of_o one_o side_n and_o the_o half_a sum_n of_o the_o measure_n of_o all_o the_o side_n multiply_v together_o shall_v be_v the_o true_a area_n or_o content_v thereof_o all_o other_o figure_n whatsoever_o of_o how_o many_o side_n soever_o they_o be_v may_v be_v reduce_v to_o triangle_n or_o to_o trapeziaes_n and_o measure_v as_o before_o which_o kind_n of_o figure_n surveyor_n and_o builder_n oftentimes_o meet_v withal_o in_o their_o operation_n problem_n xi_o for_o the_o measure_n of_o on_o oval_n the_o best_a way_n oval_n be_v to_o reduce_v it_o to_o a_o circle_n thus_o divide_v the_o distance_n on_o the_o line_n of_o number_n between_o the_o length_n and_o the_o breadth_n of_o the_o oval_a into_o two_o equal_a part_n and_o the_o middle_a point_n where_o the_o compass_n stay_v on_o shall_v be_v the_o diameter_n of_o a_o circle_n equal_a in_o area_n to_o the_o oval_a give_v example_n suppose_v a_o oval_a be_v 10_o foot_n long_o transverse_n and_o 8_o foot_n broad_a conjugate_v the_o mean_a proportion_n between_o 10_o and_o 8_o be_v 8-95_a i_o say_v that_o a_o circle_n who_o diameter_n be_v 8-95_a be_v equal_a to_o a_o oval_a of_o 8_o broad_a and_o 10_o long_a and_o how_o to_o measure_v the_o circle_n be_v show_v before_o of_o these_o figure_n if_o the_o content_a be_v 100_o than_o the_o side_n of_o these_o regular_a figure_n be_v 〈…〉_z and_o also_o so_o in_o proportion_n be_v any_o other_o quantity_n of_o content_a require_v perpendiculer-triangle_n 13._o 123._o trianguler-side_n 15._o 2._o square_n its_o side_n 10._o 0._o pantagon_n of_o five_o side_n 7._o 62._o hexagon_n of_o six_o 6._o 02._o heptagon_n of_o seven_o 5._o 26._o octagon_n of_o eight_o 4._o 55._o nonagon_n of_o nine_o 4._o 03._o decagon_n of_o ten_o 3._o 06._o half_a diameter_n or_o radius_fw-la 5._o 64._o example_n as_o thus_o i_o will_v have_v a_o triangle_n to_o contain_v 200_o what_o must_v the_o side_n be_v the_o half_a distance_n on_o the_o number_n between_o 100_o and_o 200_o shall_v reach_v from_o 15-2_a to_o 21-5_a the_o side_n require_v and_o from_o 13-123_a the_o fix_a perpendicule_a for_o a_o triangle_n who_o area_n be_v 100_o to_o 18-6_a the_o perpendicule_a of_o a_o equilatteral_a triangle_n who_o area_n be_v 200._o but_o if_o the_o side_n be_v give_v and_o you_o will_v find_v the_o area_n work_v thus_o the_o extent_n from_o the_o fixed-side_n to_o the_o given-side_n shall_v reach_v at_o two_o turn_n from_o the_o fix_a area_n to_o the_o area_n require_v the_o extent_n from_o 15-2_a to_o 21-5_a shall_v reach_v at_o twice_o repeat_v from_o 100_o to_o 200._o problem_n xii_o to_o make_v a_o oval_a equal_a to_o a_o circle_n have_v the_o diameter_n of_o the_o circle_n and_o the_o length_n or_o breadth_n of_o the_o oval_a give_v sâ—t_a one_o point_n of_o the_o compass_n in_o the_o diameter_n of_o the_o circle_n find_v out_o on_o the_o line_n of_o number_n and_o the_o other_o point_n to_o the_o oval_n length_n then_o turn_v that_o distance_n the_o contrary_a way_n from_o the_o same_o diameter-point_n and_o it_o shall_v reach_v to_o the_o breadth_n of_o the_o oval_a require_v example_n let_v the_o diameter_n of_o a_o circle_n be_v 10_o foot_n i_o will_v have_v a_o oval_a to_o contain_v as_o much_o as_o the_o circle_n and_o be_v 12_o foot_n long_o the_o query_n be_v how_o broad_a must_v it_o be_v set_a one_o point_n in_o 10_o and_o the_o other_o in_o 12_o that_o extent_n turn_v the_o other_o way_n from_o 10_o shall_v reach_v to_o 8-34_a the_o breadth_n of_o the_o oval_a require_v if_o you_o please_v to_o alter_v the_o breadth_n or_o length_n you_o shall_v soon_o find_v the_o length_n or_o breadth_n according_o to_o work_v this_o by_o the_o line_n of_o line_n you_o must_v work_v by_o the_o direction_n in_o the_o seven_o section_n of_o the_o 6_o chapter_n as_o thus_o first_o to_o find_v the_o content_a of_o the_o oval_a join_v the_o length_n and_o breadth_n in_o one_o sum_n to_o get_v the_o sum_n the_o half_a sum_n and_o difference_n and_o half_a difference_n then_o open_a the_o sector_n or_o lay_v the_o thread_n on_o 600_o to_o a_o right_a angle_n then_o count_v half_a the_o difference_n from_o the_o centre_n downward_o and_o note_v the_o place_n then_o take_v half_a the_o sum_n between_o your_o compass_n and_o set_v one_o point_n in_o the_o half-difference_n and_o extend_v the_o other_o to_o the_o other_o leg_n or_o perpendicule_a line_n and_o it_o shall_v show_v a_o point_n who_o distance_n from_o the_o centre_n be_v the_o mean_a proportional_a require_v which_o be_v the_o diameter_n of_o a_o circle_n equal_a in_o area_n to_o the_o oval_a or_o elipsis_n give_v to_o be_v measure_v as_o before_o be_v show_v to_o make_v a_o oval_a equal_a to_o a_o circle_n take_v the_o guess_v half-sum_n of_o the_o length_n and_o breadth_n of_o the_o oval_a and_o set_v one_o point_n in_o the_o diameter_n of_o the_o circle_n and_o on_o the_o other_o leg_n set_v at_o a_o right_a angle_n the_o other_o point_n shall_v show_v half_o the_o difference_n between_o the_o length_n and_o breadth_n of_o the_o oval_a then_o if_o the_o mean_v proportional_a between_o they_o be_v equal_a to_o the_o diameter_n you_o have_v wrought_v right_o if_o not_o then_o resolve_v upon_o the_o length_n or_o breadth_n of_o the_o oval_a take_v more_o or_o less_o for_o the_o breadth_n or_o length_n according_o herein_o also_o be_v see_v the_o excellency_n of_o the_o line_n of_o number_n in_o many_o operation_n problem_n xiii_o the_o length_n and_o breadth_n of_o any_o oblong_a superficies_n give_v in_o foot_n to_o find_v the_o content_a in_o yard_n as_o 9_o foot_n the_o number_n of_o foot_n in_o one_o yard_n to_o the_o length_n in_o foot_n and_o part_n so_o be_v the_o breadth_n in_o foot_n and_o part_n to_o the_o content_a in_o yard_n example_n at_o 13_o foot_n 6_o inch_n long_o and_o 7_o foot_n 6_o inch_n broad_a the_o extent_n of_o the_o compass_n from_o 9_o to_o 13_o ½_o the_o length_n shall_v reach_v the_o same_o way_n from_o 7_o ½_o the_o breadth_n to_o 11_o yard_n and_o a_o quarter_n the_o content_a note_v that_o if_o yoâ—_n measure_n by_o foot_n and_o hundred_o part_n you_o shall_v find_v this_o way_n exceed_v ready_a the_o answer_n be_v give_v in_o yard_n and_o hundred_o part_n of_o a_o yard_n but_o if_o you_o have_v a_o yard_n divide_v into_o a_o 100_o part_n to_o measure_v withal_o then_o the_o rule_n be_v thus_o as_o 1_o to_o the_o length_n oâ—_n breadth_n so_o be_v the_o breadth_n or_o length_n to_o the_o content_a in_o yard_n example_n at_o 3_o yard_n 72_o part_n broad_a and_o 5_o yard_n 82_o part_n long_o the_o extent_n of_o the_o compass_n on_o the_o line_n of_o number_n from_o 1_o to_o 3-72_a shall_v reach_v the_o same_o way_n from_o 5-82_a to_o 21_o yard_n 65_o part_n the_o content_a in_o square_a yard_n and_o 100_o part_n by_o the_o line_n of_o line_n as_o 5-82_a to_o =_o 1_o at_o 10_o the_o end_n so_o be_v =_o 3-72_a to_o 21-65_a yard_n or_o in_o the_o example_n before_o as_o 13-6_a count_v 6_o ½_o for_o 13_o be_v to_o =_o 9_o so_o be_v =_o 7_o ½_o to_o 11_o ¼_n as_o you_o count_v at_o first_o problem_n fourteen_o the_o length_n and_o breadth_n of_o any_o wall_n be_v give_v in_o foot_n and_o 100_o part_n to_o find_v how_o many_o rod_n of_o walling_n there_o shall_v be_v at_o a_o brick_n and_o a_o half_a thick_a first_o you_o must_v note_n that_o 272_o foot_n and_o a_o quarter_n make_v one_o rod_n or_o so_o many_o foot_n be_v in_o a_o rod_n second_o that_o let_v the_o wall_n be_v half_a a_o brick_n one_o brick_n two_o brick_n two_o and_o a_o half_a or_o three_o brick_n thick_a it_o be_v to_o be_v reduce_v to_o brick_n and_o a_o half_a thick_a as_o a_o standard_n thickness_n three_o note_n that_o this_o reduce_n to_o a_o brick_n and_o half_a thick_a may_v be_v at_o the_o measure_n or_o after_o the_o casting-up_a as_o you_o please_v as_o in_o the_o example_n follow_v will_v plain_o appear_v as_o thus_o for_o instance_n a_o front_n or_o side-wall_n of_o a_o house_n be_v to_o be_v measure_v wherein_o the_o celler-story_n wall_n be_v 2_o brick_n and_o a_o half_a thick_a the_o shop_n and_o first_o chamber-story_n be_v two_o brick_n thick_n the_o other_o story_n 1_o brick_n and_o a_o half_a thick_a and_o the_o gable-end_n 1_o brick_n thick_a the_o near_a way_n to_o measure_v this_o wall_n i_o conceive_v be_v thus_o 1._o the_o cellar-story_n be_v 10_o foot_n high_a but_o be_v 2_o brick_n and_o a_o half_a thick_a i_o make_v it_o 16_o foot_n 8_o inch_n high_a by_o add_v two_o three_o of_o 10_o foot_n to_o the_o 10_o foot_n high_a which_o be_v 6_o foot_n 8_o inch_n in_o all_o 16_o foot_n 8_o inch_n 2._o the_o other_o two_o story_n
be_v suppose_v 22_o foot_n but_o in_o regard_n they_o be_v two_o brick_n thick_a i_o add_v one_o three_o part_n of_o 22_o foot_n which_o be_v 7_o foot_n 4_o inch_n to_o 22_o and_o it_o make_v 29_o foot_n and_o 4_o inch_n the_o height_n of_o the_o shop_n and_o next_o story_n above_o 3._o the_o other_o two_o story_n be_v a_o brick_n and_o half_a thick_a need_v no_o alteration_n which_o suppose_v may_v be_v 19_o foot_n 4._o the_o gable-end_n or_o garret-story_n if_o â—ny_n be_v be_v but_o one_o brick_n thick_a you_o must_v take_v away_o one_o three_o part_n to_o bring_v it_o to_o a_o brick_n and_o a_o half_a also_o if_o it_o be_v a_o gable-end_n note_v it_o be_v a_o triangle_n and_o you_o must_v measure_v but_o half_a the_o height_n and_o the_o whole_a breadth_n to_o find_v the_o content_a which_o here_o may_v be_v 5_o foot_n the_o cellar_n story_n 16_o 8_o two_o next_o story_n 29_o 4_o two_o next_o story_n 19_o 0_o the_o garret_n 5_o 0_o  _fw-fr 70_o 0_o 5._o add_v all_o these_o sum_n of_o foot_n high_a together_o and_o they_o make_v 70_o then_o measure_v the_o breadth_n which_o be_v common_a to_o every_o room_n the_o outside_n go_v upright_o which_o in_o a_o double_a house_n may_v be_v 36_o or_o 40_o foot_n 6._o then_o have_v get_v the_o dimension_n right_a by_o the_o line_n of_o number_n say_v as_o 272_o ¼_n the_o foot_n in_o one_o rod_n be_v to_o 40_o foot_n the_o breadth_n of_o the_o house_n so_o be_v 70_o foot_n the_o whole_a height_n of_o every_o several_a story_n reduce_v to_o 10_o rod_n and_o 29_o part_n which_o 29_o part_n you_o may_v call_v a_o quarter_n of_o a_o rod_n and_o 10_o foot_n and_o a_o half_a the_o reason_n whereof_o be_v apparent_a thus_o as_o 100_o be_v to_o 272_o ¼_n so_o be_v 29_o to_o near_o 79_o of_o which_o 79-68_a be_v a_o quarter_n of_o a_o rod_n or_o 25_o of_o 100_o be_v a_o quarter_n likewise_o which_o by_o the_o line_n of_o number_n be_v apparent_o see_v then_o every_o 10_o part_n be_v 2_o foot_n and_o 72_o of_o a_o hundred_o which_o be_v near_o two_o and_o three_o quarter_n so_o that_o here_o 25_o be_v a_o quarter_n of_o a_o rod_n there_o be_v 4_o