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by_o this_o small_a table_n you_o may_v see_v that_o a_o board_n of_o 4_o inch_n broad_a will_v require_v 3_o foot_n thereof_o in_o length_n to_o make_v a_o foot_n square_n also_o a_o board_n of_o 5_o inch_n broad_a will_v require_v 2_o foot_n 4_o inch_n and_o 4_o five_o part_n of_o a_o inch._n the_o table_n of_o timber-measure_n give_v the_o length_n of_o a_o foot_n folid_a of_o any_o piece_n of_o timber_n or_o stone_n who_o square_a be_v under_o 8_o inch_n wherefore_o by_o the_o table_n at_o the_o begin_n of_o the_o line_n of_o timber-measure_n you_o may_v find_v that_o if_o a_o piece_n of_o timber_n be_v 1_o inch_n quare_fw-la 256_o 0_o 0_o will_v make_v a_o foot_n solid_a 2_o 36_o 0_o 0_o 3_o 16_o 0_o 0_o 4_o 9_o 0_o 0_o 5_o 5_o 9_o 0_o 6_o 4_o 0_o 0_o 7_o 2_o 11_o 0_o 8_o 2_o 3_o 0_o by_o this_o table_n which_o be_v the_o same_o in_o effect_n with_o that_o which_o stand_v at_o the_o end_n of_o the_o line_n of_o timber-measure_n you_o may_v see_v that_o a_o piece_n of_o timber_n that_o be_v 4_o inch_n square_a require_v 9_o foot_n in_o length_n to_o make_v a_o solid_a foot_n also_o a_o pitce_n of_o 5_o inch_n square_a require_v 5_o foot_n 9_o inch_n and_o 1_o 16_o part_n of_o a_o inch_n to_o make_v a_o solid_a foot_n and_o so_o of_o the_o rest_n but_o because_o these_o table_n go_v only_o to_o whole_a inch_n i_o have_v here_o add_v two_o table_n one_o for_o board_n the_o other_o for_o timber_n the_o table_n for_o board_n from_o one_o quarter_n of_o a_o inch_n to_o 6_o inch_n in_o breadth_n and_o the_o table_n for_o timber_n
same_o will_v reach_v from_o 1_o to_o 2.67_o so_o that_o 2_o inch_n 67_o 100_o part_n will_v make_v a_o foot_n solid_a of_o that_o piece_n of_o timber_n or_o stone_n this_o may_v be_v do_v another_o way_n by_o this_o analogy_n or_o proportion_n 1._o as._n 12_o to_o 30_o the_o breadth_n in_o inch_n so_o 21.6_o the_o depth_n in_o inch_n to_o a_o four_o number_n which_o here_o will_v be_v about_o 54._o 2._o as_o this_o four_o number_n 54_o be_v to_o 144_o so_o be_v 1_o to_o 2.67_o the_o length_n of_o a_o foot_n solid_a wherefore_o extend_v the_o compass_n from_o 12_o to_o 30_o the_o breadth_n that_o extent_n will_v reach_v from_o 21.6_o the_o depth_n to_o a_o certain_a place_n upon_o the_o line_n about_o 54_o where_o keep_v the_o point_n of_o the_o compass_n fast_o and_o open_v the_o other_o to_o 144_o then_o will_v this_o extent_n of_o the_o compass_n reach_v from_o 1_o to_o 2_o inch_n 67_o part_n the_o length_n of_o a_o foot_n solid_a as_o before_o ii_o in_o foot-measure_n only_o example_n 1._o let_v a_o stone_n or_o a_o piece_n of_o timber_n be_v 2_o foot_n 50_o part_n broad_a 1_o foot_n 80_o part_n deep_a and_o 15_o foot_n 25_o part_n long_o how_o many_o solid_a or_o cubical_a foot_n do_v such_o a_o piece_n contain_v the_o proportion_n be_v 1._o as_o 1_o be_v to_o 2.50_o foot_n the_o breadth_n so_o be_v 1.80_o foot_n the_o depth_n to_o 4.50_o foot_n the_o base_a in_o such_o measure_n 2._o as_o 1_o unto_o 4.50_o the_o base_a so_o 15.25_o the_o length_n to_o 68.62_o the_o content_a in_o foot_n extend_v the_o compass_n from_o 1_o to_o 2.50_o the_o breadth_n the_o same_o will_v reach_v from_o 1.80_o the_o depth_n to_o 4.50_o the_o base_a again_o extend_v the_o compass_n from_o 1_o to_o 4.50_o the_o base_a that_o extent_n will_v reach_v from_o 15.25_o the_o length_n to_o 68.62_o the_o content_a in_o foot_n example_n 2._o in_o the_o forementioned_a piece_n of_o square_a stone_n or_o timber_n be_v 2_o foot_n 50_o part_n broad_a and_o 1_o foot_n 80_o part_n deep_a let_v it_o be_v require_v to_o find_v how_o much_o thereof_o in_o length_n will_v make_v a_o foot_n the_o proportion_n be_v 1._o as_o 1_o be_v to_o 2.50_o the_o breadth_n so_o be_v 1.80_o the_o depth_n to_o 4.50_o the_o content_a of_o the_o base_a in_o foot-measure_n 2._o as_o 4.50_o the_o base_a be_v to_o 1_o so_o be_v 1_o foot_n to_o 222_o part_n the_o length_n of_o a_o foot_n solid_a wherefore_o extend_v the_o compass_n from_o 1_o to_o 2.50_o the_o breadth_n the_o same_o extent_n will_v reach_v from_o 1.80_o the_o depth_n to_o 4.50_o the_o content_a of_o the_o base_a again_o extend_v the_o compass_n from_o 4.50_o the_o base_a to_o 1_o the_o same_o will_v reach_v from_o 10_o to_o 222_o part_n the_o length_n of_o a_o cubical_a or_o solid_a foot_n of_o that_o stone_n or_o piece_n of_o timber_n iii_o in_o foot-measure_n and_o inch-measure_n together_o example_n let_v a_o square_a stone_n or_o piece_n of_o timber_n be_v 30_o inch_n broad_a 27.6_o inch_n deep_a and_o 15_o foot_n 25_o part_n long_o how_o many_o cubical_a or_o solid_a foot_n of_o stone_n or_o timber_n be_v shere_v in_o that_o piece_n the_o proportion_n be_v 1._o as_o 1_o be_v to_o 30_o inch_n the_o breadth_n so_o be_v 21.6_o inch_n the_o depth_n to_o 640_o the_o content_a of_o the_o base_a in_o inch_n 2._o as_o 144_o the_o inch_n in_o a_o foot_n superficial_a be_v to_o 648_o the_o content_a of_o the_o base_a in_o inch_n so_o be_v 15.25_o the_o length_n of_o the_o piece_n in_o foot-measure_n to_o 68_o foot_n 62_o part_n wherefore_o extend_v the_o compass_n from_o 1_o to_o 30_o the_o breadth_n the_o same_o will_v reach_v from_o 21.6_o the_o depth_n to_o 648_o the_o content_a of_o the_o base_a again_o extend_v the_o compass_n from_o 144_o to_o 648_o the_o content_a of_o the_o base_a the_o same_o extent_n will_v reach_v from_o 15.25_o the_o length_n of_o the_o piece_n to_o 68.62_o the_o solid_a content_n of_o the_o stone_n or_o timber_n in_o foot_n and_o 100_o part_n of_o a_o foot_n by_o have_v the_o same_o thing_n give_v in_o the_o same_o piece_n of_o stone_n or_o timber_n or_o in_o any_o other_o the_o work_n may_v be_v vary_v several_a way_n the_o analogy_n or_o proportion_n i_o will_v only_o give_v you_o leave_v the_o practice_n thereof_o to_o yourself_o breadth_n of_o the_o piece_n 30_o inch_n depth_n of_o the_o piece_n 21.6_o inch_n length_n of_o the_o piece_n 15.25_o foot_n the_o proportion_n 1._o as_o 144_o to_o 30_o the_o breadth_n so_o 21.6_o the_o depth_n to_o a_o four_o number_n from_o which_o four_o number_n if_o you_o extend_v your_o compass_n to_o 1_o and_o place_v one_o foot_n in_o 15.25_o the_o length_n of_o the_o piece_n the_o other_o foot_n shall_v fall_v upon_o 68.62_o the_o content_a of_o the_o stone_n or_o 2._o as_o 12_o unto_o 30_o the_o breadth_n so_o 21.6_o the_o depth_n to_o some_o four_o number_n from_o this_o four_o number_n extend_v the_o compass_n to_o 12_o that_o distance_n will_v reach_v from_o 15.25_o the_o length_n of_o the_o piece_n to_o 68.62_o the_o content_a of_o the_o piece_n chap._n xvi_o how_o to_o measure_v