will_v both_o the_o needle_n and_o index_n cut_v 68_o degree_n and_o the_o distance_n ce_fw-fr will_v be_v 8_o chain_n 72_o link_n which_o note_n in_o your_o field-book_n as_o before_o 5._o last_o place_v your_o instrument_n at_o e_z observing_z all_z the_o former_a caution_n and_o direct_v the_o sight_n to_o a_o where_o you_o shall_v find_v both_o the_o needle_n and_o index_n to_o cut_v 142_o degree_n 45_o minute_n and_o the_o measure_a distance_n ea_fw-la to_o be_v 7_o chain_n 11_o link_n which_o note_v down_o in_o your_o field-book_n and_o thus_o may_v you_o go_v about_o any_o field_n let_v it_o consist_v of_o never_o so_o many_o side_n and_o angle_n observe_v always_o this_o general_a rule_n to_o lay_v the_o index_n with_o the_o box_n and_o needle_n on_o the_o diameter_n of_o the_o table_n and_o to_o turn_v the_o table_n about_o till_o the_o needle_n hang_v direct_o over_o the_o meridian_n line_n in_o the_o card_n and_o then_o fix_v the_o table_n turn_v the_o index_n about_o till_o through_o the_o sight_n you_o espy_v the_o mark_n you_o look_v for_o then_o will_v both_o the_o index_n and_o the_o needle_n cut_v the_o degree_n which_o you_o must_v note_v in_o your_o field-book_n so_o will_v the_o collect_v note_n of_o this_o example_n stand_v as_o follow_v  _fw-fr degree_n minute_n chain_n link_n a_o 218_o 10_o 9_o 65_o b_o 298_o 30_o 9_o 28_o c_o 15_o 40_o 5_o 70_o d_o 68_o 00_o 8_o 72_o e_z 142_o 45_o 7_o 11_o have_v thus_o collect_v your_o several_a observation_n you_o may_v proceed_v to_o protract_v your_o work_n as_o be_v teach_v in_o the_o next_o chapter_n which_o differ_v nothing_o from_o that_o in_o the_o 36_o chap._n ¶_o it_o will_v be_v here_o object_v by_o the_o affector_n of_o the_o peractor_n that_o here_o it_o be_v require_v that_o the_o needle_n shall_v play_v twice_o at_o each_o observation_n to_o which_o i_o answer_v it_o be_v true_a but_o if_o you_o neglect_v the_o latter_a of_o they_o it_o be_v both_o as_o speedy_a and_o as_o exact_a as_o the_o peractor_n and_o if_o you_o have_v opportunity_n to_o observe_v both_o which_o you_o may_v convenient_o do_v it_o will_v then_o be_v better_o chap._n xxxviii_o how_o to_o protract_v any_o observation_n take_v as_o in_o the_o last_o chapter_n you_o must_v first_o rule_v your_o paper_n or_o parchment_n all_o over_o with_o parallel_a line_n or_o meridian_n as_o be_v teach_v in_o the_o 36_o chapter_n and_o upon_o one_o of_o these_o meridian_n assign_v any_o point_n at_o pleasure_n as_o a_o then_o lay_v your_o field-book_n before_o you_o place_v the_o centre_n of_o the_o protractor_n upon_o the_o point_n a_o the_o scale_n thereof_o lie_v upon_o or_o parallel_v to_o one_o of_o the_o meridians_fw-la rule_v on_o your_o paper_n and_o because_o the_o degree_n cut_v at_o a_o be_v above_o 180_o degree_n viz._n 218_o degree_n 10_o minute_n therefore_o lay_v the_o semicircle_n of_o the_o protractor_n downward_o and_o against_o 218_o degree_n 10_o minute_n of_o your_o protractor_n make_v a_o mark_n through_o which_o mark_n and_o the_o point_n a_o draw_v the_o line_n ab_fw-la contain_v 9_o chain_n 65_o link_n 2._o remove_v your_o protractor_n to_o the_o point_n b_o which_o represent_v your_o second_o station_n or_o angle_n lay_v the_o meridian_n line_n thereof_o upon_o or_o parallel_n to_o one_o of_o the_o meridian_n draw_v upon_o the_o paper_n and_o because_o the_o degree_n cut_v at_o bare_a above_o 180_o lay_v the_o semicircle_n downward_o as_o before_o and_o against_o 298_o degree_n 30_o minute_n make_v a_o mark_n and_o through_o it_o draw_v the_o line_n bc_n contain_v 9_o chain_n 28_o link_n 3._o bring_v your_o protractor_n to_o c_o and_o lay_v it_o parallel_v to_o some_o one_o of_o your_o meridian_n and_o because_o the_o degree_n observe_v at_o c_o be_v under_o 180_o namely_o 15_o degree_n 40_o minute_n lay_v the_o semicircle_n upward_o and_o against_o 15_o degree_n 40_o minute_n make_v a_o mark_n draw_v the_o line_n cd_o contain_v 5_o chain_n 70_o link_n 4._o place_v your_o protractor_n as_o before_o upon_o the_o point_n d_o with_o the_o semicircle_n upward_o and_o against_o 68_o degree_n thereof_o make_v a_o mark_n and_o draw_v the_o line_n de_fw-fr contain_v 8_o chain_n 72_o link_n last_o remove_v your_o protractor_n to_o e_o placing_z it_z as_o before_o and_o against_o 142_o degree_n 45_o minute_n which_o be_v the_o degree_n observe_v at_o your_o station_n at_o e_z make_v a_o mark_n and_o through_o it_o and_o the_o point_n e_o draw_v the_o line_n ea_fw-la which_o if_o your_o work_n be_v true_a will_v pass_v through_o the_o point_n a_o and_o will_v contain_v 7_o chain_n 11_o link_n chap._n thirty-nine_o how_o to_o find_v how_o many_o acre_n rood_n and_o perch_n be_v contain_v in_o any_o piece_n of_o land_n the_o plot_n thereof_o be_v first_o take_v by_o any_o instrument_n have_v show_v how_o to_o take_v the_o plot_n of_o any_o field_n