hundred_o part_n more_o in_o 29_o then_o thus_o the_o double_a of_o 4_o be_v 8_o or_o twice_z 4_o be_v 8_o and_o four_o time_n three_o quarter_n be_v three_o foot_n more_o of_o which_o you_o must_v abate_v somewhat_o because_o 72_o ¼_n be_v not_o 75_o which_o be_v just_a three_o quarter_n and_o all_o put_v together_o make_v ten_o rod_n one_o quarter_n ten_o foot_n and_o a_o half_a for_o if_o you_o shall_v divide_v the_o product_n of_o 40_o multiply_v by_o 70_o which_o be_v 2800_o by_o 272_o ¼_n you_o shall_v find_v the_o quotient_a to_o be_v 10_o rod_n 78_o ½_o which_o be_v as_o before_z 10_o rod_z 1_o quarter_n and_o 10_o foot_n and_o a_o half_a but_o note_v also_o by_o the_o way_n that_o when_o you_o come_v to_o take_v out_o the_o deduction_n for_o the_o door_n and_o window_n if_o any_o happen_v in_o a_o wall_n of_o two_o brick_n and_o a_o half_a or_o in_o two_o brick_n you_o must_v add_v two_o three_o or_o one_o three_o more_o to_o the_o length_n or_o breadth_n one_o way_n and_o then_o cast_v they_o up_o several_o when_o they_o be_v of_o several_a length_n or_o breadth_n you_o shall_v do_v no_o wrong_n to_o the_o workmaster_n nor_o workman_n for_o true_a arithmetic_n and_o geometry_n will_v lie_v for_o no_o man_n or_o use_v any_o kind_n of_o partiality_n this_o i_o conceive_v be_v as_o near_o a_o way_n as_o any_o such_o business_n can_v be_v perform_v but_o if_o you_o will_v measure_v every_o story_n several_o take_v account_n of_o each_o story_n several_o in_o their_o thickness_n then_o after_o it_o be_v cast_v up_o the_o best_a way_n by_o the_o rule_n to_o reduce_v it_o be_v thus_o as_o 3_o half_a brick_n for_o a_o brick_n and_o a_o half_a be_v to_o any_o other_o number_n of_o half_a brick_n thick_a over_o or_o under_o 3_o so_o be_v the_o content_a at_o that_o rate_n according_o to_o his_o content_a at_o a_o brick_n and_o a_o half_a require_v example_n 1269_o foot_n at_o 5_o half_a brick_n thick_o be_v 2115_o for_o two_o three_o of_o 1269_o which_o be_v 846_o add_v to_o 1269_o make_v 2115_o for_o the_o extent_n on_o the_o line_n of_o number_n from_o 3_o to_o 5_o shall_v reach_v the_o same_o way_n from_o 1269_o to_o 2115_o the_o number_n require_v to_o be_v find_v out_o otherwise_o thus_o to_o bring_v any_o kind_n of_o thickness_n to_o one_o brick_n and_o a_o half_a thick_a at_o one_o operation_n by_o the_o line_n of_o number_n for_o this_o purpose_n you_o must_v use_v several_a point_n as_o so_o many_o gage_n point_n as_o in_o the_o short_a table_n follow_v do_v appear_v for_o half_a a_o brick_n use_v 3-00000_a for_o 1_o brick_n 1-50000_a for_o 1_o brick_n &_o a_o half_a use_v 1-0000_a for_o 2_o brick_n 0-7500_a for_o 2_o brick_n &_o a_o half_a use_v 0-6000_a for_o 3_o brick_n 0-5000_a for_o 3_o brick_n &_o a_o half_a 0-4285_a for_o 4_o brick_n 0-3750_a for_o 4_o brick_n and_o a_o half_a 0-3333_a for_o 5_o brick_n 0-3000_a for_o 5_o brick_n and_o a_o half_a 0-2727_a for_o 6_o brick_n etc._n etc._n ad_fw-la infinitum_fw-la 0-2500_a example_n at_o the_o 6_o ordinary_a thickness_n let_v a_o wall_n be_v 30_o foot_n long_o and_o 10_o foot_n high_a and_o let_v it_o be_v suppose_v of_o any_o of_o these_o thickness_n follow_v from_o half_a a_o brick_n length_n to_o three_o brick_n length_n in_o thickness_n then_o thus_o in_o order_n increase_v etc._n etc._n first_o at_o half_a a_o foot_n for_o ½_o brick_n as_o 3_o to_o 30_o so_o be_v 10_o to_o 100_o foot_n at_o 1_o ½_o for_o 1_o brick_n as_o 15_o to_o 30_o so_o be_v 10_o to_o 200_o foot_n at_o 1_o brick_n for_o 1_o ½_o thick_a as_o 10_o to_o 30_o so_o be_v 10_o to_o 300_o foot_n at_o 1_o ½_o for_o 2_o brick_n as_o 0-75_a to_o 30_o so_o be_v 10_o to_o 400_o foot_n at_o 1_o ½_o for_o 2_o ½_o thick_a as_o 0-60_a to_o 30_o so_o be_v 10_o to_o 500_o foot_n at_o ½_o for_o 3_o brick_n as_o 0-50_a to_o 30_o so_o be_v 10_o to_o 600_o foot_n at_o 1_o ½_o for_o 3_o ½_o thick_a as_o 0-4285_a to_o 30_o so_o be_v 10_o to_o 700_o foot_n at_o 1_o ½_o for_o 4_o brick_n as_o 0-3750_a to_o 30_o so_o be_v 10_o to_o 800_o foot_n at_o 1_o ½_o and_o so_o for_o any_o other_o thickness_n as_o far_o as_o you_o please_v which_o point_n be_v find_v thus_o the_o extenâ—_n from_o the_o number_n of_o brick_n any_o wall_n be_v thick_a to_o 15_o or_o 1_o and_o ½_o shall_v reach_v the_o same_o way_n from_o 10_o or_o 1_o to_o the_o gagepoint_n require_v for_o that_o wall_n or_o wall_n of_o that_o thickness_n example_n as_o 2_o to_o 1_o ½_o so_o be_v 10_o to_o 0-750_a for_o 2_o brick_n thick_a etc._n etc._n last_o have_v the_o number_n of_o foot_n in_o the_o whole_a work_n to_o find_v how_o many_o rod_n there_o be_v say_v if_o 272_o ¼_n be_v one_o rod_n what_o shall_v any_o other_o number_n of_o foot_n make_v in_o rod_n the_o extent_n of_o the_o compass_n from_o 172_o ¼_n to_o 1_o shall_v reach_v the_o same_o way_n from_o the_o number_n of_o foot_n to_o the_o number_n of_o rod_n and_o hundred_o part_n or_o rod_n and_o quarter_n and_o foot_n as_o by_o the_o 6_o last_o mention_v example_n in_o 5269_o foot_n how_o many_o rod_n the_o extent_n from_o 272_o ¼_n to_o 1_o shall_v reach_v the_o same_o way_n from_o 5269_o to_o 19_o rod_n and_o 36_o part_n of_o a_o 100_o or_o 19_o rod_n 1_o quarter_n and_o 29_o foot_n and_o a_o quarter_n of_o a_o foot_n the_o 19_o rod_n and_o a_o quarter_n be_v see_v plain_o on_o the_o rule_n and_o 25_o be_v a_o quarter_n 36_o be_v 11_o part_n more_o for_o which_o 11_o part_n more_o i_o say_v 2_o time_n 11_o be_v 22_o foot_n and_o 11_o 3_o quarter_n of_o a_o foot_n be_v near_o 8_o foot_n which_o put_v together_o make_v 29_o foot_n as_o before_z or_o as_o the_o compass_n stand_v turn_v they_o the_o contrary_a way_n from_o the_o decimal_a part_n above_o the_o even_a quarter_n and_o it_o shall_v reach_v to_o the_o odd_a foot_n above_o the_o quarter_n require_v example_n the_o extent_n from_o 272_o ¼_n to_o 100_o or_o 1_o shall_v reach_v the_o contrary_a way_n from_o 10_o ½_o to_o 29_o foot_n the_o foot_n above_o ¼_n of_o a_o rod._n 8._o observe_v that_o the_o tyl_n the_o roof_n the_o floor_n and_o partition_n be_v measure_v by_o the_o square_a which_o be_v 10_o foot_n square_n every_o way_n or_o 100_o foot_n in_o area_n the_o chimney_n be_v usual_o do_v by_o a_o certain_a rate_n for_o a_o chimney_n or_o if_o
to_o be_v measure_v thus_o be_v the_o height_n and_o breadth_n take_v etc._n etc._n if_o a_o chimney_n stand_v single_o and_o alone_o not_o lean_v against_o or_o in_o a_o wall_n the_o usual_a way_n be_v to_o gird_v it_o about_o and_o if_o the_o jaume_n be_v but_o a_o brick_n thick_a and_o wrought_v upright_o over_o the_o mantletree_n to_o the_o floor_n then_o i_o say_v gird_v it_o about_o for_o a_o length_n and_o the_o height_n of_o the_o story_n be_v the_o breadth_n at_o a_o brick_n thick_a because_o of_o the_o gather_v together_o to_o make_v room_n for_o the_o next_o hearth_n above_o in_o the_o next_o story_n but_o if_o the_o chimney-back_a be_v a_o party-wall_n the_o wall_n be_v first_o measure_v than_o the_o breast_n and_o the_o depth_n of_o the_o two_o jaume_n be_v one_o side_n and_o the_o height_n of_o the_o story_n another_o side_n to_o be_v multiply_v together_o at_o a_o brick_n and_o a_o half_a thick_a or_o a_o brick_n thick_a according_a as_o the_o jaume_v be_v and_o nothing_o to_o be_v abate_v for_o the_o want_n between_o the_o hearth_n and_o the_o mantletree_n because_o of_o the_o with_o 's_o and_o thicken_n for_o the_o next_o hearth_n for_o measure_v the_o shaft_n of_o the_o chimney_n gird_a with_o a_o line_n round_o about_o the_o least_o place_n of_o they_o for_o one_o side_n and_o the_o height_n for_o the_o other_o side_n at_o a_o brick_n thick_a in_o consideration_n of_o the_o with_o 's_o pargitting_a and_o scaffold_n in_o measure_v of_o ceiling_n a_o foot_n broad_a and_o the_o length_n of_o the_o valley_n be_v always_o to_o be_v allow_v more_o than_o the_o whole_a roof_n also_o the_o length_n of_o the_o rafter_n foot_n above_z or_o beyond_o the_o roof_n when_o rafter_n have_v their_o usual_a pitch_n which_o be_v when_o the_o breadth_n of_o the_o house_n be_v 12_o foot_n the_o rafter_n be_v 9_o foot_n long_o which_o be_v 3_o quarter_n of_o the_o floor_n breadth_n be_v it_o more_o or_o less_o then_o i_o say_v that_o the_o content_a of_o one_o floor_n and_o half_a so_o much_o be_v the_o area_n of_o the_o whole_a roof_n in_o square_n to_o which_o be_v to_o be_v add_v the_o valley_n and_o rafter-feet_n or_o eve_n in_o tile_v and_o also_o a_o deduction_n for_o chimney-room_n and_o gutter_n if_o any_o be_v which_o work_n by_o the_o line_n of_o number_n be_v do_v at_o one_o operation_n thus_o as_o 6666_o be_v to_o the_o length_n of_o the_o house_n so_o be_v the_o breadth_n to_o the_o content_a in_o the_o roof_n example_n a_o house_n 30_o foot_n long_o and_o 20_o foot_n broad_a be_v 900_o foot_n or_o 9_o square_a for_o the_o extent_n from_o 6666_o to_o 20_o shall_v reach_v the_o same_o way_n from_o 30_o to_o 900._o also_o in_o measure_v of_o the_o roof_n as_o to_o carpenter_n work_n by_o the_o square_n there_o be_v to_o be_v allowance_n for_o those_o rafter_n in_o the_o dormer_n and_o gable-end_n on_o which_o no_o tile_n be_v lay_v as_o over-work_n for_o a_o particular_a use_n and_o convenience_n more_o than_o need_v be_v in_o a_o bare_a cover_v or_o roof_n also_o in_o measure_v of_o plasterer_n work_v in_o partition_n and_o wall_n the_o timber_n and_o quarter_n be_v not_o to_o be_v deduct_v out_o of_o the_o render_n for_o work_n only_o except_o when_o the_o workman_n find_v the_o work_n and_o stuff_v also_o then_o subtract_v a_o 6_o part_n for_o the_o quarter_n in_o the_o render_n work_n but_o in_o ceiling_n the_o summer_n which_o be_v see_v be_v always_o abate_v and_o door_n and_o window_n also_o unless_o by_o a_o due_a considerate_a or_o a_o unconsiderate_a bargain_n of_o run_a measure_n thus_o you_o have_v a_o brief_a account_n of_o the_o usual_a order_n use_v among_o workman_n in_o take_v the_o dimension_n of_o a_o house_n viz._n brickwork_n by_o the_o rod_n tile_v and_o carpenters-work_n by_o the_o square_a chimney_n usual_o by_o the_o fire_n and_o plasterer_n and_o painters-work_n by_o the_o yard_n glasier_n by_o the_o foot_n there_o be_v many_o other_o thing_n to_o be_v take_v notice_n of_o in_o the_o carpenter_n bill_n as_o lintel_n mantle-tree_n and_o tassel_n luthern_a light_n and_o other_o light_n both_o architrave_n and_o plain_a light_n sky-light_n or_o cubiloe_n modillean_n cornish_n and_o gutter_a penthouse_n cornish_n timber-front-story_n cellar-door_n and_o door-case_n the_o plank_n and_o curb_n at_o the_o cellar-stair_n dogleg-stair_n and_o open-newel-stair_n canted-stair_n count_v either_o by_o the_o step_n or_o pair_n together_o with_o the_o half_a space_n on_o the_o corner_n