stone_n or_o timber_n by_o the_o line_n by_o have_v the_o square_a of_o the_o base_a and_o the_o length_n of_o the_o piece_n give_v both_o in_o foot_n and_o inch_n measure_n how_o to_o find_v the_o length_n of_o a_o side_n of_o a_o geometrical_a square_n that_o shall_v be_v equal_a to_o any_o parallelogram_n or_o long_a square_n be_v teach_v at_o the_o latter_a end_n of_o the_o ten_o chapter_n of_o this_o book_n by_o which_o rule_n it_o may_v at_o any_o time_n be_v find_v that_o be_v do_v there_o i_o shall_v only_o here_o begin_v with_o example_n example_n 1._o there_o be_v a_o square_a piece_n of_o timber_n who_o length_n be_v 183_o inch_n and_o the_o side_n of_o the_o square_n equal_a to_o the_o base_a or_o end_n thereof_o be_v 25_o inch_n 45_o part_n how_o many_o foot_n do_v that_o piece_n contain_v 1._o as_o 41.57_o to_o 25.45_o the_o side_n of_o the_o square_a so_o be_v 183_o the_o length_n in_o inch_n to_o a_o four_o number_n 2._o and_o that_o four_o number_n to_o 68.62_o the_o content_a in_o foot_n extend_v the_o compass_n from_o 41.57_o to_o 25.45_o the_o side_n of_o the_o square_a the_o same_o will_v reach_v from_o 183_o the_o length_n to_o some_o other_o part_n of_o the_o line_n from_o whence_o if_o you_o again_o extend_v the_o same_o distance_n the_o point_n will_v rest_v upon_o 68_o foot_n 62_o 100_o part_n of_o a_o foot_n and_o so_o many_o foot_n be_v in_o the_o piece_n example_n 2._o let_v the_o side_n of_o a_o square_a equal_a to_o the_o base_a of_o a_o piece_n of_o stone_n or_o timber_n be_v 2_o foot_n 12_o part_n and_o the_o length_n of_o the_o same_o piece_n 15_o foot_n 25_o part_n how_o many_o solid_a foot_n be_v there_o in_o that_o piece_n 1._o as_o 1._o to_o 2_o foot_n 12_o part_n the_o side_n of_o the_o square_a so_o 15_o foot_n 25_o part_n the_o length_n to_o a_o four_o number_n 2._o and_o that_o four_o number_n to_o 68.62_o the_o content_a in_o foot_n extend_v the_o compass_n from_o 1_o to_o 2.12_o the_o side_n of_o the_o square_a that_o will_v reach_v from_o 15.25_o the_o length_n to_o some_o other_o number_n on_o the_o line_n from_o whence_o the_o compass_n be_v extend_v the_o movable_a point_n will_v fall_v upon_o 68.62_o the_o content_a as_o before_o example_n 3._o the_o side_n of_o a_o square_a equal_a to_o the_o base_a of_o a_o stone_n be_v 25_o inch_n 45_o part_n and_o the_o length_n of_o that_o stone_n 15_o foot_n 25_o part_n how_o many_o foot_n do_v it_o contain_v 1._o as_o 12_o to_o 25.45_o the_o square_a in_o inch_n so_o be_v 15.25_o foot_n the_o length_n to_o a_o four_o number_n 2._o and_o that_o four_o number_n to_o 68.62_o the_o content_a extend_v the_o compass_n from_o 12_o to_o 25.45_o the_o side_n of_o the_o square_a the_o same_o will_v reach_v from_o 15.25_o to_o some_o other_o point_n upon_o the_o line_n from_o whence_o the_o compass_n be_v extend_v the_o movable_a point_n will_v fall_v upon_o 68_o foot_n 62_o part_n the_o content_a of_o the_o stone_n example_n 4._o there_o be_v a_o piece_n of_o timber_n who_o side_n of_o the_o square_n of_o the_o base_a be_v 25_o inch_n 45_o part_n how_o much_o in_o length_n of_o that_o piece_n will_v make_v a_o foot_n solid_a 1._o as_o 25.45_o the_o side_n of_o the_o square_n be_v to_o 41.57_o so_o be_v 1_o foot_n to_o a_o four_o number_n 2._o and_o that_o four_o number_n to_o 6_o inch_n 67_o part_n wherefore_o extend_v the_o compass_n from_o 25.45_o the_o side_n to_o 41.57_o the_o same_o will_v reach_v from_o 1_o to_o some_o other_o point_n from_o
whence_o the_o compass_n be_v extend_v will_v reach_v to_o 6.67_o the_o length_n of_o a_o foot_n solid_a of_o that_o piece_n of_o timber_n example_n 5._o the_o length_n of_o the_o side_n of_o a_o square_a equal_a to_o the_o base_a of_o a_o piece_n of_o timber_n be_v 2_o foot_n 12_o part_n to_o find_v how_o much_o in_o length_n of_o that_o piece_n will_v make_v a_o foot_n solid_a in_o foot-measure_n as_o 2.120_o the_o side_n of_o the_o square_n be_v to_o 1.000_o so_o be_v 1.000_o to_o a_o four_o number_n and_o that_o four_o number_n to_o 0.471_o part_n of_o a_o foot_n to_o make_v a_o foot_n square_n extend_v the_o compass_n from_o 2.120_o the_o side_n of_o the_o square_n to_o 1000_o the_o same_o extent_n will_v reach_v from_o 1000_o downward_o to_o some_o other_o point_n upon_o the_o line_n and_o from_o thence_o downward_o to_o 222_o part_n of_o a_o foot_n and_o so_o much_o in_o length_n will_v make_v a_o foot_n solid_a chap._n xvii_o concern_v timber_n that_o be_v big_a at_o one_o end_n than_o at_o the_o other_o either_o round_o or_o square_a and_o how_o to_o measure_v it_o i._o for_o squared-timber_n in_o large_a timber-tree_n when_o they_o be_v square_v there_o be_v a_o great_a disproportion_n between_o the_o square_n of_o both_o end_n wherefore_o some_o do_v use_v to_o take_v the_o square_a of_o the_o middle_n of_o the_o piece_n for_o the_o mean_a or_o true_a square_n but_o this_o be_v not_o exact_a though_o much_o use_v but_o the_o best_a way_n be_v this_o find_v by_o the_o problem_n at_o the_o end_n of_o the_o ten_o chapter_n of_o this_o book_n the_o length_n of_o the_o side_n of_o a_o spuare_fw-la equal_a to_o both_o the_o end_n of_o the_o piece_n add_v these_o two_o side_n together_o and_o take_v the_o half_a thereof_o for_o the_o true_a square_n and_o with_o that_o square_n you_o may_v by_o the_o rule_n of_o the_o last_o chapter_n measure_v it_o as_o if_o it_o be_v perfect_o square_a ii_o for_o round-timber_n the_o ordinary_a way_n use_v for_o the_o measure_n of_o round_a timber_n be_v to_o gird_v it_o about_o the_o middle_n with_o a_o line_n and_o to_o take_v one_o four_o part_n thereof_o for_o the_o side_n of_o a_o square_n equal_a thereto_o but_o this_o be_v false_a though_o most_o man_n use_v it_o custom_n have_v make_v it_o bear_v the_o face_n of_o truth_n for_o it_o be_v more_o in_o measure_n than_o in_o reality_n it_o shall_v be_v but_o the_o exact_a way_n of_o measure_v of_o round_a timber_n especial_o if_o it_o be_v grow_v be_v this_o about_o the_o middle_n thereof_o in_o some_o smooth_a place_n gird_v the_o same_o about_o with_o a_o string_n then_o have_v you_o this_o proportion_n as_o 1000_o be_v to_o the_o number_n of_o inch_n about_o so_o be_v 2821_o to_o the_o length_n of_o the_o side_n of_o a_o square_n equal_a thereunto_o so_o if_o a_o tree_n be_v gird_v about_o as_o abovesaid_a shall_v contain_v in_o circumference_n 47_o inch_n 13_o part_n if_o you_o extend_v the_o compass_n from_o 1000_o to_o 47_o inch_n 13_o part_n the_o same_o extent_n will_v reach_v from_o 2821_o to_o 13_o inch_n 29_o part_n which_o be_v equal_a to_o the_o side_n of_o a_o square_n equal_a to_o that_o tree_n which_o be_v