o demonstrate_v weak_a it_o be_v much_o more_o mechanical_o weak_a error_n ever_o more_o occur_v more_o easy_o in_o the_o management_n of_o gross_a material_n than_o lineal_a design_n theorem_fw-la 5._o as_o semicircular_a arch_n or_o hemispherical_a vault_n be_v raise_v upon_o the_o total_a diameter_n be_v of_o all_o other_o the_o round_a and_o consequent_o the_o sure_a by_o the_o precedent_n theorem_fw-la so_o those_o be_v the_o graceful_a which_o keep_v precise_o the_o same_o height_n shall_v yet_o be_v distend_v one_o fourteen_o part_n long_o than_o the_o say_v entire_a diameter_n which_o addition_n of_o distent_n will_v confer_v much_o to_o their_o beauty_n and_o detract_v but_o little_a from_o their_o strength_n this_o observation_n i_o find_v in_o leon._n baptista_n alberti_fw-la but_o the_o practice_n how_o to_o preserve_v the_o same_o height_n and_o yet_o distend_v the_o end_n of_o the_o arch_n be_v in_o albert_n durer_n geometry_n who_o teach_v the_o italian_n many_o a_o excellent_a line_n of_o great_a use_n in_o this_o art._n upon_o these_o five_o theorem_n all_o the_o skill_n of_o arch_v and_o vault_a be_v ground_v as_o for_o those_o arch_n which_o our_o artisan_n call_v of_o the_o three_o and_o four_o point_n and_o the_o tuscan_a writer_n di_o terzo_n and_o di_fw-it quarto_fw-la acuto_fw-la because_o they_o always_o concur_v in_o a_o acute_a angle_n and_o do_v spring_n from_o the_o division_n of_o the_o diameter_n into_o three_o four_o or_o more_o part_n at_o pleasure_n i_o say_v â—hese_n both_o for_o the_o natural_a imbecility_n of_o the_o sharp_a angle_n itself_o and_o likewise_o for_o their_o very_a uncomliness_n ought_v to_o be_v exile_v from_o judicious_a eye_n and_o leave_v to_o their_o first_o inventor_n the_o goth_n or_o lombard_n among_o other_o relic_n of_o that_o barbarous_a age._n thus_o of_o my_o first_o partition_n of_o the_o part_n of_o every_o fabric_n into_o five_o head_n have_v go_v through_o the_o two_o former_a and_o be_v incident_o carry_v into_o this_o last_o doctrine_n touch_v arch_n and_o vault_n the_o next_o now_o in_o order_n be_v the_o apertion_n under_o which_o term_n i_o do_v comprehend_v door_n window_n staircase_n chimney_n or_o other_o conduct_n in_o short_a all_o inlet_n or_o outlet_n to_o which_o belong_v two_o general_a caution_n first_o that_o they_o be_v as_o few_o in_o number_n and_o as_o moderate_v in_o dimension_n as_o possible_o may_v consist_v with_o other_o due_a respect_n for_o in_o a_o word_n all_o open_n be_v weakning_n second_o that_o they_o do_v not_o approach_v too_o near_o the_o angle_n of_o the_o wall_n for_o it_o be_v indeed_o a_o most_o essential_a solecism_n to_o weaken_v that_o part_n which_o must_v strengthen_v all_o the_o rest_n a_o precept_n well_o record_v but_o ill_o practise_v by_o the_o italian_n themselves_o particular_o at_o venice_n where_o i_o have_v observe_v divers_a pergoli_n or_o meniana_n as_o vitruvius_n seem_v to_o call_v they_o which_o be_v certain_a ballised_a outstanding_n to_o satisfy_v curiosity_n of_o sight_n very_o dangerous_o set_v forth_o upon_o the_o very_a point_n itself_o of_o the_o muraâ—_n angle_n before_o i_o come_v to_o the_o cast_n and_o compart_v of_o the_o whole_a work_n be_v indeed_o the_o very_a definitive_a sum_n of_o this_o art_n to_o distribute_v useful_o and_o graceful_o a_o well_o choose_a plot_n i_o shall_v collect_v some_o note_n belong_v to_o these_o particular_a overture_n of_o door_n and_o window_n these_o inlet_n of_o man_n and_o light_n i_o couple_v together_o because_o i_o find_v their_o due_a dimension_n bring_v under_o one_o rule_n by_o leon._n alberti_n a_o learned_a searcher_n who_o from_o the_o school_n of_o pythagoras_n where_o it_o be_v a_o fundamental_a maxim_n that_o the_o image_n of_o all_o thing_n be_v latent_fw-la in_o number_n do_v determine_v the_o comely_a proportion_n between_o breadth_n and_o height_n namely_o the_o symmetry_n of_o two_o to_o three_o in_o their_o breadth_n and_o length_n in_o other_o the_o double_a as_o two_o to_o four_o there_o will_v indubitable_o result_v from_o either_o a_o graceful_a and_o harmonious_a contentment_n to_o the_o eye_n our_o master_n vitruvius_n seem_v to_o have_v be_v a_o extreme_a lover_n of_o luminous_a room_n and_o indeed_o i_o confess_v that_o a_o frank_a light_n can_v misbecome_v no_o edifice_n yet_o on_o the_o other_o side_n we_o must_v take_v heed_n to_o make_v a_o house_n all_o eye_n like_o argus_n which_o in_o northern_a climate_n will_v be_v too_o cold_a in_o southern_a too_o hot_a beside_o there_o be_v no_o part_n of_o building_n more_o expenceful_a than_o window_n or_o more_o ruinous_a not_o only_o for_o that_o vulgar_a reason_n as_o be_v expose_v to_o all_o wind_n and_o weather_n but_o because_o consist_v of_o so_o different_a and_o unsociable_a piece_n as_o wood_n iron_n lead_v and_o glass_n and_o those_o small_a and_o weak_a and_o easy_o shake_v of_o door_n there_o be_v this_o distinction_n some_o be_v call_v foreste_n some_o valvae_fw-la those_o as_o the_o word_n may_v seem_v to_o import_v do_v open_a outward_n these_o inward_o and_o be_v common_o of_o two_o leaf_n or_o pane_n as_o we_o call_v they_o thereby_o require_v indeed_o a_o lesser_a circle_n in_o their_o