of_o the_o open_a newel-stair_n the_o rail_n and_o ballaster_n small_a and_o great_a cornish_a outside-work_n and_o partition_n ceiling_v joyst_n and_o the_o ashler_a board_v partition_n and_o chequer-work_n back-door_n and_o door-case_n window-board_n and_o wall-timber_n plank_n in_o the_o foundation_n palce_v penthouse-floor_n and_o penthouse-roof_n fur_v the_o platform_n centering_n for_o the_o chimney_n trimmer_n girders-end_n end_n of_o brest-summer_n and_o plate_n and_o more_o the_o like_a which_o will_v come_v in_o account_n to_o be_v remember_v and_o set_v down_o according_a as_o the_o building_n be_v also_o with_o due_a allowance_n into_o the_o wall_n that_o way_n the_o end_n of_o the_o joyst_n be_v enter_v or_o lay_v in_o the_o wall_n as_o thus_o if_o it_o be_v frame_v work_n be_v only_o measure_v than_o 9_o inch_n ought_v to_o be_v allow_v into_o each_o wall_n that_o way_n the_o joyst_n end_n be_v lay_v because_o every_o joy_v if_o well_o lay_v shall_v have_v 9_o inch_n at_o least_o hold_v on_o the_o wall_n but_o if_o it_o be_v timber_n and_o board_v both_o to_o be_v measure_v than_o 6_o inch_n only_o be_v a_o competent_a allowance_n because_o the_o timber_n be_v usual_o value_v at_o one_o three_o part_n more_o than_o the_o board_v be_v also_o as_o the_o workman_n do_v think_v on_o this_o the_o workmaster_n may_v not_o forget_v to_o deduct_v for_o stair_n and_o chimney_n also_o where_o work_n and_o stuff_n be_v both_o measure_v though_o for_o work_n only_o it_o may_v be_v very_o well_o allow_v unless_o the_o better_a price_n make_v a_o allowance_n for_o it_o note_v also_o that_o by_o the_o line_n of_o number_n you_o may_v ready_o find_v the_o length_n of_o the_o hip_n and_o rafter_n in_o a_o roof_n of_o any_o largeness_n at_o true_a pitch_n by_o this_o follow_a proportion_n and_o table_n the_o breadth_n of_o the_o house_n be_v 40_o foot_n and_o the_o end_v square_a the_o length_n and_o angle_n be_v as_o in_o the_o table_n at_o the_o usual_a truâ—_n pitch_n  _fw-fr fâ—et_fw-fr 100_o par_fw-fr whole_a breadth_n 40_o 00_o half_a breadth_n 20_o 00_o rafter_n 30_o 00_o hip-rafter_n 36_o 00_o diagonal_a line_n 56_o 57_o half-diagonal_a 28_o 28_o perpendicule_a 22_o 36_o  _fw-fr  _fw-fr deg_n min._n hippolito_n angle_n at_o foot_n 38_o 22_o at_o top_n 51_o 38_o on_o the_o outside_o 116_o 12_o rafter_n angle_n at_o foot_n 48_o 10_o at_o top_n 41_o 50_o for_o any_o other_o house_n by_o the_o number_n thus_o as_o suppose_v 18_o foot_n broad_a the_o extent_n of_o the_o compass_n from_o 40_o the_o breadth_n in_o the_o table_n to_o 18_o the_o breadth_n give_v shall_v reach_v the_o same_o way_n from_o 30_o the_o rafter_n in_o the_o table_n to_o 13-50_a the_o rafter_n require_v and_o from_o 36_o the_o hip_n in_o the_o table_n to_o 16-22_a the_o hip_n require_v and_o from_o 22-36_a the_o perpendicule_a in_o the_o table_n to_o 10-06_a the_o perpendicule_a require_v and_o from_o â—6-57_a the_o diagonal_a in_o the_o table_n to_o â—5-48_a the_o diagonal_a require_v the_o angles_n be_v always_o the_o same_o in_o all_o rooss_n small_a or_o great_a as_o in_o the_o table_n be_v square_a and_o true_a pitch_n if_o you_o will_v have_v direction_n for_o bevel_n or_o taper_n frames_n to_o find_v the_o length_n and_o angle_n of_o raâ—ters_n and_o hip_n you_o may_v have_v it_o at_o large_a in_o a_o appendix_n to_o the_o mirror_n of_o architecture_n or_o vincent_n stamoâ—â—i_n print_v for_o william_n fisher_n at_o the_o postern-gate_n 1669._o by_o which_o direction_n and_o the_o sector_n you_o may_v find_v any_o thing_n that_o be_v there_o set_v down_o as_o also_o by_o the_o triangule_a quadrant_n thread_n and_o compass_n note_v also_o that_o have_v inch_n and_o foot-measure_n together_o you_o may_v present_o by_o inspection_n find_v the_o price_n of_o one_o foot_n have_v the_o price_n of_o the_o square_n and_o the_o contrary_n also_o have_v the_o 12_o inch_n on_o the_o other_o foot_n divide_v into_o 85_o part_n near_o and_o figure_v at_o every_o 8_o with_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o 10._o shall_v represent_v penny_n and_o half_a farthing_n then_o at_o any_o price_n the_o rod_n you_o have_v the_o price_n of_o one_o foot_n &_o the_o contrary_n as_o thus_o let_v every_o inch_n represent_v one_o pound_n 〈◊〉_d â—very_n 8_o
the_o point_n e_o draw_v the_o line_n ed_z of_o which_o line_n find_v the_o middle_n between_o e_z and_o d_o viz._n the_o point_n c_o then_o the_o extent_n ce_fw-fr or_o cd_o keep_v one_o point_n in_o c_o shall_v cross_v the_o ground-line_n in_o the_o point_n b_o by_o which_o and_o e_o you_o may_v draw_v the_o perpendicule_a line_n ebb_v which_o be_v but_o the_o converse_n of_o the_o former_a 6._o to_o draw_v a_o line_n parallel_n to_o another_o at_o any_o distance_n to_o the_o line_n ab_fw-la i_o will_v have_v another_o parallel_n thereto_o to_o the_o distance_n of_o ai_fw-fr take_z a_z between_o your_o compass_n and_o set_v one_o point_n in_o one_o end_n of_o the_o line_n as_o at_o a_o sweep_v the_o ark_n eif_fw-mi then_o set_v the_o compass_n in_o the_o other_o end_n as_o at_z b_o and_o sweep_v the_o ark_n gdh_n then_o just_a by_o the_o round-side_n of_o those_o ark_n draw_v a_o line_n which_o shall_v be_v the_o parallel-line_n require_v or_o thus_o take_v bc_n the_o measure_n from_o the_o point_n that_o be_v to_o cut_v the_o parallel-line_n and_o one_o end_n of_o the_o given-line_n viz._n b_o with_o this_o distance_n set_v one_o foot_n in_o a_o at_o the_o other_o end_n of_o the_o given-line_n and_o draw_v the_o arch_n at_o k_o then_o take_v all_o ab_fw-la the_o given-line_n and_o set_v one_o point_n in_o c_o cross_v the_o ark_n at_o k_o then_o c_z and_o k_o shall_v be_v point_n to_o draw_v the_o parallel-line_n by_o note_v the_o figure_n 4._o 7._o to_o make_v one_o angle_n equal_a to_o another_o the_o angle_n bac_n be_v give_v and_o i_o will_v have_v another_o angle_n equal_a unto_o it_o set_v one_o point_n of_o the_o compass_n in_o a_o and_o draw_v the_o arch_n cb_n then_o on_o the_o line_n de_fw-fr from_o the_o point_n d_o draw_v the_o like_a ark_n of_o then_o in_o that_o ark_n make_v of_o equal_a to_o cb_n then_o draw_v the_o line_n df_n it_o shall_v make_v the_o angle_n edf_n equal_a to_o the_o angle_n bac_n which_o be_v require_v 8._o to_o divide_v a_o line_n into_o any_o number_n of_o part_n let_v ab_fw-la represent_v a_o line_n to_o be_v divide_v into_o eight_o part_n on_o one_o end_n viz._n a_o draw_v a_o line_n as_o ad_fw-la to_o any_o angle_n and_o from_o the_o other_o end_n b_o draw_v another_o line_n parallel_n to_o ad_fw-la as_o be_v then_o open_a the_o compass_n to_o any_o convenient_a distance_n and_o from_o a_o and_o b_o divide_v the_o line_n ad_fw-la and_o be_v into_o eight_o part_n then_o line_n draw_v by_o a_o ruler_n lay_v to_o every_o division_n in_o the_o line_n ad_fw-la and_o be_v shall_v divide_v the_o line_n ab_fw-la in_o the_o part_n require_v note_v the_o figure_n mark_v vi_o this_o proposition_n be_v much_o easy_o wrought_v by_o the_o line_n of_o line_n on_o the_o sector_n thus_o take_v ab_fw-la between_o your_o compass_n and_o fit_v it_o over_o parral_o in_o 8_o and_o 8_o of_o the_o line_n of_o line_n then_o the_o parallel_n distance_n between_o 1_o and_o 1_o shall_v divide_v ab_fw-la into_o 8_o part_n require_v 9_o any_o three_o point_n give_v to_o bring_v they_o into_o a_o circle_n let_v abc_n be_v three_o point_n to_o be_v bring_v into_o a_o circle_n first_o set_v one_o point_v on_o a_o and_o open_v the_o other_o above_o halfway_n to_o c_o and_o sweep_v the_o part_n of_o a_o circle_n above_o and_o below_o the_o point_n a_o as_o the_o two_o arch_n at_o d_o and_o e_o not_o move_v the_o compass_n do_v the_o like_a on_o c_o as_o the_o ark_n f_o and_o g_o then_o set_v the_o compass-point_n in_o b_o and_o cross_v those_o ark_n in_o def_n and_o g_o then_o a_o rule_n lay_v from_o d_o to_z e_z and_o from_o f_o to_o g_z and_o line_n draw_v do_v inter-sect_n at_o h_n the_o true_a centre_n to_o bring_v abc_n into_o a_o circle_n 10._o any_o two_o point_n give_v in_o a_o circle_n to_o draw_v part_n of_o a_o circle_n which_o shall_v cut_v they_o and_o the_o circumference_n first_o give_v into_o two_o equal_a part_n let_v a_o and_o b_o be_v two_o point_n in_o a_o circle_n by_o which_o two_o point_n i_o will_v draw_v a_o arch_n which_o shall_v cut_v the_o whole_a circumference_n into_o two_o equal_a part_n first_o draw_v a_o line_n from_o a_o the_o point_n remote_a from_o the_o centre_n through_o the_o centre_n and_o beyond_o the_o circumference_n as_o ad_fw-la then_o draw_v another_o line_n from_o a_o to_o a_o point_n in_o the_o circumference_n perpendicule_a to_o ad_fw-la and_o cut_v the_o centre_n c_o as_o the_o line_n ae_n then_o on_o the_o point_n e_o draw_v another_o line_n perpendicule_a to_o the_o line_n ae_n till_o it_o inter-sect_n ad_fw-la at_o d_o then_o these_o three_o point_n abdella_n bring_v into_o a_o circle_n or_o arch_n by_o the_o last_o rule_n shall_v divide_v the_o circumference_n into_o two_o equal_a part_n note_v the_o figure_n 8_o where_o the_o first_o circle_n be_v cut_v into_o two_o equal_a part_n at_o f_o and_o g_o by_o part_n of_o a_o circle_n pass_v through_o the_o point_n a_o and_o b._n 11._o any_o segment_n of_o a_o circle_n give_v to_o find_v the_o diameter_n and_o centre_n of_o the_o circle_n belong_v to_o it_o let_v abc_n be_v the_o segment_n of_o a_o circle_n to_o which_o i_o will_v find_v a_o centre_n any_o where_o about_o the_o midst_n of_o the_o segment_n set_v one_o point_n of_o the_o compass_n at_o pleasure_n as_o at_o b_o on_o the_o point_n b_o at_o any_o meet_a distance_n describe_v a_o circle_n and_o note_v where_o the_o circle_n do_v cross_v the_o segment_n as_o at_o d_o and_o e_o then_o not_o stir_v the_o compass_n set_v one_o point_n in_o d_o and_o cross_v the_o circle_n twice_o as_o at_o f_o and_o i_o and_o again_o set_v one_o point_n in_o e_o and_o cross_v the_o circle_n twice_o in_o g_z and_o h_n last_o by_o the_o point_n gh_o and_o fi_n draw_v two_o line_n which_o will_v meet_v in_o the_o point_n o_o the_o centre_n require_v 12._o or_o else_o to_o find_v the_o diameter_n thus_o multiply_v the_o chord_n or_o flat-side_n of_o the_o half-segment_a viz_o ak_v 12_o by_o itself_o which_o be_v call_v square_n which_o make_v 144_o then_o divide_v that_o product_n 144_o by_o 8_o the_o line_n kb_n call_v a_o sine_fw-la the_o quotient_a which_o come_v out_o will_v be_v find_v to_o be_v 18_o then_o if_o you_o add_v 8_o the_o sine_fw-la and_o 18_o the_o quotient_a together_o it_o shall_v make_v 26_o for_o the_o diameter_n require_v to_o be_v find_v 13._o any_o segment_n of_o a_o circle_n give_v to_o find_v the_o length_n of_o the_o arch_n of_o the_o segment_n