or_o other_o enclosure_n several_a way_n and_o also_o to_o protract_v the_o same_o upon_o paper_n it_o be_v now_o necessary_a to_o show_v how_o the_o content_a thereof_o may_v be_v attain_v that_o be_v to_o say_v how_o many_o acre_n rood_n and_o perch_n any_o field_n so_o plot_v do_v contain_v in_o the_o performance_n hereof_o you_o must_v consider_v that_o the_o original_n of_o the_o mensuration_n of_o all_o superficial_a figure_n such_o as_o land_n board_n glass_n or_o the_o like_a do_v depend_v upon_o the_o exact_a measure_n of_o certain_a regular_a figure_n as_o the_o geometrical_a square_n the_o long_a square_n or_o parallelogram_n the_o triangle_n the_o trapezia_n and_o the_o circle_n therefore_o if_o any_o plot_n of_o land_n to_o be_v measure_v be_v not_o one_o of_o these_o figure_n it_o must_v before_o it_o can_v be_v measure_v be_v reduce_v into_o some_o of_o these_o form_n i_o will_v therefore_o in_o the_o first_o place_n show_v how_o to_o measure_v any_o of_o these_o figure_n several_o by_o themselves_o and_o afterward_o how_o to_o reduce_v any_o other_o irregular_a figure_n into_o some_o of_o these_o regular_a form_n and_o last_o to_o measure_v they_o by_o the_o same_o rule_n and_o first_o of_o the_o geometrical_a square_n a_o geometrical_a square_n be_v a_o figure_n consist_v of_o four_o equal_a side_n and_o angle_n as_o be_v the_o square_n abcd_v who_o side_n be_v all_o equal_a to_o the_o line_n qr_n which_o contain_v six_o equal_a part_n which_o may_v be_v attribute_v either_o to_o inch_n foot_n yard_n perch_n chain_n or_o any_o other_o measure_n whatsoever_o now_o to_o find_v the_o superficial_a content_n of_o such_o a_o square_a you_o must_v multiply_v one_o of_o the_o side_n in_o itself_o and_o the_o product_n of_o that_o multiplication_n shall_v be_v the_o content_a of_o the_o square_n example_n suppose_v the_o square_n abcd_v to_o be_v a_o piece_n of_o land_n and_o the_o side_n thereof_o to_o contain_v 6_o perch_n therefore_o multiply_v 6_o in_o itself_o and_o the_o product_n will_v be_v 36_o &_o so_o many_o perch_n do_v the_o square_a piece_n of_o land_n contain_v of_o the_o long_a square_n a_o long_a square_n be_v a_o figure_n consist_v of_o four_o side_n as_o the_o figure_n abcd_n the_o two_o opposite_a side_n whereof_o be_v equal_a as_o the_o side_n ab_fw-la and_o cd_o and_o likewise_o ac_fw-la and_o bd_o each_o of_o the_o short_a side_n contain_v 7_o perch_n and_o the_o long_a side_n 13_o perch_n to_o find_v the_o superficial_a content_n of_o this_o long_a square_n or_o parallelogram_n you_o must_v multiply_v one_o of_o the_o long_a side_n by_o one_o of_o the_o short_a and_o the_o product_n will_v show_v the_o superficial_a content_n thereof_o example_n the_o long_a side_n of_o the_o square_n contain_v 13_o perch_n and_o the_o short_a 7_o perch_n now_o if_o you_o multiply_v 13_o by_o 7_o the_o product_n will_v be_v 91_o and_o that_o be_v the_o content_a of_o the_o square_n in_o perch_n of_o the_o triangle_n although_o there_o be_v several_a kind_n of_o triangle_n yet_o in_o respect_n they_o be_v all_o measure_v by_o one_o and_o the_o same_o rule_n i_o will_v therefore_o add_v one_o example_n for_o all_o which_o be_v general_a half_n the_o length_n of_o the_o base_a be_v multiply_v by_o the_o length_n of_o the_o perpendicular_a shall_v be_v equal_a to_o the_o area_n of_o the_o triangle_n or_o half_a the_o length_n of_o the_o perpendicular_a be_v multiply_v by_o the_o whole_a base_a will_v be_v the_o content_a of_o the_o triangle_n example_n suppose_v you_o be_v to_o find_v the_o area_n or_o content_n of_o the_o triangle_n bcd_n the_o base_a thereof_o db_v contaik_v 58_o perch_n and_o the_o perpendicular_a ca_n 24_o perch_n now_o if_o you_o multiply_v 12_o which_o be_v half_a the_o length_n of_o the_o perpendicular_a ca_n by_o 58_o the_o length_n of_o the_o whole_a base_a db_n the_o product_n
and_o that_o it_o be_v require_v to_o find_v the_o area_n or_o content_n thereof_o in_o perch_n which_o to_o effect_v you_o must_v multiply_v the_o length_n by_o the_o breadth_n as_o be_v teach_v in_o the_o last_o chapter_n therefore_o the_o length_n be_v 16_o unite_v 2_o prime_n and_o the_o breadth_n 1_o unite_v 3_o prime_n 2_o second_o these_o two_o number_n multiply_v together_o shall_v produce_v the_o area_n set_a your_o number_n down_o as_o you_o be_v teach_v in_o the_o 5_o chapter_n of_o the_o 2_o book_n or_o as_o you_o see_v they_o stand_v in_o this_o example_n with_o a_o prick_n over_o the_o head_n of_o every_o fraction_n 162̇_n 13̇2̇_n 324_o 486_o 162_o 21384_o under_o these_o number_n draw_v a_o line_n and_o multiply_v they_o together_o in_o all_o respect_n as_o if_o they_o be_v whole_a number_n and_o then_o the_o work_n will_v stand_v thus_o the_o product_n of_o your_o multiplication_n be_v 21384._o now_o because_o in_o your_o two_o number_n viz._n your_o multiplicand_a