obtain_v the_o tree_n may_v be_v measure_v divers_a way_n according_a to_o the_o example_n in_o the_o last_o chapter_n chap._n xviii_o concern_v the_o measure_n of_o regular_a solid_n as_o cylinder_n globe_n cones_n and_o such_o like_a i._o of_o the_o cylinder_n a_o cylinder_n be_v a_o round_a figure_n of_o equal_a circumference_n in_o all_o part_n thereof_o as_o a_o stand_a pillar_n a_o rowling-stone_n for_o garden-walks_a etc._n etc._n to_o measure_v such_o a_o figure_n there_o be_v several_a way_n both_o by_o have_v the_o circumference_n give_v when_o it_o be_v stand_v or_o by_o have_v the_o diameter_n at_o the_o end_n thereof_o when_o it_o be_v lie_v or_o by_o have_v the_o side_n of_o a_o square_n equal_a to_o the_o base_a thereof_o i._o by_o have_v the_o diameter_n give_v example_n 1._o the_o diameter_n be_v 15_o inch_n how_o much_o in_o length_n make_v a_o foot_n as_o 15_o the_o diameter_n to_o 46.90_o so_o be_v 1_o to_o a_o four_o and_o that_o four_o to_o 9.78_o the_o length_n of_o a_o foot_n extend_v the_o compass_n from_o 15_o the_o diameter_n to_o 46.90_o that_o extent_n will_v reach_v from_o 1_o to_o another_o point_n upon_o the_o line_n and_o from_o thence_o to_o 9_o inch_n 78_o part_n the_o length_n of_o a_o foot_n solid_a example_n 2._o the_o diameter_n be_v 1_o foot_n 25_o part_n how_o much_o in_o length_n make_v a_o foot_n in_o foot-measure_n as_o 1.25_o the_o diameter_n in_o foot_n unto_o 1.128_o so_o 1_o to_o a_o four_o number_n and_o that_o to_o 8.14_o the_o length_n of_o a_o foot_n solid_a in_o foot-measure_n extend_v the_o compass_n from_o 1.25_o the_o diameter_n to_o 1.128_o the_o same_o will_v reach_v from_o 1_o to_o some_o other_o number_n and_o from_o thence_o to_o 1_o foot_n 128_o part_n of_o a_o foot_n example_n 3._o have_v the_o diameter_n 15_o inch_n and_o the_o length_n 105_o inch_n how_o many_o solid_a inches_n do_v the_o cylinder_n contain_v as_o 1.128_o to_o 15_o inch_n the_o diameter_n so_o be_v 105_o inch_n the_o length_n to_o a_o four_o number_n and_o that_o to_o 18555.34_o inch_n the_o content_a extend_v the_o compass_n from_o 1.128_o to_o 15_o the_o length_n the_o fame_n extent_n will_v reach_v from_o 105_o the_o length_n to_o some_o other_o number_n and_o from_o thence_o to_o 18555.34_o inch_n the_o content_a of_o the_o cylinder_n in_o inch_n example_n 4._o have_v the_o diameter_n 1_o foot_n 25_o part_n and_o the_o length_n 8_o foot_n 75_o part_n to_o find_v the_o content_a in_o foot_n as_o 1.128_o to_o 1.25_o the_o diameter_n so_o 8.75_o the_o length_n to_o a_o four_o and_o that_o four_o to_o 10.74_o foot_n the_o content_a extend_v the_o compass_n from_o 1.128_o to_o 1.25_o the_o diameter_n that_o extent_n will_v reach_v from_o 8.75_o the_o length_n to_o some_o other_o number_n and_o from_o that_o to_o 10_o foot_n 74_o part_n the_o content_a example_n 5._o have_v the_o diameter_n 15_o inch_n and_o the_o length_n 105_o inch_n how_o many_o foot_n do_v it_o contain_v as_o 46.90_o to_o 15_o inch_n the_o diameter_n so_o be_v 105_o inch_n the_o length_n to_o a_o four_o and_o that_o four_o to_z 10_o foot_n 74_o part_n the_o content_a extend_v the_o compass_n from_o 46.90_o to_o 15_o the_o diameter_n that_o extent_n will_v reach_v from_o 105_o the_o length_n to_o another_o number_n and_o from_o that_o to_o 10_o foot_n 74_o part_n the_o content_a example_n 6._o the_o diameter_n be_v 15_o inch_n and_o the_o length_n 8_o foot_n 75_o part_n how_o many_o foot_n do_v it_o contain_v as_o 13.54_o to_o 15_o inch_n the_o diameter_n so_o 8.75_o foot_n the_o length_n to_o a_o four_o and_o that_o four_o to_o 10.74_o the_o length_n in_o foot_n extend_v the_o compess_n from_o 3.54_o to_o 15_o the_o length_n that_o extent_n will_v reach_v from_o 8.75_o the_o length_n to_o another_o number_n and_o from_o thence_o to_o 10.74_o foot_n the_o content_a in_o foot_n ii_o by_o have_v the_o circumference_n give_v example_n 1._o the_o circumference_n of_o a_o cylinder_n be_v 47_o inch_n 13_o part_n how_o much_o thereof_o in_o length_n shall_v make_v a_o foot_n solid_a as_o 47.13_o inch_n the_o circumference_n to_o 147.36_o so_o 1_o to_o a_o four_o number_n and_o that_o to_o 9.78_o inch_n the_o length_n of_o a_o foot_n extend_v the_o compass_n from_o 47.13_o the_o circumference_n to_o 147.36_o that_o extent_n will_v reach_v from_o 1_o to_o a_o four_o number_n and_o from_o thence_o to_o 9_o inch_n 78_o part_n the_o length_n of_o a_o foot_n solid_a example_n 2._o have_v the_o circumference_n of_o a_o cylinder_n 3_o foot_n 927_o part_n to_o find_v the_o length_n of_o a_o foot_n solid_a thereof_o in_o foot-measure_n as_o 3.927_o foot_n to_o 3.545_o so_o 1_o to_o a_o four_o number_n and_o that_o to_o 815_o part_n of_o a_o foot_n the_o length_n extend_v the_o compass_n from_o 3.927_o the_o circumference_n to_o 3.545_o that_o extent_n will_v reach_v from_o 1_o to_o some_o other_o number_n and_o from_o thence_o to_o 815_o part_n of_o a_o foot_n for_o the_o length_n of_o a_o solid_a foot_n of_o that_o cylinder_n example_n 3._o the_o circumference_n of_o a_o cylinder_n be_v 47_o inch_n 13_o part_n and_o the_o length_n thereof_o 105_o inch_n how_o many_o inch_n be_v there_o in_o such_o a_o cylinder_n as_o 3.545_o to_o 47.13_o the_o circumference_n so_o 105_o inch_n the_o length_n to_o a_o four_o number_n and_o that_o to_o 18555_o the_o content_a in_o inches_n extend_v the_o compass_n from_o 3.545_o to_o 47.13_o the_o circumference_n that_o extent_n will_v reach_v from_o 105_o the_o
length_n to_o another_o number_n and_o from_o thence_o to_o 18555_o the_o number_n of_o solid_a inch_n in_o the_o cylinder_n example_n 4._o the_o circumference_n be_v 47_o inch_n 13_o part_n and_o the_o length_n 105_o inch_n as_o before_o how_o many_o solid_a foot_n in_o that_o cylinder_n as_o 147.36_o to_o 47.13_o inch_n the_o circumfer_fw-la so_o 105_o inch_n the_o length_n to_o a_o four_o number_n and_o that_o to_z 10_o foot_n 74_o part_n the_o content_a extend_v the_o compass_n from_o 147.36_o to_o 47.13_o the_o circumference_n that_o extent_n will_v reach_v from_o 105_o the_o length_n to_o another_o number_n and_o from_o that_o to_o 10_o foot_n 74_o part_n of_o a_o foot_n the_o solid_a content_n example_n 5._o let_v the_o length_n of_o the_o cylinder_n be_v 8_o foot_n 75_o part_n and_o the_o circumference_n 3_o foot_n 927_o part_n how_o many_o foot_n do_v it_o contain_v as_o 3.545_o to_o 3.927_o foot_n the_o circumference_n so_o 8.75_o foot_n the_o length_n to_o a_o four_o number_n and_o that_o to_z 10_o foot_n 74_o part_n the_o content_a extend_v the_o compass_n from_o 3.545_o to_o 3.927_o the_o same_o extent_n will_v reach_v from_o 8.75_o the_o length_n to_o 