lay_v the_o chord_n of_o the_o whole_a segment_n and_o twice_o the_o chord_n of_o half_a the_o segment_n from_o one_o point_n several_o and_o to_o the_o great_a extent_n add_v one_o three_o part_n of_o the_o difference_n between_o the_o extent_n and_o that_o sum_n of_o extent_n shall_v be_v equal_a to_o the_o archippus_n example_n 14._o to_o draw_v a_o helical_a line_n from_o any_o three_o point_n to_o several_a radiuss_n without_o much_o gibbiosity_n useful_a for_o architect_n shipwright_n and_o other_o let_v abcde_v be_v five_o point_n to_o be_v bring_v into_o a_o helical-line_n smooth_o and_o even_o without_o gibbiosity_n or_o bunch_n as_o the_o under-side_n of_o a_o arch_n or_o the_o bend_n of_o a_o ship_n or_o the_o like_a first_o between_o the_o two_o remote_a point_n of_o 3_o as_o a_o and_o c_o draw_v the_o line_n ac_fw-la then_o let_v fall_v a_o perpendicule_a from_o b_o to_o cut_v the_o line_n ac_fw-la at_o right_a angle_n and_o produce_v it_o to_o f_o draw_v the_o like_a perpendiculer-line_n from_o the_o point_n d_o to_o cut_v the_o line_n ce_fw-fr at_o right-angle_n produce_v to_o f._n i_o say_v the_o centre_n both_o for_o the_o arch_n ab_fw-la the_o lesser_a and_o bc_n the_o great_a will_v be_v find_v to_o be_v in_o the_o line_n bf_n the_o like_a on_o the_o other-side_n for_o de_fw-fr and_o cd_o the_o helical-circle_n or_o arch_n require_v but_o if_o you_o divide_v the_o arch_n abcde_v into_o 24_o or_o more_o part_n the_o several_a centre_n of_o the_o splay-line_n be_v thus_o find_v take_v the_o measure_n agnostus_n and_o lay_v it_o from_o b_o or_o d_o or_o c_o on_o the_o line_n gf_n and_o those_o point_n on_o gf_n shall_v be_v the_o several_a point_n to_o draw_v the_o splay-line_n of_o the_o arch_n and_o key-stone_n by_o chap._n iu._n of_o the_o explanation_n of_o certain_a term_n use_v in_o this_o follow_a book_n 1._o radius_fw-la or_o sine_fw-la of_o 90_o radius_fw-la or_o tangent_fw-la of_o 45_o or_o secant_fw-la of_o 00_o be_v all_o one_o and_o the_o same_o thing_n yet_o take_v respective_o in_o their_o proper_a place_n and_o be_v the_o whole_a line_n of_o sines_n or_o tangent_n to_o 45_o or_o more_o particulary_a
particular_a product_n of_o every_o double_a number_n for_o the_o three_o term_n one_o after_o another_o this_o do_v repeat_v the_o rule_n of_o three_o as_o often_o as_o there_o be_v double_a number_n the_o four_o term_v produce_v from_o those_o operation_n shall_v be_v answer_n to_o the_o question_n require_v viz._n the_o quantity_n of_o each_o man_n gain_n or_o loss_n example_n 25_o &_o 30_o a_n ox_n and_o time_n of_o feed_v multiply_v be_v 750_o 15_o &_o 40_o b_n ox_n and_o time_n of_o feed_v multiply_v be_v 600_o 20_o &_o 40_o c_n ox_n and_o day_n of_o feed_v multiply_v be_v 800_o the_o sum_n 2150_o as_o 215_o to_o 50_o so_o be_v 750_o a_n stock_n to_o 17-9_a a_n rent_n  _fw-fr 600_o b_n 13-19_a b's_n  _fw-fr 800_o c_n 18-12_a c_n 9_o to_o work_v by_o the_o line_n of_o number_n the_o extent_n of_o the_o compass_n from_o 1_o to_o 25_o shall_v reach_v the_o same_o way_n from_o 30_o to_o 750_o the_o first_o product_n of_o a_o a'_v double_a number_n or_o stock_n and_o as_o 1_o to_o 15_o so_o be_v 40_o to_o 600_o the_o product_n of_o b_n double_a number_n and_o stock_n and_o as_o 1_o to_o 20_o so_o be_v 40_o to_o 800_o the_o product_n of_o c_n double_a number_n and_o stock_n which_o three_o product_n add_v make_v 2150_o the_o first_o term_n and_o 50_o be_v the_o second_o term_n and_o 750_o 600_o and_o 800_o the_o three_o product_n several_o the_o three_o term_n then_o the_o extent_n from_o 2150_o to_o 50_o shall_v reach_v the_o same_o way_n from_o 750_o to_o 17-45_a or_o 17_o l._n 9_o s._n and_o from_o 600_o to_o 13-95_a or_o 13_o l._n 19_o s._n and_o from_o 800_o to_o 18-60_a or_o 18_o l._n =_o 12_o s._n the_o several_a answer_n require_v which_o be_v add_v together_o make_v up_o 50_o l._n the_o whole_a rent_n to_o be_v pay_v among_o they_o there_o be_v other_o rule_n of_o arithmetic_n as_o the_o rule_n call_v allegation_n medial_a and_o alternate_a and_o the_o rule_n of_o position_n or_o falsehood_n in_o the_o work_n of_o which_o be_v so_o many_o caution_n in_o order_v the_o number_n before_o you_o come_v to_o the_o proportional_a work_n that_o it_o will_v make_v the_o book_n more_o bulky_a than_o useful_a therefore_o i_o shall_v wave_v it_o and_o refer_v you_o to_o the_o particular_a book_n of_o arithmetic_n as_o that_o of_o mr._n record_n dee_n and_o mellis_n or_o that_o of_o mr._n wingate_n natural_a and_o artificial_a have_v in_o it_o plenty_n of_o example_n and_o other_o also_o as_o johnfâ—us_o jaggers_o or_o moor_n arithmetic_n any_o of_o which_o exceed_v the_o bound_n i_o intend_v for_o this_o whole_a discourse_n i_o shall_v therefore_o pass_v on_o to_o the_o rule_n of_o practice_n in_o several_a kind_n as_o measure_v superfecy_n and_o solid_n and_o rule_n of_o double_a and_o treble_a proportion_n and_o question_n of_o interest_n which_o be_v tedious_a by_o the_o pen_n without_o the_o help_n of_o particular_a table_n and_o very_o easy_a by_o the_o line_n of_o number_n as_o will_v full_o appear_v in_o the_o next_o chapter_n chap._n vii_o the_o use_n of_o the_o line_n of_o number_n in_o measure_v of_o any_o kind_n of_o superficial_a measure_n the_o measure_n that_o be_v common_o use_v in_o this_o work_n be_v a_o foot-rule_n divide_v into_o 100_o part_n or_o else_o into_o 12_o inch_n and_o those_o inch_n into_o half_n and_o quarter_n or_o 8_o part_n or_o inch_n and_o 10_o part_n but_o in_o regard_n that_o the_o number_n do_v most_o fit_o agree_v to_o the_o 100_o part_n of_o a_o foot_n it_o will_v be_v convenient_a here_o to_o show_v how_o to_o reduce_v they_o or_o any_o other_o fraction_n from_o 12_o s._n to_o 10_o s._n or_o any_o other_o whatsoever_o from_o one_o fraction_n to_o the_o other_o which_o by_o the_o line_n of_o number_n be_v quick_o do_v as_o thus_o from_o 12_o to_o 10_o reduction_n extend_v the_o compass_n from_o one_o denominator_fw-la to_o the_o other_o the_o same_o extent_n shall_v reach_v the_o same_o way_n from_o one_o numerator_n to_o the_o other_o example_n as_o 12_o to_o 10_o so_o be_v 6_o half_a of_o 12_o to_z 5_o half_n of_o 10._o again_o as_o 120_o to_o 100_o so_o be_v 30_o a_o four_o of_o 120_o to_o 25_o a_o four_o of_o 100_o which_o two_o line_n of_o inch_n and_o foot-measure_n be_v usual_o set_v together_o on_o rule_n for_o the_o ready_a way_n of_o reduction_n by_o ocular_a inspection_n only_o in_o this_o manner_n as_o in_o the_o figure_n and_o the_o like_a may_v be_v for_o any_o thing_n whatsoever_o as_o mr._n edmond_n windgate_n have_v large_o show_v in_o his_o arithmetic_n which_o line_n be_v next_o to_o the_o line_n of_o number_n on_o your_o rule_n will_v be_v very_o plain_a and_o ready_a in_o the_o use_n of_o the_o line_n of_o number_n for_o foot_n and_o inch_n or_o shilling_n and_o penny_n and_o the_o same_o rule_n of_o reduction_n serve_v for_o all_o manner_n of_o fraction_n for_o as_o the_o denominator_fw-la of_o one_o fraction_n be_v to_o the_o denominator_fw-la of_o the_o other_o which_o in_o the_o decimal_n work_n be_v always_o a_o unite_v with_o one_o two_o or_o more_o cipher_n so_o be_v the_o numerator_n of_o one_o to_o the_o numerator_n of_o the_o other_o and_o note_v that_o the_o operation_n of_o decimal_a number_n and_o their_o fraction_n be_v no_o other_o than_o whole_a number_n except_o only_o the_o cut_n off_o so_o many_o figure_n as_o there_o be_v fraction_n in_o the_o multiplicator_n and_o multiplicand_a after_o any_o multiplication_n as_o in_o the_o follow_a example_n will_v appear_v this_o be_v premise_v i_o come_v next_o to_o the_o work_n problem_n i._o the_o breadth_n of_o a_o oblong_a superficies_n give_v in_o foot_n measure_n to_o find_v how_o much_o in_o length_n make_v one_o foot_n the_o extent_n of_o the_o compass_n from_o the_o breadth_n to_o 1_o shall_v reach_v the_o same_o way_n from_o 1_o to_o the_o length_n require_v example_n at_o 7_o 10_o broad_a as_o 7_o to_o 1_o so_o be_v 1_o to_o 1_o foot_n and_o 43_o part_n the_o breadth_n give_v in_o inch_n to_o find_v how_o much_o make_v a_o foot_n as_o the_o breadth_n in_o inch_n to_o 12_o so_o be_v 12_o to_o the_o length_n of_o a_o foot_n in_o inch_n and_o 10_o part_n example_n at_o 8_o inch_n broad_a you_o must_v have_v 18_o inch_n to_o make_v a_o foot_n for_o the_o extent_n from_o 8_o to_o 12_o shall_v reach_v the_o same_o way_n from_o 12_o to_o 18._o to_o work_v these_o two_o by_o the_o line_n of_o line_n by_o inches_n as_o 1_o to_o =_o 7_o so_o be_v =_o 1_o to_o 1_o 43_o the_o length_n in_o foot-measure_n by_o inch_n as_o 12_o to_o =_o 8_o so_o be_v =_o 12_o to_o 18_o or_o else_o by_o inch_n as_o 8_o to_o =_o 12_o so_o be_v 12_o to_o =_o 18_o the_o length_n in_o inch_n problem_n ii_o have_v the_o breadth_n of_o a_o oblong_a superficies_n give_v in_o foot-measure_n to_o find_v how_o much_o be_v in_o a_o foot_n long_o this_o be_v soon_o wrought_v for_o in_o every_o foot_n long_o there_o be_v just_a as_o much_o as_o the_o breadth_n be_v either_o in_o foot-measure_n or_o inch_n for_o a_o piece_n of_o board_n half_o a_o foot_n broad_a and_o a_o foot_n long_o be_v just_o half_a a_o foot_n problem_n iii_o have_v the_o length_n and_o breadth_n in_o foot-measure_n to_o find_v the_o content_a in_o foot_n the_o extent_n from_o 1_o to_o the_o length_n shall_v reach_v the_o same_o way_n from_o the_o length_n to_o the_o content_a in_o foot_n example_n as_o 1_o to_o 1_o foot_n 50_o the_o breadth_n so_o be_v 11_o foot_n 10_o part_n the_o length_n to_o 16_o foot_n and_o 65_o part_n the_o comment_n require_v the_o breadth_n give_v in_o inch_n and_o the_o length_n in_o foot_n to_o find_v the_o content_a in_o foot_n as_o 12_o to_o the_o breadth_n in_o inch_n so_o be_v the_o length_n in_o foot_n to_o the_o content_a in_o foot_n require_v example_n at_o 9_o inch_n broad_a and_o 11_o foot_n long_o the_o extent_n from_o 12_o to_o 9_o shall_v reach_v the_o same_o way_n from_o 11_o to_o 8_o foot_n 3_o inch_n or_o ¼_n by_o the_o line_n of_o line_n as_o 11_o to_z =_o 12_o so_o be_v 9_o to_z 8_o ¼_n but_o note_v that_o in_o work_v this_o and_o many_o suchlike_a it_o will_v be_v convenient_a to_o double_v your_o scale_n in_o account_n call_v 10_o at_o the_o end_n 20_o and_o every_o single_a figure_n as_o much_o more_o as_o to_o call_v 12_o and_o 24_o etc._n etc._n so_o that_o in_o this_o operation_n the_o work_n run_v thus_o as_o 11_o take_v from_o the_o line_n of_o line_n count_v 1_o for_o 10_o as_o usual_o to_o =_o 6_o the_o half_a of_o 12_o reckon_v double_a for_o 12_o so_o be_v =_o 4_o ½_o count_v for_o 9_o to_z 8_o ¼_n between_o the_o centre_n and_o 1._o
