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so_o of_o the_o rest_n the_o line_n of_o chord_n be_v a_o line_n divide_v after_o the_o usual_a way_n of_o the_o line_n of_o chord_n from_o a_o circle_n who_o radius_fw-la be_v near_o the_o length_n of_o one_o of_o the_o leg_n this_o line_n be_v place_v upon_o each_o leg_n of_o the_o sector_n begin_v at_o the_o centre_n and_o run_v towards_o the_o end_n thereof_o it_o be_v number_v with_o 10_o 20_o etc._n etc._n to_o 60_o and_o to_o this_o line_n on_o each_o leg_n be_v set_v the_o letter_n c._n note_v this_o line_n on_o some_o sector_n run_v to_o 90_o degree_n the_o line_n of_o sines_n be_v a_o line_n of_o common_a natural_a sin_n such_o as_o we_o have_v before_o define_v only_o it_o be_v divide_v from_o a_o circle_n of_o the_o same_o radius_fw-la that_o the_o line_n of_o chord_n be_v these_o be_v also_o place_v upon_o each_o leg_n of_o the_o sector_n and_o number_v with_o the_o figure_n 10_o 20_o 30_o etc._n etc._n to_o 90_o at_o the_o end_n of_o which_o on_o each_o leg_n in_o the_o print_n annex_v be_v set_v the_o letter_n s_o the_o line_n of_o tangent_n be_v a_o line_n of_o common_a tangent_n divide_v from_o a_o circle_n of_o the_o foremention_v radius_fw-la and_o be_v place_v upon_o each_o leg_n and_o run_v to_o 45._o it_o have_v the_o number_n 10_o 20_o etc._n etc._n to_o 45_o place_v upon_o it_o with_o the_o letter_n t_o for_o tangent_fw-la beside_o this_o be_v another_o small_a line_n of_o tangent_n divide_v from_o a_o radius_fw-la of_o about_o 2_o inch_n and_o be_v place_v upon_o each_o leg_n of_o the_o sector_n it_o begin_v at_o 45_o which_o stand_v at_o the_o length_n of_o the_o radius_fw-la
set_v the_o fig._n 10_o 20_o 30_o etc._n etc._n as_o they_o be_v in_o the_o arch._n five_o if_o the_o extent_n d_o 10_o d_o 20_o d_o 30_o etc._n etc._n in_o the_o tangent-line_n be_v transfer_v to_o the_o line_n d_o b_o produce_v they_o will_v divide_v it_o into_o a_o line_n of_o secant_v which_o with_o the_o number_n 10_o 20_o 30_o etc._n etc._n may_v from_o the_o scheme_n be_v put_v upon_o any_o scale_n and_o be_v continue_v on_o the_o line_n of_o sines_n for_o after_o the_o sine_fw-la of_o 90_o the_o tangent_n begin_v hence_o you_o see_v that_o the_o secant_fw-la of_o nothing_n be_v the_o radius_fw-la or_o semi-diameter_n of_o the_o circle_n six_o if_o from_o a_o to_o each_o division_n of_o the_o arch_n b_o c_o you_o draw_v line_n they_o shall_v divide_v the_o radius_fw-la b_o d_o into_o a_o line_n of_o half_a tangent_n to_o which_o set_v the_o number_n 10_o 20_o 30_o etc._n etc._n to_o 90._o sometime_o this_o line_n be_v make_v to_o run_v as_o far_o as_o 150_o or_o 160_o deg._n which_o be_v of_o the_o same_o length_n with_o the_o whole_a tangent_fw-la of_o 75_o or_o 80_o deg._n and_o when_o it_o be_v so_o divide_v the_o line_n b_o d_o must_v be_v produce_v as_o it_o be_v for_o the_o secant_fw-la for_o the_o half_a tangent_fw-la of_o 10_o deg._n be_v the_o whole_a tangent_fw-la of_o 5_o deg._n so_o also_o the_o half_a tangent_fw-la of_o 160_o degree_n be_v the_o whole_a tangent_fw-la of_o 80_o degree_n seven_o draw_v the_o line_n a_o