and_o your_o multiplyer_n there_o be_v three_o fraction_n namely_o one_o in_o your_o multiplicand_a and_o two_o in_o your_o multiplyer_n you_o must_v therefore_o with_o a_o dash_n of_o your_o pen_n cut_v off_o the_o three_o last_o figure_n of_o the_o product_n towards_o your_o right_a hand_n and_o then_o will_v your_o product_n stand_v thus_o 21_o 384_o the_o three_o last_o figure_n whereof_o be_v the_o numerator_n of_o a_o fraction_n who_o denominator_fw-la be_v 1000_o and_o the_o other_o two_o figure_n towards_o your_o left_a hand_n be_v integer_n of_o your_o multiplication_n so_o that_o the_o sum_n of_o this_o multiplication_n be_v 21_o perch_n and_o 384_o 1000_o part_n of_o a_o perch_n which_o be_v something_o more_o than_o a_o three_o part_n of_o a_o perch_n but_o to_o express_v the_o exact_a quantity_n of_o these_o fraction_n in_o a_o business_n of_o this_o nature_n be_v superfluous_a only_o observe_v this_o one_o rule_n for_o all_o namely_o that_o if_o the_o figure_n cut_v off_o come_v near_o to_o a_o unite_v that_o be_v when_o the_o figure_n cut_v off_o be_v near_o as_o much_o as_o those_o underneath_o they_o or_o the_o first_o figure_n cut_v off_o be_v either_o 7_o 8_o or_o 9_o you_o may_v then_o increase_v your_o whole_a number_n by_o a_o unite_v and_o not_o at_o all_o regard_n the_o fraction_n but_o for_o your_o further_a practice_n take_v another_o example_n which_o let_v be_v a_o piece_n of_o land_n contain_v in_o breadth_n 5_o unite_v 6_o prime_n 3_o second_o and_o in_o length_n 15_o unite_v 4_o prime_n and_o 2_o second_o which_o place_n as_o before_o now_o if_o you_o multiply_v these_o number_n one_o by_o another_o as_o if_o they_o be_v whole_a number_n then_o will_v they_o stand_v as_o in_o the_o margin_n the_o product_n be_v 868146_o 154̇2̇_n 56̇3̇_n 4626_o 9252_o 7710_o 868146_o from_o whence_o take_v the_o four_o last_o figure_n because_o there_o be_v four_o fraction_n in_o your_o two_o number_n there_o remain_v 86_o perch_n and_o 9146_o 10000_o part_n of_o a_o perch_n now_o because_o 8146_o be_v near_o to_o 10000_o i_o add_v 1_o to_o 86_o make_v it_o 87_o perch_n dis-regarding_a the_o excess_n as_o immaterial_a in_o like_a manner_n suppose_v the_o perpendicular_a of_o a_o triangle_n shall_v contain_v 1_o unite_v 3_o prime_n 2_o second_o and_o half_o the_o length_n of_o the_o base_a shall_v contain_v 16_o unite_v 2_o prime_n these_o number_n be_v place_v as_o those_o before_o and_o multiply_v one_o by_o another_o will_v produce_v this_o product_n 21384_o from_o whence_o cut_v off_o the_o three_o last_o figure_n because_o there_o be_v three_o fraction_n in_o your_o number_n multiply_v and_o there_o will_v remain_v 21_o perch_n and_o 384_o 1000_o part_n of_o a_o perch_n which_o be_v but_o of_o small_a value_n you_o may_v reject_v chap._n xli_o how_o to_o reduce_v any_o number_n of_o perch_n into_o rood_n and_o acre_n or_o any_o number_n of_o acre_n and_o rood_n into_o perch_n by_o a_o statute_n make_v the_o 33._o of_o edw._n 1._o a_o acre_n of_o ground_n ought_v to_o contain_v 160_o square_a perch_n and_o every_o rood_n of_o land_n 40_o square_a perch_n and_o every_o perch_n be_v to_o contain_v 16_o foot_n and_o a_o half_a now_o if_o any_o number_n of_o perch_n be_v give_v to_o be_v turn_v into_o acre_n you_o must_v divide_v the_o number_n give_v by_o 160_o the_o number_n of_o perch_n contain_v in_o one_o acre_n and_o the_o quotient_n shall_v show_v you_o how_o many_o acre_n be_v contain_v in_o that_o number_n of_o perch_n and_o if_o any_o thing_n remain_v if_o it_o be_v under_o 40_o it_o be_v perch_n but_o if_o the_o remainder_n exceed_v 40_o than_o you_o must_v divide_v it_o by_o 40_o the_o number_n of_o perch_n contain_v in_o one_o rood_n and_o the_o quotient_n shall_v be_v rood_n and_o the_o remainder_n perch_v example_n let_v 5267_o perch_n be_v give_v to_o be_v reduce_v into_o acre_n first_o divide_v 5267_o by_o 160_o and_o the_o quotient_n will_v be_v 32_o and_o 147_o remain_v which_o divide_v by_o 40_o the_o quotient_n will_v be_v 3_o and_o 27_o remain_v so_o that_o the_o whole_a amount_v to_o 32_o acre_n 3_o rood_n and_o 27_o perch_n again_o let_v 5496_o perch_n be_v give_v to_o be_v reduce_v into_o acre_n first_o divide_v 5496_o by_o 160_o the_o quotient_n will_v be_v 34_o and_o 56_o remain_v which_o 56_o be_v divide_v by_o 40_o the_o quotient_n will_v be_v 1_o and_o 16_o remain_v so_o that_o the_o whole_a will_v be_v 34_o acre_n 1_o rood_n and_o 16_o perch_n to_o reduce_v acre_n into_o perch_n this_o be_v but_o the_o converse_n of_o the_o former_a for_o as_o before_o to_o reduce_v perch_n into_o acre_n you_o divide_v by_o 160_o you_o must_v now_o to_o reduce_v acre_n into_o perch_n multiply_v by_o 160._o example_n let_v 32_o acre_n 3_o rood_n and_o 27_o perch_n be_v give_v to_o be_v reduce_v into_o perch_n first_o multiply_v the_o 32_o acre_n by_o 160_o