10.74_o the_o content_a in_o foot_n example_n 6._o let_v the_o circumference_n give_v be_v 47_o inch_n 13_o part_n and_o the_o length_n 8_o foot_n 75_o part_n how_o many_o solid_a foot_n do_v the_o cylinder_n contain_n as_o 42.54_o to_o 47.13_o inch_n the_o circumference_n so_o be_v 8.75_o foot_n the_o length_n to_o a_o four_o and_o that_o four_o to_o 10.74_o foot_n the_o content_a extend_v the_o compass_n from_o 42.54_o to_o 47.13_o the_o circumference_n that_o extent_n will_v reach_v from_o 8.75_o the_o length_n to_o another_o number_n and_o from_o thence_o to_o 10_o foot_n 74_o part_n the_o content_a of_o the_o cylinder_n in_o solid_a foot_n iii_o by_o have_v the_o side_n of_o a_o square_a equal_a to_o the_o base_a or_o end_n of_o a_o cylinder_n example_n let_v the_o side_n of_o a_o square_a equal_a to_o the_o base_a or_o end_n of_o the_o cylinder_n be_v 13_o inch_n 29_o part_n and_o the_o length_n thereof_o 105_o inch_n how_o many_o square_a foot_n be_v contain_v in_o that_o cylinder_n as_o 41.57_o to_o 13.29_o inch_n the_o side_n of_o the_o square_a so_o be_v 105_o the_o length_n in_o inch_n to_o a_o four_o number_n and_o that_o to_z 10_o foot_n 47_o part_n the_o content_a of_o the_o cylinder_n in_o foot_n and_o part_n extend_v the_o compass_n from_o 41.57_o to_o 13.29_o inch_n the_o side_n of_o a_o square_n equal_a to_o the_o base_a of_o the_o cylinder_n that_o extent_n will_v reach_v from_o 105_o inch_n the_o length_n to_o another_o number_n and_o from_o thence_o to_o 10_o foot_n 47_o part_n the_o content_a of_o the_o cylinder_n in_o foot_n ii_o of_o the_o cone_n a_o cone_n be_v a_o round_a figure_n have_v for_o the_o base_a thereof_o a_o circle_n the_o side_n whereof_o rise_v from_o the_o circumference_n of_o the_o circle_n round_o about_o the_o same_o equal_o till_o it_o meet_v in_o a_o point_n just_a over_o the_o centre_n of_o the_o circle_n and_o be_v in_o the_o form_n of_o a_o spire-steeple_n and_o it_o be_v thus_o measure_v example_n 1._o let_v there_o be_v a_o cone_n the_o diameter_n of_o who_o base_a be_v 10_o inch_n and_o who_o height_n be_v 12_o inch_n i_o will_v know_v how_o many_o solid_a or_o cubical_a inch_n be_v contain_v therein_o the_o diameter_n be_v 10_o the_o content_a of_o the_o circle_n or_o base_a will_v be_v find_v to_o the_o 78_o inch_n 54_o part_n as_o by_o the_o five_o example_n in_o chap._n 13._o of_o this_o book_n the_o area_n of_o the_o base_a be_v thus_o find_v the_o proportion_n be_v as_o 3_o to_o 78.54_o inch_n the_o content_a of_o the_o base_a so_o be_v 14_o inch_n the_o height_n to_o 314_o inch_n 16_o part_n of_o a_o inch_n for_o the_o content_a of_o the_o cone_n in_o inches_n extend_v the_o compass_n from_o 3_o to_o 78.54_o the_o base_a that_o extent_n will_v reach_v from_o 12_o the_o height_n to_o 314_o inch_n 16_o part_n the_o content_a of_o the_o cone_n in_o solid_a inch_n example_n 2._o let_v the_o diameter_n of_o the_o base_a be_v 12_o inch_n as_o before_o and_o the_o length_n of_o the_o side_n he_o 13_o inch_n kow_n many_o solid_a inches_n be_v there_o in_o this_o cone_n 1._o extend_v the_o compass_n from_o 1_o to_z 5_o inch_n half_o the_o diameter_n of_o the_o base_a that_o extent_n will_v reach_v from_o 5_o to_o 25._o 2._o extend_v the_o compass_n from_o 1_o to_z 13_o the_o length_n of_o the_o side_n that_o extent_n will_v reach_v from_o 13_o to_o 169._o 3._o from_o this_o 169_o take_v the_o 25_o before_o find_v and_o there_o remain_v 144._o 4._o upon_o your_o line_n take_v half_a the_o distance_n between_o 1_o and_o 144_o and_o you_o shall_v find_v it_o to_o be_v 12_o which_o 12_o be_v the_o height_n of_o the_o cone_n so_o the_o height_n be_v have_v you_o may_v find_v the_o content_n as_o in_o the_o last_o example_n iii_o of_o spherical_a body_n a_o spherical_a body_n be_v such_o a_o body_n who_o superficies_n in_o all_o the_o part_n of_o it_o be_v equal_o distant_a from_o the_o centre_n of_o the_o body_n as_o globe_n bullet_n etc._n etc._n example_n 1._o the_o circumference_n of_o a_o globe_n or_o bullet_n be_v 28_o inch_n 28_o part_n to_o find_v the_o length_n of_o the_o diameter_n as_o 22_o to_o 7_o so_o be_v 28.28_o the_o circumference_n to_o 9_o inch_n the_o diameter_n extend_v the_o compass_n from_o 22_o to_o 7_o the_o same_o extent_n will_v reach_v from_o 28.28_o the_o circumference_n to_o 9_o inch_n the_o length_n of_o the_o diameter_n of_o that_o bullet_n example_n 2._o the_o diameter_n of_o a_o spherical_a body_n be_v give_v be_v 9_o inch_n and_o its_o circumference_n be_v 28_o inch_n 28_o part_n how_o many_o square_a inches_n be_v there_o in_o the_o superficies_n of_o that_o spherical_a body_n as_o 1_o be_v to_o 9_o inch_n the_o diameter_n so_o be_v 28.28_o inch_n the_o circumference_n to_o 254.5_o inch_n the_o superficial_a content_n extend_v the_o compass_n from_o 1_o to_z 9_o the_o diameter_n the_o same_o extent_n will_v reach_v from_o 28.28_o the_o circumference_n to_o 254_o inch_n 5_o part_n the_o superficial_a inch_n in_o this_o spherical_a body_n example_n 3._o the_o diameter_n of_o a_o spherical_a body_n be_v 9_o inch_n how_o many_o solid_a inch_n be_v therein_o contain_v 1._o as_o 1_o be_v to_o 9_o the_o diameter_n she_o be_v 9_o to_o a_o four_o number_n and_o that_o four_o number_n to_o 729_o the_o cube_fw-la of_o the_o diameter_n 2._o as_o 9_o the_o diameter_n to_o 729_o its_o cube_fw-la so_o be_v 11_o to_o 891_o inch_n the_o solid_a content_n of_o the_o spherical_a body_n extend_v the_o compass_n from_o 1_o to_o 9_o that_o extent_n will_v reach_v to_o 81_o and_o from_o 81_o to_o 729_o the_o cube_fw-la of_o the_o diameter_n then_o extend_v the_o compass_n from_o 9_o the_o diameter_n to_o 729_o its_o cube_n that_o extent_n will_v reach_v from_o 11_o to_o 891_o inch_n the_o solid_a content_n of_o the_o spherical_a body_n i_o may_v here_o add_v the_o manner_n how_o to_o measure_v other_o kind_n of_o body_n both_o regular_a and_o irregular_a as_o ellipse_n parabolas_fw-la etc._n etc._n also_o of_o prisme_n scalene_n cones_n spheroiades_n etc._n etc._n but_o these_o be_v out_o of_o the_o reach_n of_o ordinary_a artificer_n for_o who_o sake_n this_o treatise_n be_v chief_o compose_v i_o shall_v here_o conclude_v this_o treatise_n of_o the_o use_n of_o the_o line_n of_o proportion_n with_o a_o short_a supplement_n of_o gauge_v of_o vessel_n chap._n xix_o concern_v the_o gauge_v of_o vessel_n by_o the_o line_n before_o you_o can_v measure_v your_o vessel_n to_o find_v the_o content_a thereof_o in_o gallon_n or_o part_n you_o must_v find_v the_o content_a thereof_o in_o solid_a inch_n and_o to_o effect_v this_o you_o must_v find_v the_o content_a of_o two_o three_o part_n of_o a_o circle_n agreeable_a to_o the_o diameter_n at_o the_o bung_n and_o one_o three_o part_n of_o another_o circle_n agreeable_a to_o that_o of_o the_o diameter_n at_o the_o head_n these_o two_o add_v together_o and_o multiply_v by_o the_o length_n of_o the_o vessel_n that_o product_n will_v be_v the_o content_a of_o that_o vessel_n in_o inches_n example_n let_v there_o be_v a_o vessel_n who_o diameter_n at_o head_n 18_o diameter_n at_o bung_n 32_o length_n be_v 40_o and_o let_v the_o content_a thereof_o first_o in_o inches_n and_o then_o in_o gallon_n be_v require_v i._o for_o the_o two_o three_o part_n of_o the_o circle_n at_o the_o bungler_n as_o 1_o to_o this_o universal_a number_n 5236_o so_o 1024_o the_o square_a of_o the_o diameter_n at_o the_o bung_v 32_o to_o 536.166_o inch_n which_o be_v two_o