f_o and_o divide_v the_o arch_n a_o f_o into_o 8_o equal_a part_n these_o transfer_v from_o the_o arch_n to_o the_o strait_a line_n a_o f_o as_o be_v the_o chord_n divide_v it_o into_o a_o line_n of_o rhumb_n or_o point_n its_o use_n be_v to_o lay_v of_o any_o point_n of_o the_o compass_n which_o it_o do_v much_o more_o ready_a than_o a_o line_n of_o chord_n though_o that_o will_v serve_v when_o you_o have_v not_o this_o last_o you_o may_v divide_v the_o semi-diameter_n f_o d_o into_o 100_o equal_a part_n and_o it_o will_v serve_v for_o a_o line_n of_o equal_a part_n to_o take_v league_n or_o mile_n from_o not_o but_o that_o it_o may_v be_v long_o or_o short_a for_o there_o be_v no_o necessity_n of_o its_o be_v just_a of_o such_o a_o length_n this_o line_n be_v of_o great_a use_n because_o from_o it_o all_o the_o other_o line_n may_v be_v divide_v by_o suppose_v the_o radius_fw-la of_o the_o circle_n to_o contain_v any_o number_n of_o equal_a part_n as_o 1000_o 10000_o etc._n etc._n for_o then_o all_o the_o chord_n sin_n tangent_n etc._n etc._n will_v consist_v of_o a_o certain_a number_n of_o those_o part_n chap._n iii_o of_o the_o plain_a scale_n assume_v a_o line_n who_o length_n let_v be_v what_o you_o please_v suppose_v ab_fw-la on_o this_o line_n make_v a_o parallelogram_n of_o any_o breadth_n at_o pleasure_n as_o abcd_v divide_v the_o opposite_a side_n ab_fw-la and_z de_fw-la each_o into_o 10_o equal_a part_n as_o also_o add_v bc_n set_v thereto_o the_o figure_n 1_o 2_o 3_o etc._n etc._n as_o you_o see_v in_o fig._n 4._o this_o do_v draw_v from_o the_o point_v 1_o 2_o 3_o etc._n etc._n in_o the_o line_n ad_fw-la the_o several_a line_n parallel_v to_o ab_fw-la then_o from_o the_o point_n b_o 1_o 2_o 3_o etc._n etc._n in_o the_o line_n ab_fw-la to_o the_o point_n 1_o 2_o 3_o etc._n etc._n in_o the_o line_n dc_o draw_v the_o diagonal_a line_n b._n 1_o 1._o 2_o 2._o 3_o 3._o 4_o etc._n etc._n by_o which_o the_o line_n ab_fw-la or_o those_o equal_a to_o it_o be_v actual_o divide_v into_o 100_o equal_a part_n the_o reason_n of_o which_o be_v very_o clear_a from_o the_o four_o of_o the_o 6_o of_o euclid_n for_o by_o that_o proposition_n it_o will_v hold_v as_o bc_n be_v 1_o c_o he_o and_o therefore_o if_o be_v be_v 1_o 10_o of_o bc_n which_o by_o construction_n it_o be_v then_o shall_v he_o be_v 1_o 10_o of_o 1_o c_o that_o be_v 1_o 10_o of_o 1_o 10_o or_o 1_o 100_o of_o dc_o or_o ab_fw-la if_o the_o side_n ab_fw-la dc_o as_o also_o their_o parallel_n 1_o 2_o 3_o etc._n etc._n be_v produce_v and_o the_o distance_n ab_fw-la dc_o repeat_v thereon_o 1_o 2_o 3_o or_o more_o line_n as_o the_o length_n of_o the_o rule_n on_o which_o you_o put_v they_o will_v permit_v you_o have_v finish_v the_o say_v diagonal_a scale_n the_o design_n of_o which_o be_v to_o supply_v the_o use_n of_o a_o scale_n of_o equal_a part_n but_o with_o great_a exactness_n than_o any_o of_o those_o kind_n of_o scale_n can_v pretend_v to_o for_o from_o it_o may_v be_v take_v any_o number_n under_o 1000_o exact_o chap._n iu._n of_o the_o construction_n of_o the_o line_n of_o artificial_a number_n sin_n and_o tangent_n