unfolding_n and_o therefore_o much_o in_o use_n among_o the_o italian_n at_o this_o day_n but_o i_o charge_v they_o with_o a_o imperfection_n for_o though_o they_o let_v in_o as_o well_o yet_o they_o keep_v out_o worse_o of_o staircase_n to_o make_v a_o complete_a staircase_n be_v a_o curious_a piece_n of_o architecture_n the_o vulgar_a caution_n be_v these_o that_o it_o have_v a_o liberal_a light_n against_o all_o casualty_n of_o slip_n and_o fall_v that_o the_o space_n above_o the_o head_n be_v large_a and_o airy_a because_o a_o man_n do_v spend_v much_o breadth_n in_o breathe_v that_o the_o half_a pace_n be_v well_o distribute_v at_o competent_a distance_n for_o repose_v on_o the_o way_n that_o to_o avoid_v encounter_n and_o beside_o to_o gratify_v the_o beholder_n the_o whole_a staircase_n have_v no_o niggard_n latitude_n that_o be_v for_o the_o principal_a ascent_n in_o royal_a building_n at_o the_o least_o ten_o foot_n that_o the_o breadth_n of_o every_o single_a step_n or_o stair_n be_v never_o less_o than_o one_o foot_n nor_o more_o than_o eighteen_o inch_n that_o they_o exceed_v by_o no_o mean_n half_o a_o foot_n in_o their_o height_n or_o thickness_n for_o our_o leg_n do_v labour_n more_o in_o elevation_n than_o in_o distension_n that_o the_o step_n be_v lay_v where_o they_o join_v somewhat_o slope_v that_o the_o foot_n may_v in_o a_o sort_n ascend_v and_o descend_v together_o which_o though_o observe_v by_o few_o be_v a_o secret_a and_o delicate_a deception_n of_o the_o pain_n in_o mount_v last_o to_o reduce_v this_o doctrine_n to_o some_o natural_a or_o at_o least_o mathematical_a ground_n vitruvius_n borrow_v these_o proportion_n that_o make_v the_o side_n of_o a_o rectangular_a triangle_n that_o be_v three_o for_o the_o perpendicular_a from_o the_o stair-head_n to_o the_o ground_n four_o for_o the-ground-line_n itself_o or_o recession_n from_o the_o wall_n and_o five_o for_o the_o whole_a inclination_n or_o slopeness_n in_o the_o ascent_n there_o be_v likewise_o spiral_n or_o cocklestairs_n and_o sometime_o run_v about_o a_o pillar_n sometime_o vacant_a wherein_o palladio_n a_o man_n very_o expert_a in_o this_o point_n be_v wont_a to_o divide_v the_o diameter_n of_o the_o first_o fort_n into_o three_o part_n yield_v one_o to_o the_o pillar_n and_o two_o to_o the_o step_v of_o the_o second_o into_z four_z
distance_n from_o a_o point_n to_o a_o line_n be_v only_o thus_o set_v one_o point_n of_o the_o compass_n in_o the_o point_n give_v and_o open_a or_o shut_v the_o other_o be_v turn_v about_o till_o the_o other_o will_v but_o just_a touch_n or_o cleave_v the_o line_n that_o i_o call_v the_o near_a distance_n use_v i._o to_o lay_v down_o a_o line_n that_o shall_v represent_v any_o number_n of_o foot_n and_o inch_n give_v or_o require_v take_v the_o number_n give_v lateral_o from_o the_o 30_o or_o 40_o scale_n from_o the_o centre_n and_o that_o be_v the_o line_n require_v but_o if_o these_o scale_n be_v too_o great_a or_o too_o small_a then_o take_v your_o number_n of_o part_n and_o the_o length_n thereof_o lateral_o as_o for_o example_n suppose_v i_o will_v have_v 3_o inch_n to_o represent_v 30_o foot_n take_v out_o 3_o inch_n between_o your_o compass_n and_o make_v it_o a_o parallel_n in_o 30_o and_o 30_o and_o the_o 30_o scale_n be_v set_v to_o your_o desire_n use_v ii_o to_o increase_n or_o diminish_v a_o line_n to_o any_o proportion_n take_v the_o give_v line_n between_o your_o compass_n and_o make_v it_o a_o parallel_n in_o the_o part_n thereof_o then_o the_o parallel_n extent_n of_o the_o part_n you_o will_v have_v it_o increase_v or_o diminish_v to_o be_v the_o augmentation_n or_o diminution_n which_o be_v require_v example_n let_v 3_o inch_n represent_v 8_o foot_n and_o to_o the_o same_o proportion_n i_o will_v have_v 10_o foot_n or_o 5_o foot_n viz._n more_o or_o less_o take_v 3_o inch_n between_o your_o compass_n and_o make_v it_o a_o parallel_n in_o 8_o and_o 8_o on_o the_o 30_o scale_n then_o the_o parallel_n distance_n between_o 5_o and_o 5_o do_v diminish_v the_o line_n and_o the_o parallel_n between_o 10_o and_o 10_o do_v increase_v the_o line_n to_o the_o proportion_n require_v use_v iii_o to_o divide_v a_o line_n into_o any_o number_n of_o part_n or_o model_n under_o 30._o take_v the_o give_v line_n and_o make_v it_o a_o parallel_n in_o the_o part_n on_o the_o 30_o scale_n into_o which_o you_o will_v have_v it_o divide_v than_o the_o parallel_n extent_n between_o 1_o and_o 1_o shall_v divide_v the_o line_n according_o example_n let_v 4_o inch_n be_v a_o line_n to_o be_v divide_v into_o 9_o part_n take_v 4_o inch_n or_o any_o distance_n whatsoever_o and_o make_v it_o a_o parallel_n in_o 9_o and_o 9_o on_o the_o 30_o scale_n then_o the_o parallel_n distance_n between_o 1_o and_o 1_o shall_v divide_v 4_o inch_n into_o 9_o part_n require_v note_v that_o for_o more_o exactness_n and_o conveniency_n you_o may_v double_v 9_o or_o triple_a 9_o viz._n 18_o or_o 27_o and_o then_o if_o you_o make_v the_o line_n to_o be_v divide_v a_o parallel_n in_o triple_a the_o number_n you_o must_v take_v out_o 3_o in_o stead_n of_o one_o and_o that_o shall_v divide_v the_o line_n into_o the_o part_n require_v example_n i_o will_v have_v 5_o inch_n