and_o the_o product_n will_v be_v 5120_o then_o multiply_v the_o 3_o rood_n by_o 40_o the_o product_n be_v 120_o these_o two_o product_n and_o the_o 27_o perch_n be_v add_v together_o the_o sum_n will_v be_v 5267_o 5120_o 120_o 27_o 5267_o and_o so_o many_o perch_n be_v contain_v in_o the_o foresay_a number_n of_o acre_n rood_n and_o perch_n and_o thus_o much_o concern_v the_o use_n of_o master_n rathborn_v chain_n chap._n xlii_o how_o to_o cast_v up_o the_o content_a of_o any_o piece_n of_o land_n in_o acre_n rood_n and_o perch_n by_o master_n gunter'_v chain_n in_o measure_v by_o master_n gunter_n chain_v you_o be_v in_o your_o account_n only_o to_o take_v notice_n of_o chain_n and_o link_n as_o be_v before_o intimate_v in_o the_o description_n thereof_o cap._n 5._o lib._n 2._o suppose_v then_o that_o the_o figure_n b_o be_v a_o piece_n of_o land_n lie_v in_o a_o long_a square_n and_o that_o be_v measure_v by_o master_n gunter'_v chain_n shall_v contain_v in_o length_n 9_o chain_n 50_o link_n and_o in_o breadth_n 6_o chain_n 25_o link_n set_a your_o number_n down_o as_o before_o be_v teach_v and_o as_o here_o you_o see_v draw_v a_o line_n under_o they_o then_o multiply_v they_o together_o you_o shall_v find_v the_o product_n to_o be_v 593750_o 9,50_o 6,25_o 4750_o 1900_o 5700_o 593750_o from_o which_o product_v you_o must_v always_o cut_v off_o the_o five_o last_o figure_n towards_o the_o right_a hand_n with_o a_o dash_n of_o your_o pen_n then_o will_v the_o product_n stand_v thus_o 593750_o so_o be_v the_o 5_o towards_o the_o left_a hand_n complete_a acre_n and_o the_o 93750_o hundred_o thousand_o part_n of_o a_o acre_n which_o to_o reduce_v into_o rood_n and_o perch_n be_v easy_a by_o help_n of_o this_o table_n for_o if_o you_o look_v for_o 90000_o under_o the_o title_n link_n which_o be_v the_o first_o figure_n with_o cipher_n add_v link_n r._n p._n 100000_o 4_o 0_o 90000_o 3_o 24_o 80000_o 3_o 8_o 70000_o 2_o 32_o 60000_o 2_o 16_o 50000_o 2_o 0_o 40000_o 1_o 24_o 30000_o 1_o 8_o 20000_o 0_o 32_o 10000_o 0_o 16_o 9375_o 0_o 15_o 8750_o 0_o 14_o 8125_o 0_o 13_o 7500_o 0_o 12_o 6875_o 0_o 11_o 6250_o 0_o 10_o 5625_o 0_o 9_o 5000_o 0_o 8_o 4375_o 0_o 7_o 3750_o 0_o 6_o 3125_o 0_o 5_o 2500_o 0_o 4_o 1875_o 0_o 3_o 1250_o 0_o 2_o 624_o 0_o 1_o you_o shall_v find_v against_o it_o 3_o rood_n 24_o perch_n then_o look_v for_o 3750_o and_o against_o it_o you_o shall_v find_v 6_o perch_n all_o which_o be_v add_v together_o as_o here_o you_o see_v the_o area_n or_o content_n of_o the_o whole_a piece_n will_v be_v 5_o acre_n 3_o rood_n and_o 30_o perch_n a._n r._n p._n 5_o 00_o 00_o  _fw-fr 3_o 24_o  _fw-fr  _fw-fr 6_o 5_o 03_o 30_o another_o example_n suppose_v the_o base_a of_o a_o triangle_n
to_o reduce_v acre_n into_o perch_n and_o the_o contrary_a 248._o 19_o the_o use_n of_o a_o scale_n of_o reduction_n necessary_a for_o find_v the_o fraction_n part_n of_o a_o acre_n 250_o 20._o divers_a compendious_a rule_n for_o the_o ready_a cast_v up_o of_o any_o plain_a superficies_n with_o divers_a other_o compendium_n in_o survey_v by_o the_o line_n of_o number_n 251._o 21._o of_o statute_n and_o customary_a measure_n to_o reduce_v one_o to_o the_o other_o at_o pleasure_n 254._o 22._o of_o the_o lay_n out_o of_o common_a field_n into_o furlong_n 255._o 23._o of_o hill_n and_o mountain_n how_o to_o find_v the_o length_n of_o the_o horizontall_a line_n on_o which_o they_o stand_v several_a way_n 257_o 24._o of_o mountainous_a and_o uneven_a ground_n how_o to_o protract_v or_o lay_v the_o same_o down_o in_o plano_fw-la after_o the_o best_a manner_n give_v the_o area_n or_o content_n thereof_o 258._o 25._o how_o to_o take_v the_o plot_n of_o a_o whole_a manner_n by_o the_o plain_a table_n three_o several_a way_n 260._o circumferentor_n 266._o or_o peractor_n 266._o with_o the_o keep_v a_o account_n in_o your_o field-book_n after_o the_o best_a and_o most_o certain_a manner_n 270._o and_o to_o protract_v any_o observation_n so_o take_v 271._o 26._o of_o enlarge_n or_o diminish_n of_o plot_n according_a to_o any_o possible_a proportion_n by_o two_o semicircle_n mr._n rathborn_v ruler_n a_o line_n into_o 100_o part_n the_o parallelogram_n 273._o 27_o of_o convey_v of_o water_n 276._o forasmuch_o as_o the_o whole_a art_n of_o survey_v of_o land_n be_v perform_v by_o instrument_n of_o several_a kind_n and_o that_o the_o exact_a and_o careful_a make_n and_o divide_v of_o all_o such_o instrument_n be_v chief_o to_o be_v aim_v at_o i_o think_v good_a to_o intimate_v to_o such_o as_o be_v desirous_a to_o practice_v this_o art_n and_o do_v not_o ready_o know_v where_o to_o be_v furnish_v with_o necessary_a instrument_n for_o the_o performance_n thereof_o that_o all_o or_o any_o of_o the_o instrument_n use_v or_o mention_v in_o this_o book_n or_o any_o mathematical_a instrument_n whatsoever_o be_v exact_o make_v