beside_o the_o line_n before_o describe_v there_o be_v another_o sort_n which_o be_v general_o put_v on_o scale_n and_o rule_n call_v artificial_a number_n sin_n and_o tangent_n the_o rise_n and_o construction_n of_o which_o be_v from_o the_o logarithme_n for_o they_o be_v only_o the_o logarithm_n of_o the_o natural_a number_n sin_n and_o tangent_n transfer_v to_o a_o scale_n the_o line_n of_o number_n common_o know_v by_o name_n of_o gunter_n line_n be_v a_o line_n of_o geometrical_a proportion_n unequal_o divide_v into_o 9_o part_n beginning_z at_z 1_o towards_o the_o left_a hand_n and_o run_v with_o 2_o 3_o 4_o etc._n etc._n to_o 10_o which_o be_v about_o the_o middle_n of_o the_o line_n where_o another_o radius_fw-la begin_v and_o be_v continue_v to_o 100_o towards_o the_o right_a hand_n it_o have_v the_o figure_n 10_o 20_o 30_o etc._n etc._n set_v thereto_o and_o be_v divide_v just_a as_o be_v the_o first_o part_n of_o it_o the_o line_n be_v thus_o divide_v be_v call_v a_o line_n of_o number_n of_o two_o radiuse_n and_o the_o way_n of_o number_v on_o it_o thus_o for_o herein_o lie_v the_o great_a difficulty_n in_o the_o use_n of_o it_o it_o be_v as_o before_o be_v note_v divide_v into_o 18_o unequal_a part_n the_o first_o 9_o end_v about_o the_o middle_n of_o the_o scale_n and_o the_o other_o 9_o at_o the_o end_n next_o the_o right_a hand_n then_o each_o of_o these_o prime_n or_o first_o grand_a division_n be_v subdivide_v into_o 10_o other_o part_n according_a to_o the_o same_o reason_n call_v ten_o and_o again_o each_o of_o those_o ten_o be_v subdivide_v into_o 10_o other_o part_n if_o the_o length_n of_o the_o rule_n or_o scale_n will_v permit_v the_o figure_n 1_o 2_o 3_o etc._n etc._n by_o which_o the_o prime_n be_v distinguish_v be_v all_o arbitrary_a point_n and_o may_v each_o of_o they_o represent_v so_o many_o entire_a unite_v tenn_n hundred_o or_o thousand_o or_o they_o may_v also_o represent_v so_o many_o ten_o hundred_o thousand_o or_o ten_o thousand_o part_n of_o a_o unite_n if_o the_o first_o 1_o on_o the_o line_n be_v take_v for_o 1_o then_z 10_o in_o the_o middle_n of_o the_o line_n be_v 10_o as_o in_o course_n it_o fall_v but_o if_o the_o first_o 1_o be_v take_v for_o 10_o or_o 100_o than_o the_o figure_n 10_o in_o the_o middle_n of_o the_o rule_n must_v be_v take_v for_o 100_o or_o 1000_o now_o when_o the_o first_o 1_o or_o prime_n represent_v 10_o unite_v each_o ten_o in_o that_o prime_n will_v be_v 1_o and_o each_o centesme_fw-fr in_o those_o ten_o if_o there_o be_v any_o will_v be_v one_o ten_o part_n of_o a_o unite_n again_o let_v the_o first_o prime_n represent_v 100_o than_o the_o figure_n 2_o 3_o 4_o etc._n etc._n will_v denote_v 200_o 300_o 400_o etc._n etc._n and_o so_o 10_o at_o the_o end_n will_v be_v 10000_o and_o according_a to_o this_o supposition_n 1_o ten_o in_o each_o prime_n will_v be_v 10_o unite_v and_o in_o those_o ten_o each_n centesme_fw-fr will_v be_v 1._o for_o decimal_a fraction_n let_v 10_o at_o the_o end_n of_o the_o line_n next_o the_o right_a hand_n represent_v 1_o than_o each_o prime_n towards_o the_o left_a hand_n will_v be_v 1_o