put_v into_o 10_o part_n take_v 5_o inch_n between_o your_o compass_n and_o make_v it_o a_o parallel_v in_o 30_o and_o 30_o the_o triple_a of_o 10_o then_o take_v out_o parallel_v 3_o and_o 3_o the_o triple_a of_o one_o and_o that_o shall_v divide_v the_o line_n give_v be_v 5_o inch_n into_o 10_o part_n or_o model_n exact_o the_o like_a for_o any_o other_o and_o note_v as_o the_o rule_n stand_v you_o may_v take_v out_o any_o number_n of_o part_n or_o model_n whatsoever_o to_o that_o scale_n use_v iu._n any_o two_o line_n give_v to_o find_v their_o proportion_n one_o to_o another_o according_a to_o any_o other_o number_n take_v the_o line_n several_o and_o lay_v they_o lateral_o from_o the_o centre_n on_o the_o 30_o or_o the_o 40_o scale_n which_o you_o please_v and_o the_o number_n of_o foot_n and_o inch_n to_o which_o they_o reach_v shall_v show_v their_o proportion_n one_o to_o another_o of_o the_o part_n of_o the_o line_n on_o which_o they_o be_v measure_v example_n i_o have_v 2_o line_n suppose_v one_o be_v 2_o inch_n long_o and_o the_o other_o 5_o or_o any_o other_o unknown_a part_n take_v 2_o inch_n the_o measure_n of_o one_o line_n and_o measure_v it_o lateral_o on_o the_o 30_o scale_n and_o it_o give_v 5_o foot_n ½_o a_o inch_n then_o take_v out_o 5_o inch_n the_o suppose_a length_n of_o the_o other_o line_n and_o it_o give_v lateral_o 12_o foot_n 8_o inch_n then_o i_o say_v one_o line_n be_v 5_o foot_n and_o half_a a_o inch_n and_o the_o other_o be_v 12_o foot_n 8_o inch_n of_o a_o scale_n of_o 30_o foot_n in_o 11_o inch_n 3_o quarter_n length_n or_o if_o you_o conclude_v on_o the_o term_n or_o number_n of_o one_o line_n then_o make_v that_o line_n a_o parallel_n in_o the_o part_n thereof_o then_o take_v the_o other_o line_n and_o carry_v it_o parallelly_n till_o it_o stay_v in_o like_a part_n on_o both_o leg_n in_o the_o common_a line_n and_o that_o shall_v be_v the_o denomination_n of_o the_o other_o line_n use_v v._o two_o line_n be_v give_v to_o find_v a_o three_o in_o continual_a proportion_n to_o they_o take_v both_o the_o line_n and_o lay_v they_o lateral_o on_o both_o leg_n and_o note_v the_o foot_n and_o inch_n to_o which_o they_o do_v extend_v as_o in_o the_o last_o then_o take_v out_o the_o lateral_a extent_n of_o the_o second_o line_n and_o make_v a_o parallel_n in_o the_o term_n of_o the_o first_o line_n keep_v the_o common_a line_n at_o that_o open_n than_o the_o parallel_n extent_n from_o the_o term_n of_o the_o second_o line_n shall_v be_v the_o lateral_a 3d_o term_n or_o line_n in_o proportion_n example_n suppose_v i_o have_v one_o line_n 3_o foot_n long_o and_o another_o 5_o foot_n and_o i_o will_v have_v another_o to_o bear_v proportion_n to_o 5_o as_o 3_o do_v to_o 5_o increase_a be_v in_o number_n thus_o as_o 3_o be_v to_o 5_o so_o be_v 5_o to_o what_o here_o note_v that_o 3_o be_v the_o first_o number_n and_o 5_o the_o second_o the_o first_o line_n lay_v from_o the_o centre_n on_o the_o 30_o scale_n extend_v to_o 7_o foot_n 7_o inch_n and_o 5_o inch_n the_o second_o line_n give_v 12_o foot_n 8_o inch_n now_o the_o lateral_a second_o line_n viz._n 5_o or_o 12_o foot_n 8_o inch_n make_v a_o parallel_n 7_o in_o foot_n 7_o inch_n the_o term_n of_o the_o first_o line_n then_o take_v out_o the_o parallel_a extent_n from_o 12_o foot_n 8_o inch_n the_o measure_n of_o 5_o and_o it_o shall_v give_v 21_o foot_n 1_o inch_n lay_v lateral_o from_o the_o centre_n for_o a_o three_o proportional_a require_v which_o measure_v on_o the_o inch_n be_v 8_o inch_n and_o a_o three_o the_o answer_n require_v for_o as_o 3_o be_v in_o proportion_n to_o 5_o so_o be_v 5_o to_o 8_o and_o a_o three_o part_n but_o by_o the_o line_n of_o number_n have_v the_o quantity_n of_o the_o line_n give_v in_o number_n do_v thus_o the_o extent_n of_o the_o compass_n from_o the_o first_o number_n 3_o to_o the_o second_o number_n 5_o shall_v reach_v the_o same_o way_n from_o the_o second_o number_n 5_o to_o 8.33_o the_o three_o proportional_a number_n require_v use_v vi_o to_o divide_v a_o line_n in_o such_o sort_n as_o another_o line_n be_v divide_v take_v the_o whole_a line_n that_o be_v divide_v and_o lay_v it_o lateral_o on_o both_o leg_n and_o fit_v the_o line_n that_o be_v to_o be_v divide_v parallelly_n in_o the_o end_n thereof_o then_o lay_v every_o part_n of_o the_o divide_a line_n lateral_o in_o like_a manner_n as_o the_o whole_a line_n be_v lay_v and_o the_o parallel_a extent_n between_o those_o part_n shall_v divide_v the_o line_n according_o as_o for_o example_n suppose_v i_o will_v divide_v a_o line_n of_o 8_o inch_n ¼_n long_o in_o such_o sort_n as_o the_o line_n of_o circle_n on_o the_o inside_n of_o the_o 30_o scale_n be_v divide_v take_v out_o 8_o inch_n ¾_n and_o make_v it_o a_o parallel_n in_o 2_o and_o 2_o the_o divide_a line_n then_o take_v out_o the_o parallel_a distance_n from_o 3_o and_o 3_o and_o that_o shall_v give_v the_o point_n 3_o from_o the_o end_n of_o the_o line_n you_o will_v divide_v and_o so_o consequent_o all_o the_o rest_n in_o order_n as_o far_o as_o you_o please_v use_v vii_o to_o find_v a_o mean_a proportional_a between_o two_o line_n or_o number_n open_v the_o 30_o scale_n to_o a_o right_a angle_n by_o make_v lateral_a 21_o foot_n 2_o inch_n a_o parallel_n in_o 15_o foot._n then_o find_v the_o sum_n and_o half_a sum_n the_o difference_n and_o half_a difference_n between_o your_o two_o number_n and_o have_v the_o half_a sum_n between_o your_o compass_n set_v one_o point_n to_o the_o half_a difference_n count_v lateral_o on_o one_o leg_n and_o wheresoever_o the_o other_o point_n shall_v touch_v the_o common_a line_n on_o the_o other_o