by_o mr._n anthony_n thompson_n in_o hosier_n lane_n near_o smithfield_n london_n the_o complete_a surveyor_z the_o first_o book_n the_o argument_n this_o first_o book_n consist_v of_o divers_a definition_n &_o problem_n geometrical_a extract_v out_o of_o the_o write_n of_o divers_a ancient_a and_o modern_a geometrician_n as_o euclid_n ramus_n clavius_n etc._n etc._n and_o be_v here_o so_o methodical_o dispose_v that_o any_o man_n may_v gradual_o proceed_v from_o problem_n to_o problem_n without_o interruption_n or_o be_v refer_v to_o any_o other_o author_n for_o the_o practical_a performance_n of_o any_o of_o they_o only_o the_o demonstration_n be_v whole_o omit_v partly_o because_o those_o book_n out_o of_o which_o they_o be_v extract_v be_v very_o large_a in_o that_o particular_a and_o also_o for_o the_o avoid_n of_o many_o other_o proposition_n and_o theorem_n which_o have_v the_o ensue_a problem_n be_v demonstrate_v must_v of_o necessity_n have_v be_v insert_v also_o the_o figure_n will_v have_v be_v so_o encumber_v with_o multiplicity_n of_o line_n that_o the_o intend_a problem_n will_v have_v be_v thereby_o much_o darken_v and_o beside_o it_o be_v not_o my_o intent_n in_o this_o place_n to_o make_v a_o absolute_a or_o entire_a treatise_n of_o geometry_n and_o therefore_o i_o have_v only_o make_v choice_n of_o such_o problem_n as_o i_o conceive_v most_o useful_a for_o my_o present_a purpose_n and_o come_v most_o in_o use_n in_o the_o practice_n of_o survey_v and_o aught_o of_o necessity_n to_o be_v know_v by_o every_o man_n that_o intend_v to_o exercise_v himself_o in_o the_o practice_n thereof_o and_o those_o be_v chief_o such_o as_o concern_v the_o reduce_n of_o plot_n from_o one_o form_n to_o another_o and_o to_o enlarge_v or_o diminish_v they_o according_a to_o any_o assign_a proportion_n also_o divers_a of_o the_o problem_n in_o this_o book_n will_v abundant_o help_v the_o surveyor_n in_o the_o division_n and_o separation_n of_o land_n and_o in_o the_o lay_n out_o of_o any_o assign_a quantity_n whereby_o large_a parcel_n may_v be_v ready_o divide_v into_o divers_a several_n and_o those_o again_o subdivide_v if_o need_v be_v also_o for_o the_o better_a satisfaction_n of_o the_o reader_n i_o have_v perform_v divers_a of_o the_o follow_a problem_n both_o arithmetical_o and_o geometrical_o geometrical_a definition_n 1._o a_o point_n be_v that_o which_o can_v be_v divide_v a_o point_n or_o sign_n be_v that_o which_o be_v void_a of_o all_o magnitude_n and_o be_v the_o least_o thing_n that_o by_o mind_n and_o understanding_n can_v be_v imagine_v and_o conceive_v than_o which_o there_o can_v be_v nothing_o less_o as_o the_o point_n or_o prick_v note_v with_o the_o letter_n a_o which_o be_v neither_o quantity_n nor_o part_n of_o quantity_n but_o only_o the_o term_n or_o end_n of_o quantity_n and_o herein_o a_o point_n in_o geometry_n differ_v from_o unity_n in_o number_n 2._o a_o line_n be_v a_o length_n without_o breadth_n or_o thickness_n a_o line_n be_v create_v or_o make_v by_o the_o move_a or_o draw_v out_o of_o a_o point_n from_o one_o place_n to_o another_o so_o the_o line_n ab_fw-la be_v make_v by_o move_v of_o a_o point_n from_o a_o to_o b_o and_o according_a as_o this_o motion_n be_v so_o be_v the_o line_n thereby_o create_v whether_o straight_o or_o crooked_a and_o of_o the_o three_o kind_n of_o magnitude_n in_o geometry_n viz._n length_n breadth_n and_o thickness_n a_o line_n be_v the_o first_o consist_v of_o length_n only_o and_o therefore_o the_o line_n ab_fw-la be_v capable_a of_o division_n in_o length_n only_o and_o may_v be_v divide_v equal_o in_o the_o point_n c_o or_o unequal_o in_o d_o and_o the_o like_a but_o will_v admit_v of_o no_o other_o dimension_n 3._o the_o end_n or_o bound_n of_o a_o line_n be_v point_n this_o be_v to_o be_v understand_v of_o a_o finite_a line_n only_o as_o be_v the_o line_n ab_fw-la the_o end_n or_o bound_n whereof_o be_v the_o point_v a_o and_o b_o but_o in_o a_o circular_a line_n it_o be_v otherwise_o for_o there_o the_o point_n in_o its_o motion_n return_v again_o to_o the_o place_n where_o it_o first_o begin_v and_o so_o make_v the_o line_n infinite_a and_o the_o end_n or_o bound_n thereof_o undeterminate_a 4._o a_o right_a line_n be_v that_o which_o lie_v equal_o between_o his_o point_n as_o the_o right_a line_n ab_fw-la lie_v straight_a and_o equal_a between_o the_o point_v a_o and_z b_o which_o be_v the_o bound_n thereof_o without_o bow_v and_o be_v the_o short_a of_o all_o other_o line_n that_o can_v be_v draw_v between_o those_o two_o point_n 5._o a_o superficies_n be_v that_o which_o have_v only_a length_n and_o breadth_n as_o the_o motion_n of_o a_o point_n produce_v a_o line_n the_o first_o kind_n of_o magnitude_n so_o the_o motion_n of_o a_o line_n produce_v a_o superficies_n which_o be_v the_o second_o kind_n of_o magnitude_n