and_o in_o those_o prime_n each_o ten_o will_v be_v .01_o and_o in_o these_o ten_o each_o centesm_n will_v be_v .001_o part_n of_o a_o unite_n the_o division_n and_o way_n of_o number_v on_o this_o line_n be_v thus_o explain_v it_o will_v not_o i_o presume_v be_v difficult_a to_o find_v the_o point_n upon_o the_o line_n where_o any_o number_n give_v be_v represent_v for_o example_n suppose_v the_o number_n 276_o be_v propose_v for_o the_o first_o figure_n 2_o i_o account_v 2_o next_o the_o beginning_n of_o the_o line_n that_o be_v 2_o next_o the_o left_a hand_n for_o the_o second_o figure_n 7_o i_o tell_v 7_o ten_o next_o follow_v that_o be_v 7_o of_o the_o 10_o great_a division_n betwixt_o 2_o and_o 3_o then_o from_o this_o point_n i_o count_v 6_o centesm_n so_o that_o from_o 2_o to_o
take_v the_o great_a of_o the_o two_o give_v line_n and_o make_v it_o a_o parallel_n betwixt_o 10_o and_o 10_o at_o the_o end_n of_o the_o sector_n then_o take_v the_o lesser_a line_n and_o carry_v it_o parallel_v to_o the_o great_a till_o it_o rest_v in_o like_a point_n on_o each_o leg_n so_o the_o number_n of_o point_n where_o it_o rest_v be_v the_o proportion_n to_o 100_o thus_o in_o fig._n 9_o you_o will_v find_v the_o line_n aa_o in_o proportion_n to_o bb_v as_o 7_o to_o 10._o prob._n vi_o the_o length_n of_o one_o line_n be_v givan_a to_o find_v what_o length_n any_o other_o line_n be_v in_o such_o part_n suppose_v the_o line_n aa_o in_o fig._n 9_o be_v 44_o and_o i_o will_v know_v what_o the_o line_n bb_n be_v make_v the_o line_n aa_o a_o parallel_n in_o 44_o then_o take_v the_o line_n bb_n and_o carry_v it_o along_o parallel_v to_o aa_n till_o it_o rest_v in_o like_a point_n so_o the_o number_n it_o rest_v in_o which_o be_v about_o 30_o be_v the_o length_n of_o the_o line_n bb_n in_o such_o part_n as_o aa_n do_v contain_v 44._o this_o problem_n may_v be_v very_o useful_a sometime_o as_o in_o case_n the_o draught_n or_o ground-plot_n of_o any_o place_n be_v take_v and_o no_o scale_n to_o it_o then_o if_o the_o length_n of_o any_o one_o line_n or_o side_n in_o the_o draught_n be_v know_v the_o length_n of_o all_o the_o rest_n be_v easy_o find_v by_o this_o artifice_n prob._n vii_o a_o right_a line_n any_o how_o divide_v be_v give_v to_o find_v how_o to_o divide_v any_o other_o line_n of_o different_a length_n into_o the_o like_a part_n let_v ch_n in_o fig._n 10._o be_v the_o line_n give_v which_o be_v unequal_o divide_v in_o the_o point_n e._n f._n g._n and_o let_v hh_o be_v the_o other_o line_n require_v to_o be_v divide_v into_o such_o part_n as_o be_v the_o former_a lay_v the_o line_n ch_n from_o the_o centre_n on_o both_o leg_n of_o the_o sector_n and_o in_o the_o point_n where_o this_o line_n reach_v apply_v the_o short_a line_n parallel-wise_a that_o be_v open_v the_o sector_n till_o the_o short_a line_n may_v be_v apply_v betwixt_o the_o term_n of_o the_o long_a this_o do_v lay_v the_o division_n of_o the_o line_n ch_n viz._n ce_fw-fr ch_n cg_fw-mi on_o both_o leg_n of_o the_o sector_n from_o the_o centre_n than_o the_o parallel_n extent_n of_o these_o division_n viz._n ee_o ff_o gg_fw-mi be_v