but_o 43_o inch_n and_o 1_o â—_o about_o who_o four_o part_n be_v under_o 11_o so_o that_o here_o i_o may_v very_o well_o abate_v 1_o inch_n from_o the_o four_o part_n of_o the_o line_n so_o consequent_o if_o the_o rind_n be_v thin_a and_o the_o tree_n less_o a_o less_o allowance_n will_v serve_v and_o if_o the_o rind_n be_v thick_a and_o the_o tree_n large_a there_o ought_v to_o be_v more_o as_o by_o cut_v the_o rind_n away_o and_o then_o take_v the_o true_a diameter_n you_o may_v plain_o see_v this_o measure_n by_o the_o four_o part_n of_o the_o circumference_n for_o the_o side_n of_o the_o square_n and_o allowance_n for_o the_o bark_n be_v allow_v for_o as_o before_z i_o say_v will_v prove_v to_o be_v just_a one_o 5_o part_n over-measure_n especial_o consider_v this_o that_o when_o it_o be_v hew_v and_o large_a wane_v leave_v than_o the_o tree_n be_v mark_v for_o more_o measure_n sometime_o by_o 10_o foot_n in_o 60_o than_o there_o be_v before_o it_o be_v hew_v the_o reason_n be_v because_o when_o the_o tree_n be_v round_o and_o unhewn_a the_o gird_v it_o and_o count_v the_o four_o part_n for_o the_o side_n of_o the_o square_n be_v but_o very_o little_o more_o than_o the_o inscribe_v square_a and_o then_o be_v hew_v and_o that_o scarce_o to_o a_o eight_o square_n and_o measure_v with_o a_o pair_n of_o calliper_n to_o the_o extremity_n of_o that_o do_v not_o then_o allow_v the_o square_n equal_a to_o the_o circle_n for_o the_o side_n of_o the_o square_n as_o by_o the_o work_n by_o those_o several_a square_n will_v very_o plain_o appear_v which_o be_v foretell_v and_o warn_v of_o let_v those_o who_o it_o concern_v look_v to_o it_o but_o this_o be_v premise_v and_o the_o party_n agree_v the_o difference_n be_v as_o 4_o to_z 5_o the_o best_a way_n to_o measure_v round_a timber_n i_o conceive_v be_v by_o the_o diameter_n take_v with_o a_o pair_n of_o calliper_n and_o the_o length_n which_o for_o the_o just_a and_o true_a measure_n be_v large_o handle_v already_o but_o if_o this_o allowance_n be_v agree_v on_o than_o the_o proportion_n for_o it_o be_v thus_o as_o 1-526_a to_o the_o diameter_n so_o be_v the_o length_n to_o a_o four_o and_o so_o be_v that_o four_o to_o the_o content_a in_o foot_n example_n the_o extent_n from_o 1-526_a to_o 15-26_a shall_v reach_v be_v twice_o repeat_v from_o 10_o foot_n the_o length_n to_o 10_o foot_n the_o content_a require_v be_v all_o at_o one_o point_n or_o another_o example_n the_o extent_n from_o 1-526_a to_o 20_o inch_n the_o diameter_n be_v twice_o repeat_v the_o same_o way_n from_o 10_o foot_n the_o length_n shall_v reach_v to_o 17_o foot_n ¼_n the_o content_a or_o if_o you_o have_v the_o circumference_n and_o length_n then_o the_o extent_n from_o 48_o to_o the_o inch_n about_o be_v turn_v twice_o the_o same_o way_n from_o the_o length_n in_o foot_n shall_v reach_v to_o the_o content_a require_v the_o extent_n from_o 48_o to_o 62_o the_o inch_n about_o be_v turn_v twice_o from_o 10_o the_o same_o way_n shall_v reach_v to_o 17_o foot_n ¼_n the_o content_a in_o that_o measure_n thus_o you_o have_v full_a and_o complete_a direction_n for_o the_o measure_n of_o any_o round_a timber_n by_o the_o line_n of_o number_n by_o have_v the_o diameter_n and_o length_n give_v after_o any_o usual_a manner_n there_o remain_v only_o one_o general_a and_o natural_a way_n by_o find_v the_o base_a of_o the_o middle_n or_o one_o end_n by_o the_o 7_o the_o problem_n of_o superficial_a measure_n and_o then_o to_o multiply_v that_o base_a by_o the_o length_n will_v give_v the_o true_a content_a in_o foot_n or_o inch_n thus_o have_v find_v the_o base_a of_o the_o cillender_n by_o the_o seven_o or_o 10_o problem_n of_o superficial-measure_n then_o if_o you_o multiply_v that_o base_a be_v find_v in_o square_a inch_n by_o the_o length_n in_o inch_n you_o shall_v have_v the_o whole_a content_a in_o cube_n inches_n example_n suppose_v a_o cillender_n have_v 10_o inch_n for_o its_o diameter_n then_o by_o the_o seven_o or_o 10_o abovesaid_a you_o shall_v find_v the_o base_a to_o be_v 78-54_a then_o if_o you_o multiply_v 78-54_a by_o 80_o the_o suppose_a length_n in_o inch_n you_o shall_v find_v 2356-20_a cube_n inches_n which_o divide_v by_o 1728_o the_o inch_n in_o a_o cube_n foot_n show_v how_o many_o foot_n there_o be_v etc._n etc._n and_o as_o to_o the_o number_n of_o figure_n and_o the_o fraction_n cut_v off_o you_o have_v ample_a direction_n in_o the_o first_o problem_n and_o the_o three_o section_n of_o the_o sixâ—_o chapter_n problem_n vii_o how_o to_o measure_v a_o pyramid_n or_o taper_n timber_n or_o the_o section_n of_o a_o cone_n 1._o first_o get_v the_o perpendicule_a length_n of_o the_o pyramid_n or_o cone_n thus_o multiply_v half_o the_o diameter_n of_o the_o base_a ab_fw-la by_o itself_o then_o measure_v the_o side_n ad_fw-la and_o multiply_v that_o by_o itself_o then_o take_v the_o lesser_a square_n out_o of_o the_o great_a and_o the_o square_a root_n of_o the_o residue_n be_v the_o perpendicule_a altitude_n require_v viz._n db._n example_n suppose_v the_o half_a diameter_n of_o the_o base_a ac_fw-la be_v 10-25_a and_o the_o side_n dam_fw-ge 100_o ab_fw-la 10-25_a and_o 10-25_a multiply_v together_o call_v square_n make_v 105_o 0625_o dc_o 100_o multiply_v by_o 100_o call_v square_n make_v 10000_o then_o the_o lesser_a square_n 105_o 0625_o take_v out_o of_o 10000_o the_o great_a square_n the_o remainder_n be_v 9894_o 9375_o who_o square_a root_n find_v by_o the_o 8_o the_o problem_n of_o the_o six_o chapter_n be_v 99-475_a the_o true_a length_n of_o the_o line_n db_n the_o length_n or_o height_n of_o the_o cone_n then_o if_o you_o multiply_v the_o area_n or_o content_a of_o the_o base_a ac_fw-la 20-5_a which_o by_o the_o seven_o or_o 10_o of_o superficial_a measure_n be_v find_v to_o be_v 160-08_a by_o 33-158_a a_o three_o part_n of_o 99-475_a the_o whole_a height_n make_v 5308_o cut_v off_o the_o fraction_n for_o the_o true_a content_a of_o the_o cone_n who_o length_n be_v 99_o inch_n and_o near_o a_o half_a and_o who_o base_a be_v 20_o inch_n and_o a_o half_a diameter_n 2._o then_o second_o for_o the_o segment_n or_o section_n of_o a_o cone_n the_o shape_n or_o form_n of_o all_o round_a taper_n timber_n the_o true_a way_n be_v thus_o by_o the_o length_n and_o difference_n of_o diameter_n find_v the_o whole_a length_n of_o the_o cone_n which_o for_o all_o manner_n of_o timber_n as_o it_o grow_v this_o way_n be_v near_o enough_o as_o thus_o as_o the_o difference_n of_o the_o diameter_n at_o the_o the_o two_o end_n be_v to_o the_o length_n between_o the_o two_o end_n so_o be_v the_o diameter_n at_o the_o base_a to_o the_o whole_a length_n of_o the_o cone_n example_n the_o difference_n between_o the_o diameter_n ac_fw-la and_o of_o be_v 13-70_a the_o length_n ae_n be_v 66-32_a than_o the_o extent_n on_o the_o line_n of_o number_n from_o 13-70_a the_o difference_n of_o the_o diameter_n to_o 66-32_a the_o length_n between_o shall_v reach_v the_o same_o way_n from_o 20-50_a the_o great_a diameter_n to_o 99_o and_o better_o the_o length_n that_o make_v up_o the_o cone_n at_o that_o angle_n oâ—_n taper_v in_o the_o timber_n then_o if_o by_o the_o last_o rule_n you_o measure_v it_o as_o a_o cone_n of_o that_o length_n and_o also_o measure_v the_o little_a end_n or_o point_n at_o his_o length_n and_o diameter_n and_o then_o last_o this_o little_a cone_n take_v out_o of_o the_o great_a cone_n there_o remain_v the_o true_a content_a of_o the_o taper-piece_n that_o be_v to_o be_v measure_v viz._n 5246-71_a when_o 61-30_a the_o content_a of_o the_o small_a cone_n at_o the_o end_n be_v take_v out_o of_o 5308_o the_o content_a of_o the_o whole_a pyramid_n 3._o if_o this_o way_n seem_v too_o troublesome_a for_o the_o common_a use_n then_o use_v this_o be_v more_o brief_a to_o the_o content_a that_o be_v find_v out_o by_o the_o diameter_n in_o the_o midst_n of_o the_o timber_n and_o the_o length_n add_v the_o content_a of_o a_o piece_n find_v out_o by_o half_a the_o difference_n of_o diameter_n and_o one_o three_o part_n of_o the_o length_n of_o the_o whole_a piece_n and_o the_o sum_n of_o they_o two_o shall_v be_v the_o whole_a content_a require_v 4._o or_o else_o divide_v the_o length_n of_o the_o tree_n into_o 4_o or_o 5_o part_n and_o measure_v the_o middle_n of_o each_o part_n several_o and_o that_o cast_v up_o by_o his_o proper_a length_n shall_v give_v the_o content_a of_o each_o piece_n then_o the_o sum_n of_o the_o content_n of_o all_o the_o piece_n put_v together_o be_v the_o true_a content_a of_o the_o whole_a taper_n piece_n very_o near_o note_v that_o this_o curiosity_n shall_v never_o need_v to_o be_v
the_o true_a circumference_n of_o a_o round_a cillender_n five_o at_o 4_o foot_n just_o be_v set_v r_o for_o the_o circumference_n according_a to_o the_o former_a allowance_n six_o at_o 1_o foot_n 5_o inch_n 1_o 7_o fâ—rè_fw-la be_v a_o spot_n and_o close_o to_o it_o the_o letter_n w_n as_o the_o gagepoint_n for_o a_o wine-gallon_n seven_o at_o near_o 19_o inch_n or_o 1_o foot_n 7_o inch_n be_v another_o spot_n and_o close_o to_o it_o the_o letter_n a_o as_o the_o gagepoint_n of_o a_o ale-gallon_n eight_o at_o 2_o foot_n 8_o inch_n 8_o 10_o be_v a_o spot_n and_o close_o to_o it_o be_v set_v b_o for_o the_o gagepoint_n of_o a_o beer_n barrel_n and_o at_o 2_o foot_n 7_o inch_n be_v set_v a_o for_o a_o ale_n barrel_n the_o use_v whereof_o in_o order_n follow_v the_o figure_n on_o the_o left-side_n or_o fixed-edge_n be_v read_v and_o count_v as_o those_o on_o the_o right_n for_o the_o small_a 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o 10_o 11_o be_v to_o represent_v inch_n and_o the_o cut_v between_o quarter_n of_o inch_n then_o the_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o 10_o figure_n next_o somewhat_o big_a as_o to_o represent_v so_o many_o foot_n and_o the_o cut_v between_o be_v whole_a inch_n then_o 20_o 30_o 40_o 50_o 60_o 70_o 80_o 90_o 100_o 150_o for_o ten_o of_o foot_n and_o the_o part_n between_o for_o single-feet_n for_o the_o most_o part_n or_o else_o whole_a and_o half_a foot_n as_o be_v usual_a the_o use_v follow_v use_v i._o a_o piece_n of_o timber_n be_v not_o square_a or_o have_v its_o breadth_n and_o depth_n unequal_a to_o make_v it_o square_n or_o find_v the_o square_n equal_a set_v the_o breadth_n of_o the_o piece_n count_v on_o the_o rightside_n to_o the_o same_o breadth_n count_v on_o the_o left-side_n then_o right_a against_o the_o depth_n find_v on_o the_o left-side_n on_o the_o right_n or_o single-side_n be_v the_o inch_n and_o quarter_n square_v require_v example_n at_o 15_o inch_n broad_a and_o 9_o inch_n thick_a or_o deep_a set_a 9_o inch_n on_o the_o rightside_n to_o 9_o on_o the_o left_a then_o right_a against_o 15_o inch_n