the_o rafter_n require_v hip_n 2._o for_o the_o hip_n length_n count_v the_o length_n of_o the_o rafter_n last_o find_v on_o one_o leg_n and_o the_o half_a breadth_n of_o the_o house_n on_o the_o other_o leg_n and_o take_v the_o parallel_a distance_n between_o and_o measure_v it_o from_o the_o centre_n lateral_o and_o it_o shall_v be_v the_o true_a length_n of_o the_o hip_n require_v diagonal_a 3._o for_o the_o diagonal_a line_n count_v the_o half_a breadth_n of_o the_o house_n on_o both_o leg_n and_o take_v the_o parallel_a distance_n between_o and_o measure_v it_o from_o the_o centre_n and_o it_o shall_v be_v the_o length_n of_o the_o diagonal_a line_n from_o the_o corner_n to_o the_o king-post_n length_n 4._o again_o for_o the_o hip_n count_v the_o diagonal_a line_n last_o find_v on_o one_o leg_n and_o the_o perpendicular_a height_n on_o the_o other_o leg_n and_o the_o parallel_a distance_n between_o shall_v be_v the_o hip_n true_a length_n measure_v as_o before_o from_o the_o centre_n example_n in_o a_o house_n of_o 20_o foot_n wide_a at_o true_a pitch_n see_v fig._n 1._o let_v abcd_n represent_v a_o frame_n of_o a_o house_n 20_o foot_n wide_a by_o the_o half_a width_n basilius_n and_o cd_o equal_a to_o bc_n the_o whole_a width_n be_v 2_o point_n to_o draw_v the_o diagonal_a line_n by_o je_n be_v equal_a to_o ib_n and_o the_o half_a ei_o lay_v from_o e_z to_z g_z gives_z ig_v the_o true_a length_n of_o the_o rafter_n and_o gb_n or_o gc_fw-la the_o length_n of_o the_o hip_n or_o thus_o three_o quarter_n of_o cb_n viz._n ch_n give_v cf_n and_o bf_n the_o rafter_n length_n the_o same_o extent_n also_o lay_v upon_o the_o middle_a line_n from_o i_o to_o g_o gives_z cg_n and_o bg_n the_o hip_n length_n if_o be_v the_o perpendicular_a height_n e_z be_v the_o point_n of_o the_o diagonal_a line_n or_o king-post_n perpendicular_o opposite_a to_o or_o right_n under_o the_o meeting_n point_n of_o the_o 2_o hip_n and_o the_o 2_o rafter_n when_o raise_v and_o set_v in_o their_o place_n thus_o much_o for_o illustration_n what_o to_o do_v now_o for_o application_n how_o to_o do_v the_o width_n of_o the_o house_n be_v always_o give_v the_o length_n of_o the_o rafter_n or_o the_o height_n of_o the_o perpendicular_a be_v next_o resolve_v on_o which_o in_o our_o example_n be_v true_a pitch_n be_v also_o resolve_v on_o either_o by_o inspection_n as_o before_o or_o else_o by_o operation_n in_o this_o manner_n 1._o the_o breadth_n of_o the_o house_n be_v 20_o foot_n and_o the_o perpendicular_a resolve_v on_o to_o be_v 11_o foot_n 2_o inch_n ½_o to_o find_v the_o rafter_n length_n work_v thus_o open_v the_o 30_o scale_n to_o a_o right_a angle_n by_o use_v the_o seven_o count_v 10_o the_o half_a breadth_n of_o the_o house_n on_o one_o leg_n length_n and_o the_o perpendicular_a height_n 11_o 2_o ½_o on_o the_o other_o leg_n viz._n both_o on_o the_o 30_o scale_n from_o the_o centre_n and_o take_v the_o parallel_a distance_n between_o they_o on_o the_o common_a line_n and_o measure_v it_o from_o the_o centre_n and_o it_o shall_v give_v just_a 15_o foot_n the_o length_n of_o the_o rafter_n require_v cf._n 2._o but_o if_o according_a to_o the_o width_n of_o any_o frame_n you_o resolve_v on_o the_o rafter_n length_n and_o will_v have_v the_o perpendicular_a height_n of_o the_o gable_n end_n than_o thus_o count_v the_o length_n of_o the_o rafter_n from_o the_o centre_n and_o take_v the_o lateral_a extent_n thereof_o being_z 15_o between_o your_o compass_n the_o compass_n be_v so_o set_v perpendicular_a set_v one_o point_n in_o 10_o the_o half_a width_n of_o the_o frame_n and_o turn_v the_o other_o point_n parallelly_n to_o the_o common_a line_n and_o there_o it_o shall_v show_v 11_o foot_n 2_o 1_o 7_o inch_n the_o true_a height_n of_o the_o perpendicular_a at_o true_a pitch_n require_v if_o 11_o foot_n 2_o inch_n 1_o 7._o 3._o for_o the_o length_n of_o the_o hip_n count_v the_o half_a breadth_n of_o the_o house_n on_o one_o leg_n and_o the_o rafter_n length_n on_o the_o other_o leg_n and_o take_v the_o parallel_a distance_n between_o and_o it_o shall_v give_v the_o true_a length_n of_o the_o hip_n require_v length_n as_o here_o the_o parallel_a distance_n between_o 15_o and_o 10_o shall_v be_v lateral_a 8_o the_o true_a length_n of_o the_o hip_n require_v in_o a_o square_a frame_n 20_o foot_n wide_o true_a pitch_n cg_n 4._o for_o the_o diagonal_a line_n from_o the_o corner_n to_o the_o king-post_n count_n the_o half_a width_n of_o the_o house_n line_n viz._n 10_o foot_n on_o both_o leg_n and_o take_v the_o parallel_n extent_n between_o and_o it_o shall_v reach_v from_o the_o centre_n to_o 14_o foot_n 1_o inch_n â…_n be_v the_o line_n ce_fw-fr in_o the_o figure_n 5._o otherwise_o for_o the_o hip_n length_n count_v the_o half_a diagonal_a on_o one_o leg_n 24_o 1_o â…_n