take_v and_o apply_v to_o the_o line_n hh_v will_n will_v divide_v it_o into_o the_o like_a part_n as_o be_v the_o line_n ch_n chap._n viii_o of_o the_o general_n use_v of_o the_o line_n of_o chord_n sin_n tangent_n and_o secant_v by_o dispose_v and_o place_v these_o line_n as_o afore_o direct_v on_o this_o instrument_n we_o have_v scale_n to_o several_a raduis_n as_o i_o have_v already_o hint_v that_o be_v have_v a_o length_n or_o radius_fw-la give_v not_o exceed_v the_o length_n of_o the_o sector_n when_o open_v we_o can_v by_o the_o sector_n find_v the_o chord_n sine_fw-la etc._n etc._n thereto_o for_o which_o property_n this_o instrument_n be_v many_o time_n of_o great_a use_n for_o example_n suppose_v i_o will_v have_v the_o chord_n sine_fw-la or_o tangent_fw-la of_o 10_o deg_fw-la to_o a_o radius_fw-la of_o 3_o inch_n i_o take_v this_o 3_o inch_n and_o make_v it_o a_o parallel_n betwixt_o 60_o and_o 60_o on_o the_o line_n of_o chord_n then_o as_o i_o tell_v you_o before_o the_o same_o extent_n will_v reach_v from_o 45_o to_o 45_o on_o the_o line_n of_o tangent_n also_o on_o the_o other_o side_n of_o the_o sector_n the_o same_o distance_n of_o 3_o inch_n will_v reach_v from_o 90_o and_o 90_o in_o the_o line_n of_o sines_n so_o that_o if_o the_o line_n of_o chord_n be_v set_v to_o any_o radius_fw-la the_o line_n of_o tangent_n and_o sin_n be_v also_o set_v to_o the_o same_o now_o the_o sector_n be_v thus_o set_v if_o you_o take_v on_o the_o say_a line_n the_o parallel_n distance_n betwixt_o 40_o and_o 40_o it_o will_v give_v you_o the_o chord_n of_o 40_o degree_n also_o if_o you_o take_v the_o parallel_n distance_n on_o the_o line_n of_o sin_n between_o 40_o and_o 40_o it_o will_v give_v you_o the_o sine_fw-la of_o 40_o degree_n last_o if_o you_o take_v the_o parallel_n extent_n on_o the_o line_n of_o tangent_n betwixt_o 40_o and_o 40_o it_o will_v give_v you_o the_o tangent_fw-la of_o 40_o degree_n and_o all_o to_o the_o same_o radius_fw-la if_o the_o chord_n or_o tangent_fw-la of_o 70_o degree_n have_v be_v require_v then_o for_o the_o chord_n you_o must_v take_v the_o parallel_n distance_n of_o half_a the_o arch_n propose_v that_o be_v the_o chord_n of_o 35_o deg_fw-la and_o repeat_v that_o distance_n twice_o on_o the_o arch_n you_o lay_v it_o down_o on_o and_o you_o have_v the_o chord_n of_o 70_o deg_fw-la and_o for_o find_v the_o tangent_fw-la of_o 70_o deg_fw-la to_o this_o radius_fw-la you_o must_v make_v use_n of_o the_o small_a line_n of_o tangent_n for_o the_o great_a one_o run_v but_o to_o 45_o deg_fw-la we_o can_v take_v the_o parallel_n of_o 70_o deg_fw-la on_o that_o i_o therefore_o take_v the_o aforesaid_a radius_fw-la of_o 3_o inch_n and_o make_v it_o a_o parallel_n betwixt_o 45_o and_o 45_o on_o this_o small_a line_n of_o tangent_n than_o the_o parallel_n extent_n of_o 70_o deg_fw-la on_o the_o say_a line_n be_v the_o tangent_fw-la of_o 70_o deg_fw-la to_o 3_o inch_n radius_fw-la if_o you_o will_v have_v the_o secant_fw-la of_o any_o arch_n then_o take_v the_o give_v radius_fw-la and_o make_v it_o a_o parallel_n betwixt_o the_o beginning_n of_o the_o line_n of_o