or_o 1_o foot_n 3_o inch_n find_v out_o on_o the_o double_a or_o left-side_n on_o the_o right_n or_o single-side_n be_v 11_o inch_n and_o â…_n the_o square_a equal_a require_v also_o if_o you_o set_v 15_o to_o 15_o then_o right_o against_o 9_o find_v out_o on_o the_o left-side_n on_o the_o rightside_n be_v 11_o inch_n â…_n the_o square_a equal_a require_v use_v ii_o the_o side_n of_o the_o square_n give_v to_o find_v how_o much_o in_o length_n will_v make_v 1_o foot_n set_a the_o inch_n or_o foot_n &_o inch_n find_v out_o on_o the_o rightside_n to_o 1_o foot_n on_o the_o left_a then_o right_a against_o 1_o foot_n on_o the_o right_n be_v the_o inch_n or_o the_o foot_n and_o inch_n require_v to_o make_v a_o foot_n of_o timber_n example_n at_o 9_o inch_n square_a set_a 9_o inch_n find_v out_o on_o the_o rightside_n to_o 1_o foot_n on_o the_o left-side_n then_o right_a against_o 1_o on_o the_o rightside_n be_v 1_o foot_n 9_o inch_n 2_o â—_o on_o the_o left_a if_o the_o square_n be_v so_o big_a that_o the_o 1_o on_o the_o right_a fall_n beyond_o the_o end_n at_o the_o beginning_n then_o right_o against_o 10_o foot_n on_o the_o rightside_n be_v on_o the_o left_a the_o hundred_o part_n of_o a_o foot_n that_o make_v a_o foot_n of_o timber_n example_n at_o 4_o foot_n square_n set_a 4_o foot_n find_v out_o on_o the_o right_n to_z 1_o foot_n on_o the_o left_a then_o right_a against_o 10_o foot_n on_o the_o rightside_n be_v 0-063_a on_o the_o left-side_n or_o against_o 12_o foot_n you_o have_v 9_o 12_o part_n of_o 1_o inch_n the_o length_n that_o go_v to_o make_v 1_o foot_n of_o timber_n require_v use_v iii_o at_o any_o bigness_n or_o inches_n or_o foot_n and_o inch_n square_a to_o find_v how_o much_o be_v in_o 1_o foot_n long_o just_o as_o the_o rule_n stand_v even_o that_o be_v 1_o foot_n on_o the_o right_n against_o 1_o foot_n on_o the_o left_a seek_v the_o inch_n or_o foot_n and_o inch_n the_o piece_n be_v square_a on_o the_o right_n or_o single-side_n and_o just_a against_o it_o on_o the_o left_a or_o double-side_n be_v the_o answer_v require_v in_o inch_n or_o foot_n and_o inch_n example_n at_o 19_o inch_n square_a just_o against_o 1_o foot_n 7_o inch_n or_o 19_o inch_n which_o be_v all_o one_o find_v out_o on_o the_o rightside_n on_o the_o left-side_n be_v 2_o foot_n 6_o inch_n the_o quantity_n of_o timber_n in_o 1_o foot_n long_o at_o 19_o inch_n square_a which_o number_n of_o 2_o foot_n 6_o inch_n multiply_v by_o the_o length_n in_o foot_n gives_z the_o true_a content_a of_o the_o whole_a piece_n of_o timber_n require_v note_v that_o this_o be_v a_o most_o excellent_a way_n for_o great_a wood_n and_o very_o exact_a also_o note_v that_o here_o by_o inspection_n you_o may_v square_v a_o small_a number_n or_o find_v the_o square-root_n of_o a_o small_a number_n as_o thus_o the_o square_n of_o 8_o &_o ½_o be_v near_o 72_o or_o the_o square-root_n of_o 72_o be_v near_o 8_o &_o ½_o use_v iu._n the_o side_n of_o the_o square_n and_o the_o length_n of_o any_o piece_n be_v give_v to_o find_v the_o content_a in_o foot_n and_o part_n set_a the_o word_n square_a or_o 1_o foot_n always_o to_o the_o length_n find_v out_o on_o the_o left-side_n then_o right_a against_o the_o inch_n or_o foot_n and_o inch_n square_v count_v on_o the_o right_n on_o the_o left_a be_v the_o content_a require_v example_n at_o 20_o foot_n long_o and_o 15_o inch_n square_a set_a 1_o foot_n on_o the_o right_n to_o 20_o on_o the_o left_a then_o right_o against_o 1_o foot_n 3_o inch_n on_o the_o right_n be_v 31_o foot_n 2_o inch_n and_o ½_o the_o content_a note_v that_o if_o the_o piece_n be_v very_o small_a call_v the_o foot_n on_o the_o left-side_n inch_n and_o the_o part_n between_o 12_o of_o inch_n then_o the_o answer_n will_v be_v find_v on_o the_o left-side_n in_o 144_o of_o a_o foot_n example_n at_o 2_o inch_n square_a and_o 30_o foot_n long_o how_o much_o be_v there_o set_a 1_o foot_n on_o the_o right_n to_o 30_o foot_n on_o the_o left_a then_o right_a against_o 2_o foot_n on_o the_o right_n count_v as_o 2_o inch_n be_v 120_o part_n of_o a_o foot_n divide_v into_o 144_o part_n be_v just_a 10_o inch_n for_o 10_o time_n 12_o be_v 120._o but_o if_o it_o be_v â—_o great_a piece_n of_o timber_n then_o work_v thus_o set_v 1_o foot_n or_o the_o word_n square_a to_o the_o length_n on_o the_o left_a count_v the_o single_a foot_n 10_o of_o foot_n then_o right_a against_o the_o foot_n and_o inch_n square_a be_v the_o 100_o of_o foot_n require_v example_n at_o 40_o foot_n long_o and_o 4_o foot_n square_a set_a 1_o foot_n to_o 4_o foot_n count_v as_o 40_o on_o the_o left_a then_o right_a against_o 4_o foot_n on_o the_o right_n be_v 640_o the_o true_a content_a increase_v the_o 10_o to_o 100_o thus_o much_o for_o square_a timber_n though_o there_o be_v many_o other_o way_n and_o manner_n of_o work_n some_o whereof_o you_o may_v find_v in_o a_o book_n set_v forth_o under_o the_o name_n of_o the_o carpenter_n rule_n 1666_o by_o i._o brown_a and_o well_o know_v abroad_o already_o use_v v._o for_o round_a timber_n the_o middle_a diameter_n of_o any_o piece_n give_v to_o find_v how_o much_o be_v in_o a_o foot_n long_o at_o true_a measure_n set_a the_o spot_n by_o t.d._n to_o 1_o foot_n on_o the_o left_a then_o just_a against_o the_o inch_n or_o foot_n and_o inch_n diameter_n find_v on_o the_o right_n be_v the_o quantity_n of_o timber_n in_o 1_o foot_n long_o on_o the_o left-side_n require_v example_n at_o 2_o foot_n 9_o inch_n diameter_n suppose_v a_o piece_n of_o stone-pillar_n or_o garden-roul_n be_v two_o foot_n 9_o inch_n diameter_n set_v the_o spot_n by_o t.d._n just_a against_o 1_o foot_n then_o right_o against_o 2_o foot_n 9_o inch_n find_v on_o the_o right_n on_o the_o left_a be_v 6_o foot_n the_o quantity_n of_o solid_a measure_n in_o one_o foot_n long_o which_o be_v multiply_v by_o the_o length_n in_o foot_n gives_z the_o true_a content_a of_o the_o whole_a piece_n note_v that_o if_o you_o will_v have_v the_o usual_a allowance_n set_v d_o to_o 1_o instead_o of_o t.d._n use_v vi_o the_o diameter_n of_o any_o piece_n of_o timber_n give_v to_o find_v how_o much_o in_o length_n will_v make_v one_o foot_n set_a 1_o foot_n on_o the_o left_a to_o the_o inch_n diameter_n count_v on_o the_o right_n then_o right_a against_o t.d._n for_o true_a measure_n or_o d_o for_o the_o usual_a allowance_n be_v the_o answer_v require_v find_v on_o the_o
the_o fragment_n of_o a_o globe_n but_o the_o other_o be_v by_o a_o way_n find_v out_o by_o mr._n bennit_n a_o cooper_n that_o have_v long_o exercise_v the_o way_n of_o gage_v which_o be_v by_o compare_v a_o cask_n know_v and_o its_o quantity_n of_o emptiness_n to_o a_o cask_n unknown_a and_o its_o inch_n of_o emptiness_n as_o follow_v first_o by_o the_o line_n of_o number_n and_o artificial_a line_n of_o segment_n to_o find_v the_o quantity_n of_o gallon_n that_o any_o vessel_n want_v of_o be_v full_a at_o any_o number_n of_o inch_n from_o the_o inside_n of_o the_o bunghole_n to_o the_o superficies_n of_o the_o liquor_n which_o be_v usual_o call_v inch_n dry_a extend_v the_o compass_n on_o the_o line_n of_o number_n from_o the_o inch_n and_o ten_o diameter_n at_o the_o bung_n to_o 100_o on_o the_o line_n of_o segment_n the_o same_o extent_n apply_v the_o same_o way_n from_o the_o inch_n and_o part_n dry_a shall_v reach_v to_o a_o four_o number_n on_o the_o line_n of_o artificial_a segment_n which_o four_o number_n you_o must_v keep_v or_o if_o you_o will_v you_o may_v use_v the_o inch_n wet_a lay_v the_o same_o extent_n from_o the_o inch_n wet_a and_o that_o also_o will_v on_o the_o segment_n give_v a_o four_o number_n which_o you_o must_v likewise_o keep_v then_o second_o as_o the_o extent_n from_o 1_o to_o the_o whole_a content_a of_o the_o vessel_n in_o wine_n or_o ale-gallon_n so_o be_v the_o four_o number_n keep_v to_o the_o gallon_n of_o emptiness_n or_o fullness_n that_o it_o want_v of_o be_v full_a or_o the_o quantity_n of_o gallon_n in_o the_o vessel_n example_n of_o a_o canary-pipe_n who_o diameter_n at_o bung_n be_v 28_o inch_n and_o 7_o and_o full_a content_n be_v gallon_n 116_o ½_o at_o 12_o inch_n dry_a or_o 16_o inch_n and_o 7_o ten_o wet_v the_o extent_n of_o the_o compass_n from_o 28-7_a to_o 100_o at_o the_o end_n of_o the_o line_n of_o segment_n shall_v reach_v the_o same_o way_n from_o 12_o the_o inch_n dry_a to_o 39_o ½_o on_o the_o line_n of_o segment_n for_o a_o four_o or_o from_o 16-7_a wet_a to_o 60_o 2_o 10_o on_o the_o segment_n for_o his_o four_o also_o which_o two_o 4th_n keep_v then_o second_o the_o extent_n from_o 1_o to_o 116_o ½_o the_o whole_a content_a in_o gallon_n shall_v reaâ—h_v from_o 39_o the_o dry_a four_o on_o the_o line_n of_o number_n to_o 46_o 3_o 10_o for_o the_o gallon_n dry_a or_o want_v or_o the_o same_o extent_n shall_v reach_v the_o same_o way_n on_o the_o line_n of_o number_n from_o 60_o 2_o 10_o the_o four_o number_n for_o wet_a to_o 70_o gallon_n and_o 2_o ten_o in_o the_o vessel_n at_o 16_o inch_n and_o 7_o ten_o wet_a which_o two_o number_n put_v together_o make_v up_o 116_o gallon_n and_o a_o half_a the_o full_a content_a the_o like_a manner_n of_o work_v serve_v for_o any_o cask_n whatsoever_o and_o the_o near_o the_o vessel_n want_v of_o be_v half_o empty_a the_o more_o near_o to_o the_o truth_n will_v your_o work_n be_v and_o the_o most_o error_n in_o very_o round_o and_o swell_a cask_n when_o the_o emptiness_n be_v not_o above_o one_o or_o two_o inch_n but_o in_o vessel_n near_o to_o cillender_n it_o will_v give_v the_o answer_n very_o true_a and_o as_o ready_o as_o any_o way_n whatsoever_o observe_v also_o that_o if_o you_o use_v the_o segment_n in_o take_v the_o want_n you_o must_v abate_v of_o the_o gallon_n find_v till_o you_o come_v to_o the_o 2_o three_o of_o the_o half_a diameter_n that_o be_v to_o say_v the_o rule_n say_v there_o be_v more_o want_v than_o indeed_o there_o be_v and_o that_o somewhat_o considerable_a about_o the_o first_o 6_o inch_n in_o a_o vessel_n of_o 30_o inch_n diameter_n so_o that_o i_o find_v a_o table_n make_v as_o a_o mean_a between_o the_o superficial_a and_o solid_a segment_n will_v do_v the_o work_n the_o true_a and_o best_a of_o any_o other_o or_o else_o use_v the_o mean_a diameter_n and_o mean_a part_n of_o emptiness_n find_v thus_o take_v the_o equade_v diameter_n from_o the_o diameter_n at_o the_o bung_n and_o note_v the_o difference_n then_o half_a this_o difference_n take_v from_o the_o inch_n and_o part_n empty_a give_v the_o mean_a emptiness_n then_o use_v the_o mean_a diameter_n and_o mean_a emptiness_n instead_o of_o the_o other_o and_o the_o work_n be_v more_o exact_a the_o other_o way_n of_o mr._n bennits_n invention_n be_v thus_o first_o you_o