and_o the_o perpendicular_a 11_o foot_n 2_o in._n inches_n 2_o 2_o 7_o on_o the_o other_o leg_n hip._n and_o the_o parallel_n distance_n between_o measure_v from_o the_o centre_n will_v give_v the_o true_a hip_n length_n 18_o viz._n the_o line_n cg_n thus_o much_o for_o the_o rafter_n and_o hip_n in_o square_a frames_n at_o true_a pitch_n the_o measure_n of_o who_o angle_n and_o length_n be_v as_o follow_v  _fw-fr  _fw-fr  _fw-fr f._n inc._n 180_o part_n f._n 100_o part_n deg_n min._n the_o breadth_n of_o the_o house_n be_v 20_o f._n and_o the_o two_o end_n square_a and_o the_o rafter_n it_o true_a pitch_n the_o length_n and_o angle_n of_o the_o frame_v be_v cf_n rafter_n 15_o 00_o 00_o 15_o 000_o rafter_n top_n foot_n 41_o 50_o cg_n hippolito_n 18_o 00_o 00_o 18_o 000_o 48_o 10_o if_o perpendicular_a 11_o 02_o 17_o 11_o 180_o hip_n top_n foot_n 51_o 38_o be_v ½_o diagonal_a 14_o 01_o 68_o 14_o 140_o 38_o 22_o by_o half_a breadth_n 10_o 00_o 00_o 10_o 000_o  _fw-fr  _fw-fr  _fw-fr bc_n whole_a breadth_n 20_o 00_o 00_o 20_o 000_o difference_n 13_o 16_o kg_n near_o dist._n 16_o 07_o 50_o 16_o 624_o  _fw-fr  _fw-fr  _fw-fr bd_o who_o will_v diagoras_n 28_o 03_o 36_o 28_o 281_o outsid_n ang_fw-mi 116_o 12_o the_o rule_n to_o find_v the_o angle_n of_o the_o rafter_n and_o hip_n 1._o for_o the_o angle_n that_o the_o rafter_n make_v at_o foot_n and_o head_n with_o the_o raising-piece_n and_o king-post_n do_v thus_o when_o the_o rule_n stand_v square_a in_o the_o 30_o scale_n then_o lay_v any_o straight_a piece_n to_o the_o compass_n point_v when_o one_o stand_v in_o the_o half_a breadth_n and_o the_o other_o point_n in_o the_o perpendicular_a then_o if_o you_o apply_v a_o bevel_n several_o to_o that_o straight_a edge_n and_o the_o 30_o scale_n at_o each_o end_n the_o one_o be_v the_o angle_n at_o foot_n 48_o 10_o and_o the_o other_o the_o angle_n at_o head_n viz._n 41_o 50._o 2._o for_o the_o angle_n at_o head_n or_o king-post_n and_o foot_n of_o the_o hip_n do_v thus_o the_o 30_o scale_n be_v set_v square_a set_v one_o point_n of_o the_o compass_n in_o the_o half_a diagonal-line_n on_o one_o 30_o scale_n and_o the_o other_o point_n in_o the_o perpendicular_a on_o the_o other_o 30_o scale_n and_o to_o those_o point_n lay_v any_o straight_a piece_n than_o set_v a_o bevel_n to_o that_o straight_a piece_n and_o each_o 30_o scale_n and_o the_o one_o shall_v be_v the_o angle_n at_o foot_n 38_o 22_o and_o the_o other_o the_o angle_n at_o the_o head_n viz._n 51_o 38_o than_o this_o last_o angle_n double_v and_o the_o difference_n between_o the_o angle_n at_o head_n and_o foot_n viz._n 13_o 16_o add_v make_v 1â—6_n 12_o the_o angle_n of_o the_o outside_n of_o the_o hip-rafter_n in_o a_o square_a frame_n at_o true_a pitch_n require_v 3._o a_o general_n way_n to_o find_v this_o outward_a angle_n of_o the_o hip_n or_o mould_n be_v thus_o in_o any_o pitch_n first_o you_o must_v find_v the_o near_a distance_n from_o one_o corner_n to_o the_o opposite_a hippolito_n set_v up_o or_o from_o a_o point_n in_o the_o raise_n piece_n as_o far_o from_o one_o corner_n as_o the_o house-end_n be_v broad_a as_o the_o point_n k_n or_o l_o in_o the_o figure_n may_v be_v do_v thus_o count_n the_o length_n of_o the_o hip_n on_o one_o leg_n and_o also_o take_v it_o between_o your_o compass_n and_o count_v the_o breadth_n of_o the_o house_n on_o the_o other_o leg_n and_o set_v one_o point_n of_o the_o compass_n in_o the_o hip_n length_n on_o one_o leg_n and_o the_o other_o point_n in_o the_o breadth_n of_o the_o house_n on_o the_o other_o leg_n alter_v the_o 30_o scale_n but_o not_o your_o compass_n then_o when_o the_o 30_o scale_n be_v so_o set_v one_o represent_v the_o raising-piece_n and_o the_o other_o the_o hip_n set_v up_o then_o the_o near_a distance_n from_o the_o breadth_n
of_o the_o house_n on_o one_o 30_o scale_n to_o the_o common-line_n on_o the_o other_o 30_o scale_n be_v the_o near_a distance_n require_v be_v measure_v from_o the_o centre_n 167½_a then_o take_v the_o whole_a diagonal_a line_n viz._n cl_n or_o bk_n from_o the_o centre_n lateral_o and_o make_v it_o a_o parallel_n in_o the_o near_a distance_n last_o find_v and_o that_o shall_v set_v the_o 30_o scale_n to_o the_o angle_n of_o the_o outside_n of_o the_o hip_n require_v which_o you_o may_v measure_v in_o degree_n thus_o take_v parallel_v 15_o as_o the_o rule_n stand_v and_o lay_v it_o from_o the_o centre_n and_o it_o shall_v reach_v to_o 116_o degree_n on_o the_o chord_n next_o one_o 30_o scale_n the_o angle_n of_o the_o outside_n of_o the_o hip_n require_v example_n and_o more_o brief_o in_o a_o house_n 20_o foot_n broad_a the_o house-end_n be_v 20_o foot_n broad_a the_o whole_a diagonal_a line_n be_v 28_o foot_n 3_o inch_n â…œ_n the_o hip-rafter_n 18_o foot._n take_v 18_o foot_n between_o your_o compass_n and_o set_v one_o point_n in_o 20_o and_o open_a or_o shut_v the_o rule_n till_o the_o other_o fit_v 18_o than_o the_o near_a distance_n from_o 20_o to_o the_o common-line_n will_v be_v 16_o foot_n 7_o inch_n 4_o 8._o then_o take_v 28_o foot_n 3_o inch_n â…_n the_o whole_a diagonal_a line_n and_o make_v it_o a_o parallel_n in_o 16_o foot_n 7_o inch_n ½_o the_o near_a distance_n and_o the_o 30_o scale_n be_v set_v