secant_v that_o be_v 0_o and_o 0_o so_o the_o parallel_n distance_n betwixt_o 40_o and_o 40_o or_o 70_o and_o 70_o on_o the_o say_v secant_fw-la line_n will_v give_v you_o the_o secant_fw-la of_o 40_o or_o 70_o degree_n to_o the_o radius_fw-la of_o three_o inch_n after_o this_o manner_n may_v the_o chord_n sine_fw-la or_o tangent_fw-la of_o any_o arch_n be_v find_v provide_v the_o radius_fw-la can_v be_v make_v a_o parallel_n betwixt_o 60_o and_o 60_o on_o the_o line_n of_o chord_n or_o betwixt_o the_o small_a tangent_fw-la of_o 45_o or_o secant_fw-la of_o 0_o deg_fw-la but_o if_o the_o radius_fw-la be_v so_o large_a that_o it_o can_v be_v make_v a_o parallel_n betwixt_o 45_o and_o 45_o on_o the_o small_a tangent_n than_o no_o tangent_fw-la of_o any_o arch_n above_o 45_o degree_n can_v be_v find_v nor_o the_o secant_fw-la of_o no_o arch_n at_o all_o to_o such_o a_o radius_fw-la because_o all_o secant_v exceed_v the_o radius_fw-la or_o semi-diameter_n of_o the_o circle_n if_o the_o converse_n of_o any_o of_o these_o thing_n be_v require_v that_o be_v if_o i_o require_v the_o radius_fw-la to_o which_o a_o give_a line_n be_v the_o chord_n sine_fw-la tangent_fw-la or_o secant_fw-la of_o any_o arch_n suppose_v of_o 40_o deg_fw-la then_o it_o be_v but_o make_v that_o line_n if_o it_o be_v a_o chord_n a_o parallel_n on_o the_o line_n of_o chord_n betwixt_o 40_o and_o 40_o so_o will_v the_o sector_n stand_v at_o the_o radius_fw-la that_o be_v the_o parallel_n extent_n betwixt_o 60_o and_o 60_o on_o the_o say_a chord-line_n be_v the_o radius_fw-la and_o so_o if_o it_o be_v a_o sine_fw-la tangent_fw-la or_o secant_fw-la it_o be_v but_o make_v it_o a_o parallel_n betwixt_o the_o sine_fw-la tangent_fw-la or_o secant_fw-la of_o 40_o deg_fw-la according_a as_o it_o be_v give_v then_o will_v the_o distance_n of_o 90_o and_o 90_o on_o the_o sin_n if_o it_o be_v a_o tangent_fw-la the_o extent_n of_o 45_o and_o 45_o on_o the_o tangent_n and_o if_o it_o be_v a_o secant_fw-la the_o extent_n or_o distance_n between_o 0_o and_o 0_o on_o the_o say_a line_n of_o secant_v will_v be_v the_o radius_fw-la hence_o you_o see_v it_o be_v very_o easy_a to_o find_v the_o chord_n sine_fw-la or_o tangent_fw-la to_o any_o radius_fw-la which_o be_v the_o great_a benefit_n and_o advantage_n we_o receive_v from_o this_o ingenious_a contrivance_n and_o of_o great_a use_n it_o be_v not_o only_o in_o delineat_v and_o project_v on_o paper_n but_o even_o on_o the_o ground_n in_o lay_v out_o any_o parcel_n in_o any_o form_n or_o shape_n for_o by_o a_o sector_n of_o two_o or_o three_o foot_n long_o when_o open_v we_o can_v furnish_v ourselves_o with_o scale_n big_a or_o little_a enough_o for_o any_o purpose_n or_o design_n whatsoever_o chap._n ix_o of_o some_o particular_a uses_n of_o the_o circular_a line_n viz._n chord_n sin_n etc._n etc._n prob._n i._n a_o angle_n be_v give_v to_o find_v the_o number_n of_o degree_n it_o contain_v suppose_v i_o will_v find_v the_o quantity_n of_o the_o angle_n a_o in_o fig._n 11._o take_v any_o distance_n but_o so_o short_a that_o it_o may_v cut_v both_o side_n of_o the_o angle_n with_o this_o