be_v to_o fill_v a_o ordinary_a cask_n of_o a_o competent_a magnitude_n as_o 60_o or_o 100_o gallon_n of_o a_o mean_a form_n between_o a_o spheriord_n or_o roundish_n form_n and_o a_o cillenderical_a form_n or_o else_o fill_v two_o cask_n of_o each_o form_n and_o learn_v the_o true_a content_a and_o diameter_n of_o that_o mean_a vessel_n or_o rather_o of_o both_o those_o vessel_n and_o the_o vessel_n be_v full_a draw_v off_o with_o a_o true_a gallon-measure_n and_o on_o the_o draw_v off_o every_o gallon_n take_v the_o exact_a quantity_n of_o inch_n and_o 10_o part_n that_o the_o draw_v off_o of_o every_o gallon_n make_v in_o the_o emptiness_n or_o dryness_n of_o that_o mean_a vessel_n or_o rather_o both_o those_o vessel_n at_o least_o until_o you_o have_v draw_v off_o the_o half_a quantity_n of_o the_o vessel_n which_o number_n of_o gallon_n draw_v off_o and_o the_o inch_n and_o ten_o part_n of_o emptiness_n or_o fullness_n or_o dryness_n or_o wetness_n you_o must_v draw_v into_o a_o table_n or_o insert_v they_o on_o a_o rule_n make_v the_o inch_n as_o equal_a part_n and_o the_o gallon_n and_o his_o proportional_a part_n of_o a_o gallon_n the_o unequal_a part_n then_o with_o the_o line_n of_o number_n and_o this_o mean_a table_n or_o rather_o two_o table_n or_o scale_n which_o you_o may_v put_v on_o a_o rule_n as_o mr._n bennit_n have_v do_v you_o may_v find_v out_o the_o want_n of_o any_o cask_n whatsoever_o either_o spherioid_a or_o cillender-like_a as_o follow_v this_o measure_a cask_n on_o the_o scale_n or_o table_n for_o method_n sake_n and_o avoid_v tautology_n i_o shall_v call_v the_o first_o cask_n and_o the_o vessel_n or_o cask_n who_o want_n you_o will_v know_v i_o shall_v call_v the_o second_o cask_n then_o the_o proportion_n be_v thus_o as_o the_o diameter_n at_o the_o bung_n of_o the_o second_o cask_n be_v to_o the_o bung_a diameter_n of_o the_o first_o cask_n which_o be_v always_o fix_v so_o be_v the_o inch_n dry_a of_o the_o second_o cask_n to_o a_o four_o on_o the_o line_n of_o number_n which_o four_o number_n seek_v on_o the_o inch_n of_o your_o table_n or_o scale_n on_o the_o opposite-part_n of_o your_o scale_n or_o table_n give_v a_o 5_o number_n which_o you_o must_v keep_v then_o as_o the_o whole_a content_a of_o the_o first_o cask_n be_v to_o the_o whole_a content_a of_o the_o second_o cask_n so_o be_v the_o first_o number_n keep_v to_o the_o number_n of_o gallon_n the_o vessel_n want_v of_o be_v full_a at_o so_o many_o inch_n dry_a example_n there_o be_v such_o a_o scale_n make_v on_o purpose_n for_o victualler_n use_v to_o measure_v what_o they_o want_v of_o a_o barrel_n of_o ale_n be_v put_v into_o a_o beer-barrel_n which_o scale_n i_o shall_v here_o use_v to_o try_v this_o former_a example_n by_o suppose_v as_o before_o a_o canary_n pipe_v want_v 12_o inch_n of_o be_v full_a and_o the_o content_a 116_o ½_o gallon_n and_o 28_o inch_n and_o 7_o ten_o diameter_n at_o bung_n the_o extent_n on_o the_o line_n of_o number_n from_o 28-7_a to_o 22-5_a shall_v reach_v from_o 12_o to_o 9-4_a then_o just_a against_o 9_o inch_n and_o 4_o ten_o on_o that_o barrel_n scale_n i_o find_v 14_o gallon_n of_o beer_n which_o be_v 17_o gallon_n and_o a_o half_a of_o wine_n be_v the_o 5_o number_n to_o be_v keep_v then_o the_o extent_n from_o 44_o the_o content_a of_o a_o barrel_n in_o wine-gallon_n to_o 116_o ½_o the_o content_a of_o a_o canary-pipe_n in_o the_o same_o gallon_n shall_v reach_v the_o same_o way_n from_o 17_o ½_o the_o number_n keep_v to_o 46_o and_o near_o a_o half_a the_o gallon_n want_v at_o 12_o inch_n dry_a in_o the_o canary_n pipe_n and_o 46_o gallon_n and_o 3_o quart_n be_v the_o number_n mr._n bennit_n find_v in_o a_o canary-pipe_n by_o measure_v at_o 12_o inch_n dry_a thus_o you_o have_v a_o account_n of_o the_o two_o easy_a mechanic_n way_n to_o discover_v the_o want_v of_o cask_n very_o applicable_a and_o ready_a and_o experiment_v to_o be_v propè_fw-la verum_fw-la the_o gallon_n want_v in_o a_o barrel_n at_o every_o inch_n and_o quarter_n  _fw-fr beer_n gall._n wine_n gall_n  _fw-fr  _fw-fr  _fw-fr gall_n pi_fw-la 100_o gall_n pi_fw-la 100_o gall_n 1000_o  _fw-fr  _fw-fr 0_o 0_o 40_o 0_o 0_o 49_o 0_o 0612_o  _fw-fr  _fw-fr 0_o 1_o 20_o 0_o 1_o 47_o 0_o
from_o a_o but_o if_o you_o remove_v to_o e_o the_o thread_n fall_v on_o 3_o on_o the_o shadow_n the_o altitude_n will_v be_v but_o one_o three_o part_n of_o the_o distance_n ea_fw-la from_o hence_o you_o may_v observe_v that_o observe_v at_o c_o and_o at_z d_o where_o the_o thread_n fall_v on_o 1_o and_o on_o 2_o the_o distance_z between_z c_z and_o d_o be_v equal_a to_o the_o altitude_n so_o likewise_o at_o d_o and_o at_z e_z and_o so_o by_o consequence_n at_o 1_o ½_o and_o 2_o ½_o and_o 3_o ½_o or_o any_o other_o equal_a part_n this_o be_v a_o excellent_a easy_a way_n the_o like_a will_v be_v if_o you_o observe_v at_o d_o and_o c_o look_v up_o to_o f_o where_o the_o altitude_n of_o be_v twice_o the_o distance_n ac_fw-la use_v v._o another_o way_n by_o the_o line_n of_o shadow_n at_o one_o station_n measure_v any_o distance_n as_o foot_n yard_n ii_o or_o the_o like_a from_o any_o object_n as_o suppose_v from_o a_o to_o d_o be_v 200_o foot_n and_o look_v up_o to_o b_o the_o thread_n cut_v the_o stroke_n by_o 2_o on_o the_o line_n of_o shadow_n then_o by_o the_o line_n of_o number_n say_v as_o 2_o the_o part_n cut_v be_v to_o 1_o so_o be_v 200_o the_o distance_n measure_v to_o 100_o the_o height_n or_o suppose_v i_o measure_v any_o other_o uneven_a number_n and_o the_o thread_n fall_v between_o 00_o on_o the_o loose-piece_n and_o 1_o on_o the_o shadow_n common_o call_v contrary_a shadow_n the_o rule_n be_v always_o thus_o as_o the_o part_n cut_v by_o the_o thread_n be_v to_o 1_o so_o be_v the_o measure_a distance_n to_o the_o height_n require_v be_v less_o than_o the_o measure_a distance_n but_o when_o the_o thread_n fall_v between_o 1_o and_o 90_o at_o the_o head_n call_v right_a shadow_n then_o the_o rule_n go_v thus_o as_o 1_o to_o the_o part_n cut_v by_o the_o thread_n so_o be_v the_o measure_a distance_n to_o the_o height_n be_v always_o more_o than_o the_o measure_a distance_n from_o the_o foot_n of_o the_o object_n to_o the_o station_n use_v vi_o another_o way_n by_o the_o line_n of_o shadow_n and_o the_o sun_n shine_v when_o the_o sun_n shine_v find_v his_o altitude_n and_o also_o as_o the_o thread_n lie_v see_v what_o division_n on_o the_o line_n of_o shadow_n be_v cut_v by_o the_o thread_n and_o then_o straightway_o measure_v the_o shadow_n length_n on_o the_o ground_n and_o if_o the_o sun_n be_v under_o 45_o degree_n high_a the_o shadow_n be_v long_o than_o the_o length_n of_o that_o object_n which_o cause_v the_o shadow_n but_o if_o the_o sun_n be_v above_o 45_o degree_n high_a than_o the_o object_n be_v long_o than_o the_o shadow_n and_o the_o operation_n be_v thus_o by_o the_o line_n of_o number_n only_o with_o a_o pair_n of_o compass_n the_o height_n of_o the_o sun_n be_v under_o 45_o say_v as_o the_o part_n cut_v by_o the_o thread_n on_o the_o shadow_n be_v to_o 1_o so_o be_v the_o shadow_n measure_v to_o the_o height_n require_v the_o height_n of_o the_o sun_n be_v above_o 45_o say_v as_o 1_o to_o the_o part_n cut_v by_o the_o thread_n on_o the_o line_n of_o shadow_n so_o be_v the_o measure_n of_o the_o shadow_n to_o the_o height_n in_o the_o same_o part_n use_v vii_o to_o find_v a_o inaccessible_a altitude_n by_o the_o quadrat_fw-la and_o shadow_n otherwise_o observe_v the_o altitude_n at_o both_o station_n and_o count_v the_o observe_v altitude_n at_o both_o station_n on_o the_o quadrat_fw-la or_o shadow_n according_a as_o it_o happen_v to_o be_v either_o above_o or_o under_o 45_o degree_n and_o take_v the_o lesser_a out_o of_o the_o great_a note_v the_o remainder_n for_o the_o first_o term_n and_o the_o divisor_n to_o divide_v the_o distance_n between_o the_o station_n increase_v with_o cipher_n if_o need_v be_v and_o the_o quotient_n be_v the_o answer_v require_v but_o by_o the_o line_n of_o number_n work_v thus_o the_o extent_n from_o the_o difference_n to_o 1_o shall_v reach_v the_o same_o way_n from_o the_o measure_a distance_n to_o the_o height_n require_v example_n figure_n ii_o let_v abcde_v represent_v the_o object_n and_o three_o station_n let_v the_o line_n ac_fw-la represent_v the_o altitude_n the_o point_n b_o one_o station_n 50_o foot_n from_o a_o d_o another_o station_n 100_o foot_n from_o a_o or_o 50_o from_o b_o and_o e_z another_o station_n 73_o foot_n from_o d_o or_o 173_o foot_n from_o a_o all_o which_o measure_v you_o need_v not_o know_v before_o but_o only_o bd_o and_o de_fw-fr also_o the_o angle_n at_o b_o 63-27_a and_o his_o compliment_n count_v the_o other_o way_n be_v the_o angle_n at_o c_o 26_o degree_n 33_o minute_n the_o angle_n at_o d_o 45_o and_o his_o compliment_n so_o also_o the_o angle_n at_o e_o 30_o and_o his_o compliment_n 60._o now_o mind_n the_o operation_n by_o either_o of_o these_o first_o lay_v the_o thread_n on_o 26-33_a and_o in_o the_o quadrat_fw-la it_o cut_v 50_o lay_v the_o thread_n on_o 45_o and_o in_o the_o shadow_n or_o quadrat_fw-la it_o cut_v 100_o or_o 1_o or_o if_o you_o lay_v the_o thread_n to_o 60_o then_o in_o the_o shadow_n it_o cut_v 173._o the_o difference_n between_o 173_o and_o 100_o be_v 73._o then_o as_o 73_o the_o difference_n in_o tangent_n between_o the_o two_o observation_n be_v to_o the_o distance_n in_o foot_n 73_o so_o be_v radius_fw-la 100_o or_o the_o side_n of_o the_o quadrat_fw-la to_o 100_o the_o hight_v require_v again_o for_o the_o two_o near_a observation_n who_o difference_n of_o tangent_n be_v 50._o as_o 50_o the_o difference_n in_o tangent_n to_o 50_o foot_n the_o measure_a distance_n so_o be_v 100_o the_o side_n of_o the_o quadrat_fw-la to_o 100_o the_o height_n again_o last_o by_o the_o observation_n at_o b_o &_o e_o the_o difference_n of_o tangent_n be_v 123._o as_o 123_o the_o difference_n in_o tangent_n to_o 123_o the_o measure_a distance_n so_o be_v 100_o the_o radius_fw-la or_o side_n of_o the_o quadrat_fw-la to_o 100_o the_o height_n require_v or_o in_o the_o first_o figure_n the_o angle_n at_o the_o top_n be_v 33-30_a and_o 48-15_a and_o the_o measure_a distance_n 88_o foot_n and_o a_o half_a the_o difference_n in_o tangent_n will_v be_v 45-8_a then_o as_o 45-8_a to_o 100_o the_o side_n of_o the_o quadrat_fw-la so_o be_v 88_o ½_o the_o measure_a distance_n to_o 194_o the_o altitude_n require_v this_o way_n be_v general_a for_o any_o station_n though_o both_o of_o right_a shadow_n or_o both_o of_o contrary_n or_o mix_v of_o right_a and_o contrary_a and_o do_v by_o the_o line_n of_o number_n or_o by_o multiplication_n and_o division_n also_o note_v