three_o part_n of_o the_o content_a of_o the_o circle_n at_o the_o bungler_n wherefore_o extend_v the_o compass_n from_o 1_o to_o 5236_o the_o same_o extent_n will_v reach_v from_o 1024_o the_o square_n of_o 32_o the_o diameter_n at_o the_o bung_n to_o 536.166_o inch_n the_o content_a of_o two_o three_o part_n of_o the_o circle_n at_o the_o bungler_n ii_o for_o one_o three_o part_n of_o the_o circle_n at_o the_o head_n as_o 1_o to_o this_o general_a number_n 2618_o so_o be_v 324_o the_o square_a of_o the_o diameter_n at_o head_n 18_o to_o 84.823_o inch_n which_o be_v one_o three_o part_n of_o the_o content_a of_o the_o circle_n at_o the_o head_n wherefore_o extend_v the_o compass_n from_o 1_o to_o 2618_o the_o same_o extent_n will_v reach_v from_o 324_o the_o square_n of_o 18_o the_o diameter_n at_o the_o head_n to_o 84.823_o inch_n the_o content_a of_o one_o three_o part_n of_o the_o diameter_n at_o the_o head_n iii_o for_o the_o number_n of_o square_a inch_n in_o the_o vessel_n add_v these_o two_o number_n 536.166_o and_o 84.823_o they_o make_v 620.989_o  _fw-fr 40_o which_o multiply_v by_o 40_o the_o length_n of_o the_o vessel_n produce_v 24839.560_o and_o so_o many_o square_a inch_n be_v contain_v in_o such_o a_o vessel_n who_o diameter_n at_o the_o head_n be_v 18_o inch_n at_o the_o bung_v 32_o inch_n and_o be_v 40_o inch_n long_o iv_o for_o the_o content_a in_o wine_n or_o ale_n gallon_n divide_v this_o number_n 24839.56_o by_o 231_o for_o wine_n 282_o for_o ale_n and_o the_o quotient_n shall_v tell_v you_o the_o number_n of_o gallon_n and_o part_n of_o a_o gallon_n wine_n  _fw-fr gall_n part_n 231_o 24839.56_o 107.52_o  _fw-fr 231_o  _fw-fr  _fw-fr 1739_o  _fw-fr  _fw-fr 1617._o  _fw-fr  _fw-fr 1225_o  _fw-fr  _fw-fr 1155_o  _fw-fr  _fw-fr 706_o  _fw-fr  _fw-fr 693_o  _fw-fr  _fw-fr 13_o  _fw-fr ale_n  _fw-fr gall_n part_n 282_o 24839.56_o 88.08_o  _fw-fr 2256_o  _fw-fr  _fw-fr 2279_o  _fw-fr  _fw-fr 2256_o  _fw-fr  _fw-fr 2356_o  _fw-fr  _fw-fr 2256_o  _fw-fr  _fw-fr 100_o  _fw-fr by_o this_o work_n you_o may_v perceive_v that_o this_o vessel_n contain_v 107_o gallon_n 53_o part_n of_o wine_n measure_n 88_o gallon_n 08_o part_n of_o ale_n measure_n how_o to_o multiply_v and_o divide_v by_o the_o line_n be_v teach_v in_o the_o second_o and_o three_o chapter_n of_o this_o book_n and_o therefore_o be_v needless_a here_o to_o repeat_v it_o again_o but_o i_o choose_v rather_o to_o do_v it_o arithmetical_o for_o the_o better_a illustration_n and_o for_o the_o satisfaction_n of_o such_o as_o have_v a_o delight_n in_o number_n how_o to_o measure_v board_n glass_n timber_n stone_n etc._n etc._n by_o a_o line_n of_o equal_a part_n draw_v from_o the_o centre_n of_o a_o two-foot_n joynt-rule_n all_o proportion_n that_o may_v be_v wrought_v upon_o a_o straight_a ruler_n by_o the_o line_n of_o proportion_n or_o number_n the_o same_o may_v be_v wrought_v by_o a_o line_n of_o equal_a part_n draw_v from_o the_o centre_n of_o a_o open_a joint_n and_o whereas_o this_o line_n of_o equal_a part_n be_v number_v from_o the_o centre_n of_o the_o rule_n towards_o the_o end_n thereof_o by_o 1_o 2_o 3_o 4_o etc._n etc._n to_o 10_o that_o these_o figure_n as_o in_o the_o other_o line_n do_v sometime_o signify_v themselves_o only_o sometime_o 1_o 2_o 3_o etc._n etc._n do_v signify_v 10_o 20_o 30_o etc._n etc._n sometime_o 100_o 200_o 300_o etc._n etc._n according_a to_o the_o quality_n of_o the_o question_n propound_v by_o this_o line_n you_o may_v also_o multiply_v divide_v work_v the_o rule_n of_o proportion_n and_o perform_v divers_a thing_n which_o the_o line_n of_o number_n perform_v and_o some_o other_o which_o that_o will_v not_o but_o i_o shall_v here_o only_o show_v you_o how_o board_n glass_n timber_n stone_n etc._n etc._n may_v be_v thereby_o measure_v which_o i_o shall_v do_v in_o these_o follow_a proposition_n and_o i._o for_o superficial_a measure_n as_o board_n glass_n etc._n etc._n i._o in_o inch-measure_n prop._n i._n a_o plank_n be_v 27_o inch_n broad_a and_o 263_o inch_n long_o how_o many_o square_a inch_n be_v contain_v therein_o as_o i_o be_v to_o 27_o so_o be_v 263_o to_o 7101._o take_v in_o your_o compass_n the_o distance_n from_o the_o centre_n to_o 27_o the_o breadth_n upon_o your_o line_n of_o equal_a part_n with_o this_o distance_n set_v one_o foot_n in_o 10_o at_o the_o end_n of_o the_o line_n and_o open_v the_o rule_n till_o the_o other_o foot_n fall_v in_o 10_o on_o the_o other_o leg_n of_o the_o rule_n the_o rule_n thus_o stand_v take_v with_o your_o compass_n the_o distance_n between_o 263_o on_o one_o leg_n of_o the_o rule_n to_o 263_o on_o the_o other_o leg_n this_o distance_n will_v reach_v from_o the_o centre_n of_o the_o rule_n to_o 7101_o and_o so_o many_o square_a inch_n be_v in_o that_o piece_n prop._n 2._o if_o a_o board_n or_o plank_n or_o piece_n of_o pavement_n or_o of_o glass_n be_v 20_o inch_n broad_a how_o much_o thereof_o in_o length_n shall_v make_v a_o foot_n square_a as_o 20_o be_v to_o 144_o so_o 1_o to_o 7.2_o take_v 144_o out_o of_o your_o line_n of_o equal_a part_n from_o the_o centre_n and_o set_v one_o foot_n in_o 20_o open_v the_o other_o leg_n till_o the_o other_o compass_n point_n fall_n in_o 20_o also_o the_o rule_n thus_o stand_v take_v the_o distance_n between_o 10_o and_o 10_o and_o that_o distance_n will_v reach_v from_o the_o centre_n of_o the_o rule_n to_o 7_o inch_n 2_o 10_o part_n of_o a_o inch_n and_o so_o much_o in_o length_n will_v make_v a_o foot_n square_n ii_o in_o foot-measure_n prop._n 3._o a_o room_n be_v 52_o foot_n broad_a and_o 110.5_o foot_n long_o how_o many_o square_a foot_n be_v there_o in_o that_o room_n as_o 52_o be_v to_o 1_o so_o be_v 110.5_o to_o 5746._o take_v in_o your_o compass_n 52_o the_o breadth_n with_o this_o distance_n open_v the_o ruler_n in_o 10_o and_z 10_o