to_o the_o angle_n require_v for_o if_o you_o take_v out_o parallel_v 15_o the_o chord_n of_o 60_o and_o measure_v it_o latteral_o from_o the_o centre_n it_o shall_v reach_v to_o 116_o the_o angle_n in_o degree_n and_o minute_n require_v note_n if_o the_o whole_a breadth_n and_o whole_a diagonal_a line_n be_v too_o large_a for_o your_o compass_n than_o the_o half_a breadth_n and_o half_a diagonal_a will_v do_v as_o well_o take_v the_o half_a length_n of_o the_o hip_n also_o between_o your_o compass_n and_o on_o the_o scale_n also_o and_o that_o shall_v set_v the_o scale_n to_o the_o same_o angle_n as_o before_o use_v xv._n to_o find_v the_o length_n and_o angle_n of_o the_o rafter_n and_o hip_n or_o sleeper_n in_o bevel_v frames_n at_o any_o piton_n 1._o for_o the_o length_n of_o the_o rafter_n set_v the_o 30_o scale_n square_a then_o count_v the_o half_a length_n of_o the_o bevel-end_n on_o one_o leg_n rafter_n be_v always_o more_o there_o than_o the_o half_a breadth_n and_o the_o perpendicular_a resolve_v on_o on_o the_o other_o leg_n then_o the_o parallel_n distance_n between_o measure_v lateral_o shall_v be_v the_o length_n of_o the_o rafter_n require_v and_o a_o ruler_n lay_v to_o the_o two_o point_n of_o the_o compass_n to_o set_v on_o the_o 30_o scale_n and_o a_o bevel_n set_v as_o before_o in_o square_a frames_n be_v show_v shall_v give_v the_o angle_n at_o head_n and_o foot_n require_v 2._o for_o the_o hip_n length_n count_v the_o rafter_n length_n on_o one_o leg_n and_o the_o half_a breadth_n of_o the_o bevel-end_n of_o the_o house_n more_o by_o half_a the_o number_n of_o inches_n bevel_v on_o the_o other_o leg_n and_o take_v the_o parallel_n distance_n between_o hip_n and_o measure_v it_o lateral_o from_o the_o centre_n and_o it_o shall_v be_v the_o length_n of_o the_o long_a hip-rafter_n and_o for_o the_o short_a hip_n count_v less_o than_o the_o bevel-end_n by_o half_a the_o number_n of_o inches_n bevel_v and_o that_o shall_v be_v the_o short_a hip_n require_v 3._o the_o inch_n or_o foot_n and_o inch_n of_o bevel_v be_v give_v to_z and_o how_o much_o one_o corner_n be_v under_o and_o the_o other_o over_o 90_o degree_n or_o just_o square_a open_v the_o 30_o scale_n frame_n and_o take_v the_o breadth_n of_o the_o house_n over_o at_o the_o near_a distance_n between_o your_o compass_n from_o the_o 30_o scale_n from_o the_o centre_n lateral_o and_o make_v it_o a_o parallel_n in_o 15_o and_o 15_o for_o 60_o of_o the_o chord_n then_o take_v the_o foot_n and_o inch_n bevel_v from_o the_o same_o 30_o scale_n lateral_o and_o carry_v it_o parallelly_n till_o it_o stay_v in_o like_a part_n then_o just_a against_o it_o on_o the_o degree_n or_o tangent_n be_v the_o degree_n and_o minute_n require_v that_o one_o corner_n be_v more_o and_o the_o other_o less_o than_o 90_o degree_n 4._o to_o find_v the_o diagonal_a line_n take_v the_o distance_n in_o the_o chord_n to_o the_o degree_n above_o or_o under_o 90_o last_o sound_n from_o the_o centre_n lateral_o and_o make_v it_o a_o parallel_n in_o 15_o and_o 15_o and_o then_o the_o 30_o scale_n be_v set_v to_o the_o angle_n the_o end_n be_v over_o diagonal-line_n or_o under_o 90_o degree_n then_o count_v the_o whole_a or_o half_a bevel-end_n on_o both_o leg_n and_o the_o parallel_n distance_n between_o shall_v show_v the_o length_n of_o the_o whole_a or_o half_a diagonal_a line_n measure_v from_o the_o centre_n note_v that_o when_o the_o rule_n stand_v at_o the_o blunt_a angle_n it_o give_v the_o long_a diagonal_a line_n and_o when_o it_o stand_v at_o the_o sharp_a angle_n it_o give_v the_o short_a diagonal_a line_n 5._o by_o the_o diagonal_a line_n and_o perpendicular_a to_o find_v the_o hip_n length_n and_o the_o angle_n at_o head_n and_o foot_n of_o the_o hip_n or_o sleeper_n count_n the_o half_a diagonal_a on_o one_o leg_n hip_n and_o the_o perpendicular_a height_n on_o the_o other_o leg_n the_o 30_o scale_n be_v square_a than_o the_o parallel_n distance_n between_o shall_v be_v the_o length_n of_o the_o hip_n require_v be_v long_o or_o short_a as_o the_o diagonal_a line_n be_v also_o a_o rule_n lay_v to_o the_o two_o point_n of_o the_o compass_n measure_v the_o parallel_n extent_n hip_n and_o a_o bevel_n lay_v to_o the_o rule_n and_o the_o two_o 30_o scale_n at_o each_o end_n give_v the_o angle_n at_o head_n and_o foot_n of_o hip_n require_v to_o find_v the_o near_a distance_n from_o the_o corner_n of_o the_o rombus_n to_o the_o opposite_a hippolito_n set_v up_o in_o his_o true_a place_n 6._o count_n the_o length_n of_o the_o hip_n on_o one_o 30_o scale_n and_o take_v that_o distance_n also_o between_o your_o compass_n lateral_o count_v also_o on_o the_o other_o 30_o scale_v the_o length_n of_o the_o bevel-end_n and_o there_o set_v one_o point_n of_o the_o compass_n and_o open_a or_o shut_v the_o rule_n till_o the_o other_o point_n fall_v in_o the_o length_n of_o the_o hip-rafter_n first_o count_v then_o one_o 30_o scale_n represent_v the_o raising-piece_n distance_n and_o the_o other_o the_o hip_n set_v up_o then_o the_o near_a distance_n from_o the_o breadth_n of_o the_o frame_n over_o at_o the_o bevel-end_n and_o more_o or_o less_o by_o half_a the_o foot_n and_o inch_n bevel_v to_o the_o common_a line_n