that_o you_o may_v find_v this_o difference_n in_o tangent_n or_o secant_v by_o the_o natural_a tangent_n or_o natural_a secant_v on_o the_o sector_n and_o the_o scale_n of_o equal_a part_n belong_v to_o they_o thus_o take_v the_o distance_n between_o the_o compleplement_n of_o the_o two_o observation_n on_o the_o great_a or_o lesser_a line_n of_o tangent_n as_o be_v most_o convenient_a and_o measure_v this_o distance_n in_o the_o line_n of_o line_n or_o equal_a part_n equal_a to_o that_o radius_fw-la and_o that_o shall_v be_v the_o difference_n in_o tangent_n require_v the_o like_a for_o the_o secant_v also_o by_o the_o artificial_a number_n sin_n and_o tangent_n you_o may_v come_v by_o this_o difference_n in_o tangent_n or_o secant_v very_o well_o thus_o just_o right_a against_o the_o tangent_fw-la of_o the_o co-altitude_n count_v on_o the_o line_n of_o tangent_n in_o the_o line_n of_o number_n be_v one_o number_n and_o against_o the_o tangent_fw-la of_o the_o compliment_n of_o the_o other_o angle_n be_v the_o other_o number_n only_o with_o this_o caution_n that_o if_o the_o tangent_fw-la be_v above_o 45_o then_o take_v the_o distance_n from_o 45_o to_o the_o tangent_fw-la as_o it_o be_v count_v backward_o with_o compass_n and_o set_v the_o same_o the_o increase_a way_n from_o 1_o on_o the_o number_n to_o the_o other_o number_n require_v then_o the_o lesser_a take_v from_o the_o great_a leave_v the_o difference_n in_o tangent_n that_o be_v require_v in_o the_o same_o manner_n the_o sin_n count_v from_o 90_o and_o lay_v the_o contrary_a way_n from_o 1_o increase_a will_v give_v the_o difference_n in_o secant_v to_o measure_v the_o baâ—â—_n and_o hypothenusa_n by_o number_n only_o use_v viii_o another_o pretty_a way_n by_o scale_n and_o compass_n without_o arithmetic_n from_o t._n s._n then_o draw_v the_o line_n that_o the_o thread_n make_v on_o the_o board_n then_o measure_v from_o your_o stand_n to_o the_o foot_n of_o the_o object_n and_o take_v the_o number_n of_o foot_n or_o yard_n from_o any_o scale_n and_o lay_v it_o from_o the_o right_a angle_n on_o the_o other_o line_n and_o raise_v a_o perpendicule_a from_o thence_o to_o the_o plumb-line_n make_v by_o the_o thread_n and_o
that_o shall_v be_v the_o altitude_n require_v be_v measure_v on_o the_o same_o scale_n example_n let_v abgd_n represent_v the_o board_n end_n or_o trencher_n and_o on_o that_o let_v ab_fw-la be_v one_o straight_a line_n and_o agnostus_n another_o perpendicule_a to_o it_o in_o the_o point_n a_o knock_v in_o one_o pin_n and_o in_o b_o or_o any_o where_o towards_o the_o end_n another_o on_o the_o pin_n at_o a_o hang_v a_o thread_n and_o plummet_n and_o stand_v at_o ay_o any_o convenient_a station_n look_v up_o by_o the_o two_o pin_n at_o b_o and_o a_o till_o they_o bourn_n in_o a_o right_a line_n with_o the_o point_n h_n the_o object_n who_o height_n be_v to_o be_v measure_v then_o the_o plummet_n play_v well_o and_o even_o make_v a_o point_n just_a therein_o and_o draw_v the_o line_n ad_fw-la as_o the_o thread_n show_v then_o have_v measure_v the_o distance_n from_o g_o the_o foot_n of_o the_o object_n to_o i_o the_o station_n take_v it_o from_o any_o first_o scale_n and_o lay_v it_o from_o a_o to_o g_o then_o on_o the_o point_n g_o raise_v a_o perpendicule_a to_o agnostus_n till_o it_o intersect_v the_o plumb-line_n ad_fw-la then_o i_o say_v the_o distance_n cd_o measure_v on_o the_o same_o scale_v you_o take_v ac_fw-la from_o shall_v be_v equal_a to_o the_o altitude_n gh_o which_o be_v require_v use_v ix_o the_o same_o work_n at_o two_o station_n but_o if_o you_o can_v come_v to_o measure_v from_o i_o the_o first_o station_n to_o g_o v_n then_o measure_v from_o i_o to_o k_o and_o have_v observe_v at_o i_o and_o draw_v the_o plumb-line_n ad_fw-la take_v the_o measure_n between_o i_o and_o k_o the_o two_o station_n from_o any_o fit_a scale_n of_o equal_a part_n and_o lay_v it_o on_o the_o line_n ac_fw-la from_o a_o to_o c_o viz._n 79_o part_n and_o in_o the_o point_n c_o knock_v another_o pin_n and_o hang_v the_o thread_n and_o plummet_n thereon_o and_o observe_v careful_o where_o this_o last_o plumb-line_n do_v cross_v the_o other_o as_o suppose_v at_o e_o then_z from_o e_z let_v fall_v a_o perpendicule_a to_o the_o line_n ac_fw-la which_o line_n ac_fw-la shall_v be_v the_o height_n gh_o require_v or_o thus_o the_o near_a distance_n from_o e_o to_o ac_fw-la be_v the_o height_n require_v viz._n 120_o of_o the_o same_o part_n that_o ik_fw-mi be_v 79_o note_v the_o figure_n and_o behold_v that_o acfe_n the_o small_a figure_n on_o the_o board_n be_v like_a and_o proportional_a to_o aa_n gh_n the_o great_a figure_n vi_o other_o way_n there_o be_v as_o by_o a_o bowl_n of_o water_n or_o a_o glass_n or_o a_o plash_n of_o water_n or_o a_o square_a but_o these_o set_v down_o be_v as_o convenient_a and_o ready_a as_o any_o whatsoever_o as_o in_o the_o next_o figure_n you_o may_v see_v the_o way_n by_o the_o glass_n and_o square_a as_o thus_o let_v c_o represent_v a_o glass_n a_o bowl_n or_o plash_n of_o water_n wherein_o the_o eye_n at_o a_o see_v the_o picture_n or_o reflection_n of_o the_o object_n e._n then_o by_o the_o line_n of_o number_n as_o cb_n the_o measure_n from_o your_o foot_n to_o the_o glass_n be_v to_o ab_fw-la the_o height_n from_o your_o eye_n to_o the_o ground_n at_o your_o foot_n so_o be_v the_o measure_n from_o c_o to_o d_o to_o the_o height_n de._n see_v figure_n vi._n again_o to_o find_v a_o distance_n by_o the_o square_n that_o be_v not_o overlong_a let_v c_o represent_v the_o upper-corner_n of_o a_o square_a hang_v on_o a_o staff_n at_o f_o then_o the_o one_o part_n of_o the_o square_n direct_v to_o e_o and_o the_o other_o to_o a._n the_o proportion_n will_v hold_v by_o the_o line_n of_o number_n as_o favorina_n 11-37_a to_o fc_v 50_o so_o be_v fc_fw-mi 50_o to_o fe_o 220._o that_o be_v so_o many_o time_n as_o you_o find_v of_o in_o fc_n so_o many_o time_n be_v fc_a in_o fe_o and_o the_o like_a note_v that_o you_o must_v conceive_v afe_a to_o be_v the_o ground_n or_o base-line_n in_o this_o operation_n by_o the_o square_a c_o be_v the_o top_n of_o a_o upright_a staff_n 5_o foot_n long_o call_v 50_o for_o fraction_n sake_n use_v x._o to_o find_v a_o distance_n not_o approachable_a by_o the_o triangule_a quadrant_n first_o i_o plant_v my_o triangule_a quadrant_n set_v upon_o a_o three_o legged_a staff_n and_o ball_n socket_n right_a over_o the_o place_n a_o and_o then_o bring_v the_o index_n with_o two_o sight_n in_o it_o lay_v or_o fasten_v to_o the_o centre_n of_o the_o triangule_a quadrant_n right_a over_o the_o line_n of_o sines_n and_o line_n cut_v 90_o at_o the_o head_n the_o index_n and_o sight_n so_o place_v hold_v it_o there_o and_o bring_v it_o and_o the_o instrument_n together_o till_o you_o see_v the_o mark_n at_o c_o through_o the_o two_o sight_n by_o help_n of_o the_o ball-socket_n and_o then_o there_o keep_v it_o then_o remove_v the_o index_n only_o to_o 0-60_a on_o the_o loose-piece_n which_o make_v a_o right_a angle_n and_o set_v up_o a_o mark_n in_o that_o line_n at_o any_o convenient_a distance_n as_o suppose_v at_o b_o 102_o foot_n from_o a_o then_o remove_v the_o instrument_n to_o b_o and_o lay_v the_o index_n on_o the_o centre_n and_o 0-60_a on_o the_o loose-piece_n direct_v the_o sight_n to_o a_o the_o first_o station_n by_o help_n of_o a_o mark_n leave_v there_o on_o purpose_n then_o remove_v the_o sight_n till_o you_o see_v the_o mark_n at_o c_o and_o note_v exact_o on_o what_o degree_n the_o index_n fall_v as_o here_o on_o 60_o count_v from_o 060_o on_o the_o loose-piece_n or_o on_o 30_o count_v from_o the_o head_n which_o be_v the_o angle_n at_o b_o and_o at_o c._n then_o by_o the_o artificial_a number_n sin_n and_o tangent_n on_o the_o edge_n say_v as_o the_o sine_fw-la of_o 30_o the_o angle_n at_o c_o to_o 102_o the_o measure_a distance_n count_v on_o the_o number_n so_o be_v the_o sine_fw-la of_o 60_o the_o angle_n at_o b_o to_o 117_o on_o the_o number_n the_o distance_n require_v so_o also_o be_v 90_o the_o angle_n at_o a_o to_o 206_o the_o distance_n from_o b_o to_o c._n or_o by_o the_o line_n and_o sin_n on_o the_o quadrant-side_n as_o it_o lie_v thus_o as_o the_o measure_n of_o 102_o take_v from_o any_o scale_n as_o the_o line_n of_o line_n double_v to_o the_o =_o sine_fw-la of_o 30_o lay_v the_o index_n or_o a_o thread_n to_o the_o near_a distance_n so_o be_v the_o =_o sine_fw-la of_o 60_o to_o 117_o measure_v latteral_o on_o the_o same_o line_n of_o line_n and_o so_o be_v the_o =_o sine_fw-la of_o 90_o to_o 206_o the_o distance_n from_o b_o to_o c._n so_o also_o if_o you_o observe_v at_o b_o and_o at_z d_o only_o you_o must_v be_v sure_a to_o set_v your_o instrument_n at_o one_o station_n at_o the_o same_o situation_n as_o at_o the_o other_o as_o a_o look_a back_n from_o station_n to_o station_n will_v do_v it_o and_o the_o same_o way_n of_o work_n will_v serve_v for_o as_o the_o sine_fw-la of_o 20_o to_o 110_o so_o be_v the_o sine_fw-la of_o 40_o to_o 206._o and_o so_o be_v the_o sine_fw-la of_o 120_o to_o the_o line_n dc_o 278_o etc._n etc._n use_v xi_o to_o find_v a_o breadth_n and_o a_o distance_n at_o any_o two_o station_n iu._n let_v ab_fw-la be_v two_o mark_n as_o two_o corner_n of_o a_o house_n or_o wall_n and_o let_v the_o breadth_n between_o they_o be_v demand_v and_o their_o distance_n from_o c_o and_o d_o the_o two_o station_n first_o set_v up_o two_o mark_n at_o the_o two_o station_n then_o set_v up_o the_o instrument_n at_o c_o set_v the_o fiducial_a line_n on_o the_o rule_n to_o d_o the_o other_z mark_n then_o direct_v the_o sight_n exact_o to_o b_o and_o to_o a_o observe_v the_o angle_n dcb_n 45_o and_o dca_n 113-0_a as_o in_o the_o figure_n second_o remove_v the_o instrument_n to_o d_o the_o other_z station_n and_o set_v the_o fiducial-line_n of_o the_o quadrant_n viz._n the_o line_n of_o line_n and_o sines_n direct_o to_o c_z then_o fix_v it_o there_o and_o remove_v the_o index_n and_o sight_n to_o a_o and_o to_z b_o to_o get_v the_o angle_n cda_n 42-30_a and_o cdb_v 109-0_a then_o observe_v that_o the_o 3_o angle_n of_o every_o triangle_n be_v equal_a to_o 180_o degree_n have_v get_v the_o angle_n at_o c_o 113_o and_o the_o angle_n at_o d_o 42-30_a by_o consequence_n as_o you_o take_v 155_o the_o sum_n of_o the_o angle_n at_o c_o and_o d_o out_z of_o 180_o then_o there_o remain_v 24-30_a the_o angle_n at_o a._n so_o also_o take_v 109_o and_o 45_o from_o 180_o rest_v 26_o the_o angle_n at_o b_o then_o also_o take_v 45_o the_o angle_n bcd_v out_o of_o 113_o the_o angle_n dca_n rest_v 68_o degree_n the_o angle_n bca_n in_o like_a manner_n take_v 42-30_a from_o 109_o the_o angle_n at_o d_o rest_v