and_o be_v capable_a of_o two_o dimension_n namely_o length_n and_o breadth_n and_o so_o the_o superficies_n abcd_n may_v be_v divide_v in_o length_n from_o a_o to_o b_o and_o also_o in_o breadth_n from_o a_o to_o c._n 6._o the_o extreme_n of_o a_o superficies_n be_v line_n as_o the_o extreme_n or_o end_n of_o a_o line_n be_v point_n so_o the_o extreme_n or_o bound_n of_o a_o superficies_n be_v line_n and_o so_o the_o extreme_n or_o end_n of_o the_o superficies_n abcd_n be_v the_o line_n ab_fw-la bd_o dc_o and_o ca_n which_o be_v the_o term_n or_o limit_n thereof_o 7._o a_o plain_a superficies_n be_v that_o which_o lie_v equal_o between_o his_o line_n so_o the_o superficies_n abcd_n lie_v direct_a and_o equal_o between_o his_o line_n and_o whatsoever_o be_v say_v of_o a_o right_a line_n the_o same_o be_v also_o to_o be_v understand_v of_o a_o plain_a superficies_n 8._o a_o plain_a angle_n be_v the_o inclination_n or_o bow_v of_o two_o line_n the_o one_o to_o the_o other_o the_o one_o touch_v the_o other_o &_o not_o be_v direct_o join_v together_o as_o the_o two_o line_n ab_fw-la and_z bc_n incline_v the_o one_o to_o the_o other_o and_o touch_v one_o another_o in_o the_o point_n b_o in_o which_o point_n by_o reason_n of_o the_o inclination_n of_o the_o say_a line_n be_v make_v the_o angle_n abc_n but_o if_o the_o two_o line_n which_o touch_v each_o other_o be_v without_o inclination_n and_o be_v draw_v direct_o one_o to_o the_o other_o than_o they_o make_v no_o angle_n at_o all_o as_o the_o line_n cd_o and_o de_fw-fr touch_v each_o other_o in_o the_o point_n d_o and_o yet_o they_o make_v no_o angle_n but_o one_o continue_v right_a line_n ¶_o and_o here_o note_n that_o a_o angle_n common_o be_v sign_v by_o three_o letter_n the_o middlemost_a whereof_o show_v
will_v be_v 696_o and_o that_o be_v the_o area_n or_o content_n of_o the_o triangle_n or_o if_o you_o multiply_v 24_o the_o whole_a length_n of_o the_o perpendicular_a by_o 29_o the_o length_n of_o half_a the_o base_a the_o product_n will_v be_v 696_o as_o before_o or_o again_o if_o you_o multiply_v 58_o the_o whole_a length_n of_o the_o base_a by_o 24_o the_o whole_a length_n of_o the_o perpendicular_a the_o product_n will_v be_v 1392_o the_o half_a whereof_o be_v 696_o the_o area_n or_o content_n of_o the_o triangle_n as_o before_o of_o the_o trapezia_n a_o trapezia_n be_v a_o figure_n consist_v of_o four_o unequal_a side_n and_o as_o many_o unequal_a angle_n as_o be_v the_o figure_n abcd._n to_o measure_v this_o trapezia_n you_o must_v first_o draw_v the_o diagonall_a line_n bd_o for_o by_o this_o mean_v the_o figure_n be_v reduce_v into_o two_o triangle_n as_o adb_v and_o cdb_n then_o if_o you_o let_v fall_v the_o perpendicular_o from_o the_o point_v a_o and_o c_o you_o may_v measure_v they_o by_o the_o last_o example_n as_o two_o triangle_n the_o sum_n whereof_o be_v add_v together_o will_v be_v the_o area_n or_o content_n of_o the_o whole_a trapezia_n example_n have_v draw_v the_o line_n bd_o and_o so_o reduce_v the_o trapezia_n into_o two_o triangle_n and_o let_v fall_v the_o perpendicular_o ae_n and_o cf_n upon_o the_o line_n bd_o which_o be_v the_o common_a base_a to_o both_o the_o triangle_n you_o may_v find_v the_o area_n of_o the_o whole_a trapezia_n thus_o suppose_v the_o perpendicular_a cf_n be_v 102_o perch_n the_o perpendicular_a ae_n 118_o perch_n and_o the_o base_a bd_o which_o be_v common_a to_o both_o triangle_n 300_o perch_n now_o if_o according_a to_o former_a direction_n you_o multiply_v 300_o the_o base_a by_o 59_o half_a the_o perpendicular_a ae_n the_o product_n will_v be_v 17700_o for_o the_o content_a of_o the_o triangle_n abdella_n in_o like_a manner_n if_o you_o multiply_v 300_o the_o base_a by_o 51_o half_o the_o perpendicular_a fc_n the_o product_n will_v be_v 15300_o for_o the_o content_a of_o the_o triangle_n bcd_n now_o if_o you_o add_v the_o content_n of_o these_o two_o triangle_n together_o namely_o 17700_o and_o 15300_o the_o sum_n of_o they_o will_v be_v 33000_o and_o that_o be_v the_o content_a of_o the_o whole_a trapezia_n abcd._n but_o this_o work_n may_v be_v perform_v with_o more_o brevity_n thus_o in_o respect_v the_o base_a bd_o be_v common_a to_o both_o the_o triangle_n you_o may_v therefore_o add_v the_o two_o perpendicular_o together_o the_o half_a of_o which_o be_v multiply_v by_o the_o whole_a base_a the_o product_n will_v show_v the_o content_a of_o the_o whole_a trapezia_n example_n the_o two_o perpendicular_o 118_o and_o 102_o be_v add_v together_o the_o sum_n of_o they_o be_v 220_o the_o half_a whereof_o be_v 110_o this_o number_n be_v multiply_v by_o 300_o the_o whole_a length_n of_o the_o common_a base_a give_v 33000_o the_o content_a of_o the_o whole_a trapezia_n or_o you_o may_v multiply_v the_o sum_n of_o the_o perpendicular_o by_o the_o length_n of_o the_o base_a and_o half_o