of_o the_o other_o 30_o scale_n be_v measure_v from_o the_o centre_n shall_v be_v the_o near_a distance_n require_v to_o find_v the_o angle_n on_o the_o outside_n of_o the_o hip._n 7._o to_o find_v the_o outside_o angle_n of_o the_o long_a hip_n take_v the_o short_a diagonal_a line_n between_o your_o compass_n and_o make_v it_o a_o parallel_n in_o the_o near_a distance_n belong_v to_o that_o hip_n and_o the_o 30_o scale_n will_v be_v set_v to_o the_o angle_n require_v and_o to_o measure_v it_o take_v parallel_v 15_o and_o 15_o and_o measure_v it_o lateral_o from_o the_o centre_n in_o the_o chord_n angle_n and_o you_o shall_v have_v the_o measure_n of_o the_o angle_n require_v example_n in_o a_o house_n of_o 20_o foot_n over_o at_o near_a distance_n and_o 4_o foot_n or_o 48_o inch_n bevel_v out_o of_o square_n see_v fig._n ii_o let_v abcd_n represent_v a_o frame_n 20_o foot_n over_o and_o 4_o foot_n bevel_v the_o bevel_n end_n b_o c_o be_v long_o than_o right_o over_o by_o 5_o inch_n for_o if_o you_o set_v the_o rule_n square_a and_o take_v the_o parallel_n extent_n from_o 20_o the_o measure_n over_o to_z 4_o foot_n the_o measure_n of_o bevel_v and_o measure_v it_o lateral_o you_o shall_v find_v it_o reach_v lateral_o to_o 20_o foot_n and_o 5_o inch_n the_o true_a length_n of_o the_o bevel-end_n the_o bevel-end_n be_v 20_o 5_o inch_n the_o perpendicular_a resolve_v on_o which_o at_o true_a pitch_n ought_v to_o be_v about_o 11_o foot_n 5_o inch_n fere_n then_o first_o for_o the_o rafter_n length_n 1._o set_a the_o 30_o scale_n square_a and_o set_v one_o point_n in_o 11_o 5_o the_o perpendicular_a and_o the_o other_o point_n in_o 10_o foot_n 2_o ½_o the_o half_a bevel-end_n and_o to_o the_o compass-point_n lay_v a_o rule_n and_o to_o the_o rule_n and_o the_o 30_o scale_n at_o both_o end_v set_v a_o bevel_n angle_n and_o one_o shall_v be_v the_o angle_n at_o foot_n and_o the_o other_o at_o the_o angle_n at_o the_o top_n of_o the_o rafter_n
and_o the_o one_o angle_n will_v be_v 42_o degree_n for_o the_o foot_n and_o the_o other_o 48_o for_o the_o top_n of_o the_o rafter_n and_o the_o measure_n between_o the_o compass_n measure_v from_o the_o centre_n shall_v be_v 15_o foot_n 4_o inch_n the_o rafter_n length_n require_v as_o by_o inspection_n on_o the_o 40_o and_o 30_o scale_n you_o may_v see_v 2._o for_o the_o long_a hip_n set_v one_o point_n in_o 12_o foot_n 2_o inch_n and_o ½_o more_o by_o two_o foot_n than_o 10_o foot_n 2_o inch_n ½_o the_o half_a bevel-end_n and_o the_o other_o point_n in_o 15_o 4_o the_o rafter_n length_n and_o measure_v it_o from_o the_o centre_n it_o give_v 19_o foot_n 6_o inch_n the_o long_a hip._n again_o length_n set_v one_o point_n in_o 8_o foot_n 2_o inch_n ½_o 2_o foot_n short_a than_o the_o half_a bevel-end_n by_o 2_o foot_n the_o half_a of_o 4_o foot_n the_o bevel_v and_o the_o other_o point_n in_o 15_o foot_n the_o rafter_n length_n and_o measure_v it_o from_o the_o centre_n it_o give_v 17_o foot_n 4_o inch_n the_o length_n of_o the_o short_a hip._n 3._o for_o the_o length_n of_o both_o diagonal_a line_n set_v the_o scale_n of_o 30_o to_o the_o angle_n of_o the_o frame_n at_o each_o corner_n and_o the_o measure_n from_o the_o half_a or_o the_o whole_a bevel-end_n take_v parallelly_n shall_v be_v the_o length_n require_v of_o the_o half_a or_o whole_a diagonal_a line_n according_a as_o you_o take_v the_o whole_a or_o half_a bevel-end_n as_o here_o in_o our_o example_n the_o blunt-end_n be_v 101_o gr_n 30_o min._n or_o 11_o 30_o more_o than_o 90_o gr_n therefore_o take_v the_o distance_n from_o the_o centre_n to_o 101_o diagonal-line_n 30_o on_o the_o chord_n and_o make_v it_o a_o parallel_n in_o 15_o and_o 15_o the_o chord_n of_o 60_o then_o be_v the_o 30_o scale_n set_v to_o the_o angle_n of_o the_o blunt-end_n of_o the_o frame_n and_o the_o parallel_n distance_n between_o 20-5_a the_o whole_a bevel-end_n give_v 31_o foot_n 6_o inch_n the_o whole_a diagonal_a line_n b_o l_o or_o the_o parallel_n between_o 10_o 2_o ½_o give_v 15-09_a inch_n b_o e_o the_o half_a again_o the_o sharp_a end_n be_v 78_o 30_o 11_o degr_n 30_o less_o than_o 90_o than_o the_o lateral_a chord_n of_o 78_o 30_o make_v a_o parallel_n chord_n of_o 60_o at_o 15_o then_o be_v the_o rule_n set_v to_o the_o sharp_a end_n of_o the_o frame_n for_o the_o short_a diagonal_a line_n and_o the_o parallel_n distance_n between_o 10-2_a ½_o give_v 12._o foot_n 11._o the_o half_a or_o 25_o foot_n 10_o inch_n the_o whole_a diagonal_a line_n c_o f_o who_o half_a be_v c_o e_o the_o short_a whole_a and_o half_o diagonal_a line_n 4._o for_o the_o hip_n length_n and_o angle_n at_o foot_n and_o head._n set_a the_o 30_o scale_n square_a and_o count_v the_o short_a half_a diagonal_a 12_o foot_n 11_o on_o one_o leg_n and_o the_o perpendicular_a 11_o foot_n 5_o on_o the_o other_o leg_n length_n than_o the_o compass_n point_v so_o set_v lay_v a_o rulâ—_n and_o take_v the_o bevel_n at_o both_o end_n and_o it_o shall_v give_v the_o two_o angle_n at_o head_n and_o foot_n of_o the_o short_a hip_n and_o the_o same_o distance_n of_o the_o compass_n point_n shall_v be_v 17_o foot_n 4_o inch_n fere_n angle_n the_o hip-lenght_n as_o before_o and_o the_o angle_n at_o the_o top_n 48_o 30_o