it_o so_o rest_v take_v the_o distance_n between_o 110.5_o and_o 110.5_o on_o either_o side_n that_o distance_n apply_v to_o the_o centre_n of_o the_o rule_n will_v reach_v to_o 5746_o and_o so_o many_o square_a foot_n be_v in_o that_o room_n prop._n 4._o a_o plank_n be_v 2_o foot_n 25_o part_n broad_a how_o much_o in_o length_n thereof_o shall_v make_v a_o foot_n square_n as_o 2.25_o the_o breadth_n be_v to_o 1_o or_o 10_o so_o be_v 10_o to_o 44_o the_o length_n of_o a_o foot_n take_v in_o your_o compass_n the_o distance_n from_o the_o centre_n of_o your_o rule_n to_o 1_o then_o set_v one_o foot_n in_o 2.25_o and_o open_v the_o other_o leg_n till_o the_o other_o compass_n point_n fall_n in_o 2.25_o on_o the_o other_o side_n the_o rule_n thus_o stand_v take_v the_o distance_n between_o 10_o and_o 10_o that_o distance_n apply_v from_o the_o centre_n of_o the_o rule_n will_v reach_v to_o 44_o part_n of_o a_o foot_n and_o so_o much_o in_o length_n will_v make_v a_o foot_n iii_o in_o yard-measure_n prop._n 5._o a_o room_n be_v hang_v with_o tapestry_n contain_v 130_o yard_n 25_o part_n in_o compass_n and_o in_o depth_n 5_o yard_n 20_o part_n how_o many_o yard_n of_o tapestry_n be_v in_o that_o room_n as_o 1_o to_o 5.20_o so_o 130.25_o to_o 677.4_o take_v 5.20_o in_o your_o compass_n and_o that_o distance_n put_v over_o in_o 10_o and_o 10_o the_o rule_n thus_o stand_v take_v the_o distance_n between_o 130.25_o and_o 130.25_o on_o each_o leg_n of_o the_o rule_n that_o distance_n will_v reach_v from_o the_o centre_n of_o the_o rule_n to_o 677_o yard_n 4_o ten_o of_o a_o yard_n ii_o for_o solid-measure_n as_o timber_n stone_n etc._n etc._n by_o the_o line_n of_o equal_a part_n i._o in_o inch-measure_n prop._n i._n a_o piece_n of_o timber_n be_v 30_o inch_n broad_a 21_o inch_n 6_o part_n deep_a and_o 183_o inch_n long_o how_o many_o foot_n be_v contain_v in_o that_o piece_n of_o timber_n 1._o as_o 1_o be_v to_o 30_o so_o be_v 21.6_o to_o 648._o take_v the_o distance_n from_o the_o centre_n to_o 30_o then_o set_v one_o foot_n in_o 10_o and_o open_v the_o rule_n till_o the_o other_o compass_n point_n fall_n in_o 10_o on_o the_o other_o leg_n of_o the_o rule_n then_o take_v the_o distance_n between_o 21.6_o and_o 21.6_o that_o distance_n will_v reach_v from_o the_o centre_n of_o the_o rule_n to_o 648_o the_o content_a of_o the_o base_a or_o end_n of_o the_o piece_n of_o timber_n in_o inch_n then_o 2._o as_o 1728_o the_o number_n of_o inch_n in_o a_o foot_n solid_a be_v to_o 648_o the_o content_a of_o the_o base_a so_o be_v 183_o inch_n the_o length_n to_o 68_o foot_n 62_o part_n the_o content_a in_o foot_n take_v in_o your_o compass_n the_o distance_n from_o the_o centre_n to_o 1728_o with_o this_o distance_n set_v one_o foot_n in_o 648_o and_o
count_v 7_o of_o the_o intermediate_v division_n and_o that_o point_n be_v the_o place_n upon_o the_o rule_n represent_v 37._o example_n 2._o let_v it_o be_v require_v to_o find_v the_o place_n of_o 134_o upon_o the_o line_n for_o your_o first_o figure_n 1_o count_v 1_o upon_o the_o line_n for_o your_o second_o figure_n 3_o count_v three_o of_o the_o grand_a division_n and_o for_o the_o three_o figure_n 4_o count_v 4_o of_o the_o small_a intermediate_v division_n and_o that_o very_a point_n be_v the_o place_n upon_o the_o line_n represent_v 134._o again_o to_o find_v the_o place_n represent_v 308._o for_o your_o first_o figure_n 3_o count_v the_o 3_o upon_o the_o line_n for_o your_o second_o figure_n o_o which_o be_v a_o cipher_n count_n none_o of_o the_o grand_a division_n but_o for_o your_o last_o figure_n 8_o count_v 8_o of_o the_o intermediate_v division_n and_o that_o point_n shall_v be_v the_o place_n upon_o the_o line_n represent_v 308._o example_n 3._o let_v it_o be_v require_v to_o find_v the_o place_n of_o 1350._o for_o your_o first_o figure_n 1_o take_v 1_o on_o the_o middle_n of_o the_o line_n for_o your_o second_o figure_n 3_o take_v the_o figure_n 3_o upon_o the_o line_n upward_o for_o the_o 5_o count_v five_o of_o the_o grand_a intermediate_v division_n and_o that_o be_v the_o place_n of_o 1350._o again_o to_o find_v the_o place_n of_o 1626._o for_o your_o first_o figure_n 1_o count_v the_o 1_o on_o the_o middle_n of_o the_o line_n for_o your_o second_o figure_n 6_o count_v the_o figure_n 6_o upon_o the_o line_n upward_o then_o for_o your_o three_o figure_n 2_o count_v two_o of_o the_o grand_a division_n and_o for_o your_o last_o figure_n 6_o estimate_v six_o ten_o part_n of_o the_o next_o grand_a division_n which_o be_v something_o more_o than_o half_a the_o distance_n because_o 6_o be_v more_o than_o half_a 10_o and_o that_o be_v the_o point_n upon_o the_o line_n represent_v 1626._o note_v by_o these_o example_n last_o mention_v you_o may_v perceive_v that_o the_o figure_n 1.2.3.4.5.6.7.8.9_o do_v sometime_o signify_v themselves_o alone_o sometime_o 10.20.30_o etc._n etc._n sometime_o 100.200.300_o etc._n etc._n as_o the_o work_n perform_v thereby_o shall_v require_v the_o first_o figure_n of_o every_o number_n be_v always_o that_o which_o be_v here_o set_v down_o and_o the_o rest_n of_o the_o figure_n be_v to_o be_v supply_v according_a as_o the_o nature_n of_o the_o question_n shall_v require_v and_o by_o this_o variation_n and_o change_n of_o the_o power_n of_o these_o number_n from_z 1_o to_z 10_o or_o 100_o or_o 1000_o any_o proportion_n either_o arithmetial_a or_o geometrical_a may_v be_v wrought_v one_o whereof_o i_o will_v insert_v for_o your_o better_a exercise_n of_o number_v on_o the_o rule_n by_o the_o often_o practice_n whereof_o you_o will_v find_v the_o work_n facile_a and_o delightful_a which_o shall_v be_v this_o follow_v prop._n 2._o have_v two_o number_n give_v to_o find_v as_o many_o more_o as_o you_o please_v which_o shall_v be_v in_o continual_a proportion_n one_o to_o the_o other_o as_o the_o two_o number_n give_v be_v for_o the_o work_n of_o this_o proposition_n this_o be_v the_o rule_n place_n one_o foot_n of_o the_o compass_n in_o the_o first_o give_v number_n on_o the_o line_n and_o extend_v the_o other_o foot_n to_o the_o second_o give_v number_n then_o may_v you_o turn_v the_o compass_n from_o that_o second_o number_n to_o a_o three_o from_o that_o three_o to_o a_o four_o from_o that_o four_o to_o a_o five_o a_o six_o a_o seven_o etc._n etc._n towhat_o number_n of_o place_n you_o please_v example_n 1._o let_v the_o two_o give_v number_n be_v 2_o and_o 4._o place_n one_o foot_n of_o your_o compass_n in_o 2_o and_o extend_v the_o other_o foot_n to_o 4_o than_o that_o foot_n which_o now_o stand_v in_o 2_o be_v turn_v about_o will_v reach_v from_o 4_o to_o 8_o and_o