that_o product_n will_v be_v the_o content_a of_o the_o trapezia_n also_o of_o irregular_a figure_n how_o to_o reduce_v they_o into_o triangle_n or_o trapezias_n and_o to_o cast_v up_o the_o content_a thereof_o let_v abcdefgh_n be_v the_o figure_n of_o a_o field_n draw_v upon_o your_o plain_a table_n or_o otherwise_o protract_v upon_o paper_n according_a to_o any_o of_o the_o former_a direction_n in_o regard_n that_o the_o field_n be_v irregular_a that_o be_v to_o say_v it_o be_v neither_o square_n triangle_n or_o trapezia_n it_o must_v therefore_o before_o it_o can_v be_v measure_v be_v reduce_v into_o some_o of_o these_o form_n which_o to_o effect_n do_v thus_o draw_v line_n from_o one_o angle_n to_o other_o as_o the_o line_n ad_fw-la db_fw-la ae_n of_o and_o fh_n then_o will_v the_o whole_a figure_n be_v reduce_v into_o six_o triangle_n as_o 1._o the_o triangle_n bcd_v 2._o the_o triangle_n adb_n 3._o the_o triangle_n ade_n 4._o the_o triangle_n aef_n 5._o the_o triangle_n afh_v 6._o the_o triangle_n fgh_n these_o six_o triangle_n be_v measure_v several_o according_a to_o the_o former_a direction_n and_o the_o content_n of_o they_o all_o add_v together_o into_o one_o sum_n will_v show_v the_o area_n or_o content_n of_o the_o whole_a field_n as_o suppose_v the_o triangle_n bcd_n shall_v contain_v 72_o perch_n suppose_v the_o triangle_n adb_n shall_v contain_v 84_o perch_n suppose_v the_o triangle_n ade_n shall_v contain_v 110_o perch_n suppose_v the_o triangle_n aef_n shall_v contain_v 121_o perch_n suppose_v the_o triangle_n afh_n shall_v contain_v 165_o perch_n suppose_v the_o triangle_n fgh_n shall_v contain_v 66_o perch_n these_o six_o number_n be_v add_v together_o make_v 618_o perch_n and_o that_o be_v the_o area_n or_o content_n of_o the_o whole_a field_n in_o perch_n but_o for_o a_o abreviation_n of_o this_o work_n you_o need_v not_o to_o find_v the_o area_n of_o every_o triangle_n but_o of_o every_o trapezia_n as_o be_v before_o teach_v for_o the_o figure_n be_v as_o well_o divide_v into_o trapezias_n as_o triangle_n namely_o into_o the_o trapezias_n abcd_v adef_n afgh_n by_o this_o mean_v you_o need_v but_o to_o find_v the_o area_n or_o content_n of_o these_o three_o trapezias_n which_o will_v abreviate_v nigh_o half_a of_o the_o arithmetical_a work_n for_o if_o you_o measure_v the_o three_o trapezias_n several_o as_o have_v be_v teach_v in_o this_o chapter_n you_o shall_v find_v the_o trapezia_n abcd_v to_o contain_v 156_o perch_n the_o trapezia_n adef_n to_o contain_v 231_o perch_n the_o trapezia_n afgh_v to_o contain_v 231_o perch_n these_o three_o number_n be_v add_v together_o produce_v 618_o exact_o agree_v with_o the_o former_a ¶_o here_o note_n that_o at_o any_o time_n when_o you_o reduce_v any_o irregular_a plot_n into_o triangle_n your_o number_n of_o triangle_n will_v be_v less_o by_o two_o then_o the_o number_n of_o the_o side_n of_o your_o plot_n as_o in_o this_o figure_n the_o plot_n consist_v of_o 8_o side_n and_o you_o see_v it_o be_v reduce_v into_o 6_o triangle_n of_o the_o circle_n the_o proportion_n of_o the_o circumference_n of_o any_o circle_n be_v to_o its_o diameter_n as_o 7_o to_o 22._o now_o to_o find_v the_o area_n or_o content_n of_o any_o circle_n you_o must_v multiply_v the_o diameter_n thereof_o in_o itself_o and_o multiply_v that_o sum_n by_o 11_o which_o product_v be_v divide_v by_o 14_o shall_v give_v you_o the_o area_n of_o the_o circle_n example_n in_o this_o circle_n abcd_v let_v the_o diameter_n thereof_o db_n be_v 28_o which_o multiply_v in_o itself_o give_v 784_o this_o number_n multiply_v by_o 11_o give_v 8624_o which_o be_v divide_v by_o 14_o the_o quotient_n will_v be_v 616_o and_o that_o be_v the_o area_n of_o the_o circle_n the_o circumference_n of_o a_o circle_n be_v give_v to_o find_v the_o diameter_n multiply_v the_o circumference_n by_o 7_o and_o divide_v the_o 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any_o other_o because_o it_o give_v the_o content_a of_o any_o superficies_n measure_v therewith_o in_o its_o small_a denomination_n namely_o in_o perch_n and_o part_n of_o perch_n so_o that_o when_o any_o superficies_n be_v cast_v up_o and_o bring_v to_o perch_n it_o may_v easy_o be_v reduce_v into_o rood_n and_o acre_n now_o for_o their_o sake_n that_o affect_v this_o chain_n i_o will_v show_v the_o use_n thereof_o and_o afterward_o of_o master_n gunter'_v chain_n leave_v every_o man_n to_o take_v his_o choice_n and_o use_v that_o which_o like_v he_o best_o suppose_v that_o the_o figure_n b_o be_v a_o piece_n of_o land_n lie_v in_o a_o long_a square_n which_o be_v measure_v by_o master_n rathborn_v chain_n shall_v contain_v in_o length_n 16_o unite_v 2_o prime_n and_o in_o breadth_n 1_o unite_v 3_o prime_n 2_o second_o