and_o at_o foot_n 41_o 30_o his_o compliment_n again_o set_v one_o point_n in_o 15_o foot_n 10_o the_o long_a half_a diagonal_a and_o the_o other_o point_n in_o 11_o foot_n 5_o inch_n the_o perpendicular_a and_o lay_v a_o rule_n to_o they_o and_o set_v the_o bevel_n to_o both_o end_n and_o you_o shall_v find_v 54_o gr_n the_o angle_n at_o the_o top_n and_o 36_o the_o angle_n at_o foot_n and_o the_o distance_n between_o the_o compass_n lay_v from_o the_o centre_n give_v 19_o foot_n 6_o inch_n the_o long_a hip._n 5._o for_o the_o outside_o angle_n of_o both_o hip_n the_o long_a first_o take_v 19_o 6_o between_o your_o compass_n the_o hip_n length_n from_o the_o 30_o scale_n set_v one_o point_n in_o 20_o foot_n 5_o the_o bevel-end_n and_o close_o the_o rule_n till_o the_o other_o point_n touch_v 19_o 6_o the_o hip_n length_n hip._n then_o take_v the_o near_a distance_n from_o 18_o foot_n 5_o 2_o foot_n less_o than_o 20_o foot_n 5_o the_o breadth_n of_o the_o bevel-end_n of_o the_o frame_n to_o the_o other_o 30_o scale_n and_o it_o be_v the_o near_a distance_n from_o the_o point_n of_o the_o rombus_n a_o to_o the_o hip_n bg_n set_v up_o 15_o foot_n 9_o inch_n then_o tâ—ke_v out_o 25_o foot_n 10_o the_o short_a diagonal_a and_o make_v it_o a_o parallel_n in_o 15_o 9_o the_o near_a distance_n and_o then_o the_o 30_o scale_n be_v set_v to_o the_o angle_n require_v for_o the_o outside_o of_o the_o long_a hippolito_n be_v 110_o degree_n for_o parallel_v 15_o measure_v lateral_o on_o the_o chord_n give_v 1100._o 6._o for_o the_o outside_o angle_n of_o the_o short_a hip._n take_v 17_o foot_n 4_o inch_n between_o your_o compass_n and_o set_v one_o point_n in_o 20_o foot_n 5_o the_o bevel-end_n and_o open_a or_o shut_v the_o rule_n till_o the_o other_o point_n reach_v 17_o 4_o on_o the_o other_o 30_o scale_n hip_n then_o the_o near_a distance_n from_o 22_o foot_n 2_o foot_n more_o than_o the_o breadth_n of_o the_o bevel-end_n of_o the_o frame_n to_o the_o common_a line_n on_o the_o other_o 30_o scale_n and_o that_o shall_v be_v the_o near_a distance_n from_o l_o to_o cg_n the_o short_a hip_n set_v up_o which_o be_v 18_o foot_n 9_o inch_n then_o take_v out_o 15_o foot_n 10_o inch_n the_o half_o great_a diagonal_a line_n because_o 31_o foot_n 8_o inch_n be_v more_o than_o the_o scale_n of_o 30_o and_o make_v it_o a_o parallel_n in_o 9_o 00_o the_o half_a of_o 18_o foot_n and_o the_o 30_o scale_n be_v set_v to_o the_o angle_n require_v viz._n 122_o degree_n for_o if_o you_o take_v out_o parallel_v 15_o and_o measure_v it_o in_o the_o chord_n lateral_o it_o shall_v be_v 122_o the_o angle_n require_v note_v here_o by_o the_o way_n the_o length_n of_o the_o bevel-end_n and_o the_o two_o diagonal_a line_n and_o the_o half_n of_o they_o and_o the_o blunt_a and_o sharp_a angle_n of_o the_o frame_n be_v give_v by_o the_o draught_n of_o the_o frame_n and_o they_o be_v first_o know_v the_o work_n be_v half_o do_v but_o if_o not_o give_v then_o use_v these_o direction_n which_o will_v help_v you_o to_o see_v the_o reason_n of_o plain_a triangle_n and_o the_o use_n of_o the_o scale_n use_v xvi_o to_o find_v the_o rafter_n hip_n and_o angle_n in_o bevel_n and_o taper_n frames_n be_v broad_a at_o one_o end_n than_o the_o other_o see_v fig._n iii_o first_o when_o the_o frame_n be_v broad_a at_o one_o end_n than_o the_o other_o than_o the_o middle_a breadth_n be_v to_o be_v the_o guide_n for_o the_o rafter_n length_n and_o the_o perpendicular_a to_o be_v equal_a to_o the_o middle_a rafter_n perpendicular_a on_o both_o end_n though_o one_o pair_n of_o rafter_n be_v long_o than_o another_o and_o the_o roof_n in_o wind_v thereby_o which_o wind_v may_v many_o time_n be_v remedy_v by_o some_o convenient_a artifice_n or_o other_o as_o the_o ingenious_a workman_n will_v soon_o perceive_v as_o thus_o in_o brief_a let_v abcd_n represent_v the_o frame_n of_o a_o house_n bevel_v at_o both_o end_n and_o broad_a by_o 2_o foot_n at_o one_o end_n than_o the_o other_o as_o here_o in_o the_o bevel_a figure_n be_v 20_o foot_n on_o one_o side_n and_o 24_o foot_n on_o the_o other_o side_n at_o one_o bevel-end_n 10_o foot_n 4_o inch_n and_o at_o the_o other_o end_n 8_o foot_n 1_o inch_n ½_o but_o at_o the_o near_a distance_n over_o only_o 10_o foot_n and_o 8_o foot._n first_o for_o your_o more_o apparent_a satisfaction_n draw_v the_o true_a form_n of_o the_o frame_n by_o as_o large_a a_o scale_n as_o you_o convenient_o can_v with_o the_o side_n and_o angle_n as_o exact_v as_o you_o can_v as_o abcd_v then_o draw_v the_o middle_a line_n of_o quite_o through_o the_o length_n and_o gh_a through_o the_o breadth_n of_o the_o frame_n perpendicular_a one_o to_o the_o other_o then_o measure_n gh_o as_o suppose_v 9_o foot_n then_o lay_v off_o the_o half_a of_o gh_o from_o h_n to_o i_o and_o k_o then_o take_v out_o ¾_n of_o gh_a and_o lay_v it_o from_o k_o and_o i_o to_o l_o 6_o foot_n 9_o inch_n for_o a_o pair_n of_o middling_a rafter_n for_o this_o taper-house_n lg_n be_v the_o common_a perpendicular_a at_o the_o middle_n and_o both_o end_n of_o the_o roof_n 5_o foot_n 0_o inch_n â…_n of_o a_o inch._n and_o for_o the_o principal_a rafter_n frame_n or_o other_o rafter_n open_v the_o 30_o scale_v square_a and_o