from_o 8_o to_o 16_o from_o 16_o to_o 32_o from_o 32_o to_o 64_o from_o 64_o to_o 128._o but_o when_o your_o compass_n stand_v in_o 64_o if_o you_o turn_v they_o about_o yet_o far_o they_o will_v fall_v beyond_o the_o end_n of_o the_o line_n wherefore_o you_o must_v place_v one_o foot_n in_o some_o other_o 64_o near_a the_o begin_n of_o the_o line_n and_o then_o the_o other_o foot_n will_v reach_v to_o 128_o and_o from_o 128_o to_o 256_o and_o from_o 256_o to_o 512_o and_o from_o 512_o to_o 1024_o but_o here_o it_o will_v go_v off_o of_o your_o line_n again_o whereof_o as_o before_o you_o must_v choose_v another_o 512_o near_a the_o begin_n of_o the_o line_n and_o there_o place_v you_o compass_n they_o will_v reach_v to_o 1024._o from_o 1024_o to_o 2048_o from_o 2048_o to_o 4096_o etc._n etc._n example_n 2._o but_o if_o the_o give_v number_n be_v 10_o and_o 9_o decrease_a then_o place_v one_o foot_n in_o 10_o at_o the_o end_n of_o line_n and_o extend_v the_o other_o downward_o to_o 9_o the_o same_o extent_n will_v reach_v still_o backward_o to_o 8.1_o or_o 8_o 1_o 10_o and_o from_o 8.1_o to_o 7.29_o and_o still_o backward_o from_o 7.29_o to_o 6.56_o likewise_o if_o the_o two_o first_o number_n have_v be_v as_o 1_o to_o 9_o the_o three_o proportional_a will_v have_v be_v 81_o the_o four_o 729_o and_o the_o five_o 656_o with_o the_o same_o extent_n of_o the_o compass_n again_o let_v the_o two_o number_n be_v 10_o and_o 12_o place_z one_o foot_n in_o 10_o and_o extend_v the_o other_o to_o 12_o that_o extent_n will_v reach_v from_o 12_o to_o 14.4_o and_o from_o thence_o to_o 17.28_o but_o if_o the_o number_n be_v 1_o and_o 12_o than_o the_o three_o proportional_a will_v be_v 144_o and_o the_o four_o 1728_o and_o all_o with_o the_o same_o extent_n of_o the_o compass_n chap._n ii_o multiplication_n by_o the_o line_n in_o multiplication_n the_o proportion_n be_v this_o as_o 1_o upon_o the_o line_n be_v to_o one_o of_o the_o number_n to_o be_v multiply_v so_o be_v the_o other_o of_o the_o number_n to_o be_v multiply_v to_o the_o product_n of_o they_o which_o be_v the_o number_n seek_v example_n 1._o let_v it_o be_v require_v to_o multiply_v 5_o by_o 7._o the_o proportion_n be_v as_o 1_o ∶_o to_o 5_o ∷_o so_o be_v 7_o ∶_o to_o 35._o therefore_o set_v one_o foot_n of_o your_o compass_n in_o 1_o and_o extend_v the_o other_o foot_n to_o 5_o with_o that_o extent_n of_o the_o compass_n place_n one_o foot_n in_o 7_o and_o the_o other_o foot_n will_v fall_v upon_o 35_o which_o be_v the_o product_n example_n 2._o let_v it_o be_v require_v to_o multiply_v 32_o by_o 9_o the_o proportion_n be_v as_o 1_o ∶_o to_o 9_o ∷_o so_o 32_o ∶_o to_o 288._o set_a one_o foot_n in_o one_o and_o extend_v the_o other_o foot_n to_o 9_o that_o same_o extent_n will_v reach_v from_o 32_o to_o 288_o which_o be_v the_o product_n or_o sum_n of_o 32_o be_v multiply_v by_o 9_o otherwise_o set_v one_o foot_n in_o 1_o and_o extend_v the_o other_o to_o 32_o the_o same_o extent_n will_v reach_v from_o 9_o to_o 288_o as_o before_o example_n 3._o let_v it_o be_v require_v to_o multiply_v 8_o 75_o 100_o by_o 5_o 45_o 100_o the_o analogy_n or_o proportion_n be_v as_o 1_o ∶_o to_o 8.75_o ∷_o so_o 6.45_o ∶_o to_o 56.44_o fere_n set_a one_o foot_n in_o 1_o and_o extend_v the_o other_o to_o 8.75_o the_o same_o extent_n apply_v forward_o upon_o the_o line_n will_v reach_v from_o 6.45_o to_o 56.44_o fere_n or_o if_o you_o set_v one_o foot_n in_o 1_o and_o extend_v the_o other_o to_o 6.45_o the_o same_o extent_n will_v reach_v from_o 8.75_o to_o 56.44_o almost_o namely_o to_o 43_o ¾_n as_o before_o chap._n iii_o division_n by_o the_o line_n in_o division_n three_o thing_n be_v to_o be_v mind_v viz._n the_o dividend_n or_o number_n to_o be_v divide_v the_o divisor_n the_o number_n by_o which_o the_o dividend_n be_v divide_v the_o quotient_a which_o be_v the_o number_n seek_v and_o so_o often_o as_o the_o divisor_n be_v contain_v in_o the_o dividend_n so_o often_o do_v the_o quotient_a contain_v unity_n for_o the_o work_n of_o division_n this_o be_v the_o analogy_n as_o the_o divisor_n be_v to_o unity_n or_o 1._o so_o be_v the_o dividend_n to_o the_o quotient_a example_n 1._o let_v it_o be_v require_v to_o divide_v 35_o by_o 7._o the_o proportion_n be_v as_o 7_o ∶_o to_o 1_o ∷_o so_o 35_o ∶_o to_o 5._o set_a one_o foot_n of_o the_o compass_n in_o 7_o and_o extend_v the_o other_o foot_n downward_o to_o one_o that_o same_o extent_n will_v reach_v from_o 35_o downward_o to_o 5_o which_o be_v the_o quotient_a and_o so_o many_o time_n be_v 7_o contain_v in_o 35._o otherwise_o extend_v the_o compass_n upward_o from_o 7_o to_o 35_o that_o same_o extent_n will_v reach_v upward_o from_o 1_o to_o 5_o
it_o will_v fall_v out_o of_o the_o line_n wherefore_o you_o must_v set_v one_o foot_n in_o 37.8_o in_o the_o low_a part_n of_o the_o line_n and_o then_o the_o other_o will_v fall_v upon_o 176.74_o and_o thus_o you_o must_v do_v in_o other_o case_n whenever_o your_o compass_n point_n go_v beyond_o your_o line_n chap._n fourteen_o ii_o of_o the_o triangle_n a_o triangle_n be_v a_o figure_n consist_v of_o three_o side_n and_o three_o angle_n the_o long_a side_n whereof_o we_o call_v the_o base_a and_o a_o line_n draw_v from_o the_o angle_n opposite_a to_o the_o base_a we_o call_v the_o perpendicular_a to_o measure_v triangle_n there_o be_v several_a way_n i_o will_v only_o show_v you_o one_o or_o two_o to_o be_v do_v by_o the_o line_n example_n 1._o there_o be_v a_o triangle_n who_o base_a be_v 14_o foot_n and_o his_o perpendicular_a 6_o foot_n i_o will_v know_v how_o many_o square_a foot_n be_v contain_v in_o this_o triangle_n the_o proportion_n be_v as_o 2_o be_v to_z 6_o the_o perpendicular_a so_o be_v 14_o the_o base_a to_o 42_o the_o area_n or_o as_o 1_o be_v to_z 3_o half_o the_o base_a so_o be_v 14_o to_o 42_o the_o area_n or_o as_o 2_o be_v to_z 6_o the_o perpendicular_a so_o be_v 7_o half_a the_o base_a to_o 42_o the_o area_n or_o as_o 1_o be_v to_z 6_o the_o perpendicular_a so_o be_v 14_o the_o base_a to_o 84_o the_o double_a area_n all_o these_o way_n produce_v the_o same_o effect_n but_o the_o first_o be_v the_o best_a wherefore_o the_o base_a of_o your_o triangle_n be_v 14_o and_o the_o perpendicular_a 6_o extend_v the_o compass_n from_o 2_o to_o 6_o the_o same_o extent_n will_v reach_v from_o 14_o to_o 42_o the_o area_n iii_o of_o the_o trapezia_n a_o trapezia_n be_v any_o right_n line_v figure_n consist_v of_o four_o unequal_a side_n and_o as_o many_o angle_n for_o the_o measure_n of_o it_o you_o must_v first_o reduce_v it_o into_o two_o triangle_n by_o draw_v a_o line_n or_o diameter_n from_o one_o