will_v reach_v from_o 12_o chain_n 50_o link_n to_o 11_o acre_n 37_o part_n 4._o have_v the_o base_a and_o perpendicular_a of_o a_o triangle_n give_v in_o perch_n to_o find_v the_o content_a in_o acre_n as_o 320_o to_o the_o perpendicular_a so_o the_o length_n of_o the_o base_a to_o the_o content_a in_o acre_n so_o in_o the_o triangle_n labernele_n if_o the_o line_n bd_o be_v take_v for_o the_o perpendicular_a of_o the_o triangle_n than_o the_o length_n of_o the_o base_a be_v 50_o perch_n and_o the_o perpendicular_a 36_o 2_o 5_o the_o area_n will_v be_v find_v to_o be_v 5_o acre_n 22_o part_n which_o be_v 2_o rood_n 30_o perch_n then_o if_o you_o extend_v the_o compass_n from_o 320_o to_o 36_o 2_o 5_o the_o perpendicular_a the_o same_o extent_n will_v reach_v from_o 50_o the_o length_n of_o the_o base_a to_o 5_o acre_n 22_o part_n 5._o the_o base_a and_o perpendicular_a of_o a_o triangle_n be_v give_v in_o chain_n to_o find_v the_o content_a in_o acre_n as_o 20_o to_o the_o 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have_v at_o each_o end_n thereof_o a_o semicircle_n be_v inferior_a to_o none_o but_o the_o instrument_n be_v very_o chargeable_a and_o the_o use_n thereof_o very_o intricate_a and_o tedious_a i_o shall_v whole_o omit_v to_o speak_v any_o more_o of_o it_o there_o be_v another_o way_n also_o which_o master_n rathborn_a use_v which_o be_v with_o a_o ruler_n by_o he_o invent_v for_o that_o purpose_n which_o will_v indifferent_a well_o reduce_v a_o plot_n from_o one_o bigness_n to_o another_o according_a to_o some_o particular_a proportion_n the_o make_n of_o this_o ruler_n be_v so_o well_o know_v and_o the_o use_n thereof_o so_o apparent_a that_o i_o shall_v not_o need_v to_o say_v any_o thing_n concern_v the_o description_n or_o use_n of_o it_o i_o only_o intimate_v that_o there_o be_v such_o a_o ruler_n that_o those_o which_o please_v may_v have_v it_o make_v another_o way_n be_v by_o one_o line_n divide_v into_o 100_o or_o 1000_o equal_a part_n only_o which_o by_o the_o help_n of_o 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manor-house_n according_a to_o its_o symmetry_n or_o situation_n with_o all_o other_o house_n of_o note_n also_o all_o water-mil_n windmill_n and_o whatsoever_o else_o be_v necessary_a that_o may_v be_v put_v into_o your_o plot_n without_o confusion_n for_o far_a explanation_n of_o what_o have_v be_v deliver_v in_o this_o chapter_n i_o have_v here_o add_v the_o figure_n of_o a_o small_a manor_n which_o will_v be_v sufficient_a for_o example_n sake_n chap._n liii_o how_o to_o find_v whether_o water_n way_n be_v convey_v from_o a_o spring_n head_n to_o any_o appoint_a place_n there_o be_v a_o instrument_n call_v a_o water-levell_a for_o the_o performance_n hereof_o the_o make_v whereof_o be_v sufficient_o know_v now_o if_o it_o be_v require_v to_o know_v whether_o water_n may_v be_v convey_v in_o pipe_n or_o trench_n from_o a_o spring_n head_n to_o any_o determinate_a place_n observe_v the_o follow_a direction_n place_v your_o water-levell_a at_o some_o convenient_a distance_n from_o the_o spring_n head_n in_o a_o right_a line_n towards_o the_o place_n to_o which_o the_o water_n be_v to_o be_v convey_v as_o at_o 30_o 40_o 60_o or_o 100_o yard_n distant_a from_o the_o springhead_n then_o have_v in_o a_o readiness_n two_o long_v straight_a pole_n which_o you_o may_v call_v your_o station_n staff_n divide_v into_o foot_n inch_n and_o part_n of_o inch_n from_o the_o bottom_n upward_o be_v thus_o provide_v cause_n one_o who_o you_o may_v call_v your_o first_o assistant_n to_o set_v up_o one_o of_o the_o say_a staff_n at_o the_o spring_n head_n and_o require_v another_o which_o you_o may_v call_v your_o second_o assistant_n to_o erect_v the_o other_o staff_n beyond_o your_o instrument_n at_o 30_o 40_o 60_o or_o 100_o yard_n forward_o towards_o the_o place_n to_o which_o the_o water_n shall_v be_v convey_v these_o station_n staff_n be_v erect_v perpendicular_a and_o your_o water-levell_a in_o the_o mid_a way_n precise_o horizontall_a go_v to_o the_o end_n of_o the_o level_n and_o look_v through_o the_o sight_n cause_v your_o first_o assistant_n to_o move_v a_o leaf_n of_o paper_n up_o and_o down_o your_o station_n staff_n till_o through_o the_o sight_n you_o see_v the_o very_a edge_n thereof_o and_o then_o by_o some_o know_a sign_n or_o sound_n intimate_v to_o he_o that_o the_o paper_n be_v then_o in_o its_o true_a position_n then_o let_v this_o first_o assistant_n note_n against_o what_o number_n of_o foot_n inch_n and_o part_n of_o a_o inch_n the_o edge_n of_o the_o paper_n rest_v which_o he_o must_v note_v down_o in_o a_o paper_n then_o your_o water-levell_a remain_v immovable_a go_v to_o the_o other_o end_n thereof_o and_o look_v through_o the_o sight_n towards_o your_o other_o station_n staff_n cause_n your_o second_o assistant_n to_o move_v a_o leaf_n of_o paper_n along_o the_o staff_n till_o you_o see_v the_o very_a edge_n thereof_o