opposite_a angle_n to_o another_o the_o long_a way_n then_o from_o the_o two_o angle_n opposite_a to_o this_o line_n let_v fall_v two_o perpendicular_o so_o be_v the_o trapezia_n divide_v into_o two_o triangle_n the_o manner_n how_o to_o measure_v it_o be_v this_o example_n there_o be_v a_o trapezia_n who_o diameter_n be_v 12.34_o and_o one_o perpendicular_a be_v 4.20_o the_o other_o 5.07_o i_o will_v know_v the_o content_a or_o area_n thereof_o the_o two_o perpendicular_o add_v together_o make_v 9.27_o then_o the_o analogy_n be_v as_o 2_o be_v to_o 9.27_o the_o sum_n of_o the_o perdendiculars_n so_o be_v 12.34_o the_o base_a to_o 57.19_o the_o area_n there_o be_v as_o many_o way_n to_o measure_n trapezias_n as_o in_o the_o last_o example_n i_o give_v you_o for_o triangle_n but_o this_o be_v the_o best_a and_o here_o note_v that_o if_o you_o be_v to_o measure_v any_o irregular_a piece_n of_o what_o nature_n soever_o whether_o land_n board_n glass_n pavement_n or_o the_o like_a your_o best_a and_o exact_a way_n be_v to_o reduce_v they_o to_o trapezias_n and_o measure_v they_o as_o before_o be_v teach_v four_o of_o regular_a figure_n of_o 5_o 6_o 8_o 10_o or_o 20_o equal_a side_n these_o figure_n by_o geometrician_n be_v call_v regular_a polygon_n and_o the_o way_n to_o measure_v they_o be_v by_o add_v all_o the_o cider_n together_o then_o measure_v the_o length_n from_o the_o centre_n of_o the_o figure_n to_o the_o middle_n of_o one_o of_o the_o side_n by_o the_o help_n of_o these_o two_o you_o may_v find_v the_o area_n of_o the_o figure_n example_n let_v there_o be_v a_o regular_a polygon_n of_o 11_o equal_a side_n each_o side_n be_v 7_o inch_n and_o let_v the_o length_n of_o the_o line_n from_o the_o centre_n to_o the_o middle_n of_o one_o of_o those_o side_n be_v 12_o inch_n add_v all_o the_o side_n together_o they_o make_v 77_o then_o as_o 2_o be_v to_o 77_o the_o sum_n of_o the_o side_n so_o be_v 12_o the_o length_n of_o the_o line_n from_o the_o middle_n of_o the_o figure_n to_o 462_o the_o content_a of_o the_o figure_n chap._n xv._o the_o use_n of_o the_o line_n apply_v to_o solid-measure_a such_o as_o timber_n stone_n etc._n etc._n timber_n and_o stone_n be_v usual_o measure_v by_o the_o same_o rule_n or_o measure_n as_o board_n and_o glass_n be_v namely_o by_o foot_n and_o inch_n therefore_o such_o a_o rule_n as_o be_v mention_v in_o the_o begin_n of_o the_o ten_o chapter_n be_v fit_a for_o this_o business_n also_o before_o we_o come_v to_o show_v the_o way_n of_o measure_v of_o stone_n or_o timber_n it_o will_v be_v necessary_a to_o premise_v thus_o much_o that_o the_o base_a or_o end_n of_o every_o piece_n of_o timber_n or_o stone_n be_v or_o must_v be_v suppose_v either_o exact_o square_a that_o be_v every_o side_n alike_o or_o else_o one_o of_o the_o side_n long_o than_o the_o other_o wherefore_o the_o first_o thing_n to_o be_v do_v be_v to_o find_v the_o area_n or_o superficial_a content_n of_o the_o base_a or_o end_n of_o any_o piece_n of_o timber_n or_o stone_n to_o be_v measure_v which_o may_v be_v do_v several_a way_n either_o in_o inch-measure_n as_o by_o the_o first_o example_n of_o the_o first_o part_n of_o the_o ten_o chapter_n or_o in_o foot-measure_n by_o the_o first_o example_n in_o the_o second_o part_n of_o the_o same_o chapter_n or_o both_o in_o foot-measure_n and_o inch-measure_n as_o in_o the_o first_o example_n of_o the_o three_o part_n of_o the_o same_o ten_o chapter_n and_o therefore_o need_v not_o be_v here_o repeat_v again_o wherefore_o we_o will_v proceed_v to_o our_o intend_a purpose_n of_o measure_v first_o by_o inch-measure_n only_o second_o by_o foot-measure_n only_o and_o three_o by_o both_o together_o as_o we_o do_v before_o in_o the_o measure_n of_o board_n etc._n etc._n 1._o in_o inch-measure_n only_o example_n 1._o there_o be_v a_o piece_n of_o timber_n 30_o inch_n broad_a 21_o inch_n 6_o part_n deep_a and_o 183_o inch_n long_o how_o many_o square_a inch_n be_v there_o in_o this_o solid_a piece_n of_o timber_n the_o proportion_n be_v 1._o as_o 1_o unto_o 30_o inch_n the_o breadth_n so_o be_v 21.6_o inch_n the_o depth_n to_o 648_o inch_n the_o content_a of_o the_o base_a of_o the_o piece_n 2._o as_o 1_o unto_o 648_o the_o content_a of_o the_o base_a so_o be_v 183_o inch_n the_o length_n of_o the_o piece_n to_o 118584_o the_o solid_a content_n in_o inches_n wherefore_o extend_v the_o compase_n from_o 1_o to_o 30_o the_o breadth_n the_o same_o will_v reach_v from_o 21.6_o the_o depth_n to_o 648_o the_o content_a of_o the_o base_a again_o extend_v the_o compass_n from_o 1_o to_o 648_o the_o content_a of_o the_o base_a that_o extent_n will_v reach_v from_o 183_o the_o length_n to_o 118584_o inch_n the_o solid_a content_n but_o so_o many_o place_n of_o figure_n can_v well_o be_v estimate_v upon_o your_o line_n except_o it_o be_v very_o large_a but_o by_o follow_v example_n you_o shall_v have_v your_o desire_n accomplish_v exact_o and_o easy_o example_n 2._o to_o find_v the_o content_a of_o the_o same_o piece_n of_o timber_n in_o foot-measure_n the_o dimension_n be_v give_v in_o inch_n and_o part_n the_o proportion_n be_v 1._o as_o 1_o to_o 30_o the_o breadth_n so_o 21.6_o the_o depth_n to_o 648_o the_o content_a of_o the_o base_a as_o before_o 2._o as_o 1728_o the_o number_n of_o solid_a inch_n in_o a_o foot_n of_o timber_n be_v to_o 648_o the_o content_a of_o the_o base_a so_o be_v 183_o the_o length_n in_o inch_n to_o 68_o foot_n and_o 62_o 100_o part_n of_o a_o foot_n as_o before_o wherefore_o as_o before_o extend_v the_o compass_n from_o 1_o to_o 30_o the_o breadth_n the_o same_o will_v reach_v from_o 21.6_o the_o depth_n to_o 648_o the_o content_a of_o the_o base_a as_o before_o again_o extend_v the_o compass_n from_o 1728_o to_o 648_o the_o base_a the_o same_o extent_n will_v reach_v the_o same_o way_n from_o 183_o the_o length_n to_o 68.62_o the_o content_a of_o the_o piece_n of_o timber_n in_o foot_n and_o part_n that_o be_v 68_o foot_n and_o above_o half_a a_o foot_n example_n 3._o let_v a_o square_a stone_n or_o piece_n of_o timber_n be_v 30_o inch_n broad_a and_o 21_o inch_n six_o part_n deep_a how_o much_o in_o length_n shall_v make_v a_o foot_n square_n of_o that_o piece_n of_o timber_n or_o stone_n you_o may_v find_v the_o content_a of_o the_o base_a as_o in_o the_o last_o example_n to_o be_v 648_o inch_n then_o the_o proportion_n be_v as_o 648_o the_o content_a of_o the_o base_a be_v to_o 1728_o the_o inch_n in_o a_o foot_n so_o be_v 1_o to_o 2_o inch_n 67_o part_n the_o length_n of_o a_o foot_n solid_a therefore_o extend_v the_o compass_n from_o 648_o the_o base_a to_o 1728_o the_o