the_o line_n of_o proportion_n the_o construction_n and_o use_v of_o the_o line_n of_o proportion_n by_o help_n whereof_o the_o hard_a question_n of_o arithmetic_n &_o geometry_n as_o well_o in_o break_a as_o whole_a number_n be_v resolve_v by_o addition_n and_o substraction_n by_o edm_n wingate_n gent._n nulla_fw-la dies_fw-la sine_fw-la n1_fw-la london_n print_v by_o john_n dawson_n 1628._o ¶_o the_o preface_n have_v not_o many_o month_n ago_o publish_v a_o discourse_n declare_v the_o nature_n and_o use_n of_o the_o logarithmeticall_a table_n and_o observe_v the_o table_n of_o number_n there_o to_o be_v too_o small_a for_o ordinary_a use_n not_o give_v indeed_o without_o much_o difficulty_n the_o logarithme_n of_o any_o number_n that_o exceed_v 1000_o i_o have_v invent_v this_o tabular_a scale_n or_o line_n of_o proportion_n by_o mean_n whereof_o as_o i_o take_v it_o you_o shall_v find_v that_o defect_n full_o supply_v this_o instrument_n yield_v you_o the_o resolution_n of_o the_o 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point_n of_o the_o common_a line_n α_n ω_n where_o the_o last_o figure_n happen_v to_o fall_v be_v the_o point_n that_o represent_v the_o number_n give_v example_n 2_o 3_o 7_o 2_o be_v give_v i_o demand_v the_o point_n upon_o the_o common_a line_n that_o represent_v the_o same_o number_n 2_o the_o first_o figure_n direct_v i_o to_o the_o three_o fraction_n sign_v by_o the_o figure_n 2_o 3_o the_o second_o figure_n lead_v i_o to_o the_o hundred_o mark_v upon_o that_o fraction_n by_o the_o figure_n 3_o for_o 7._o the_o three_o figure_n i_o count_v seven_o ten_o of_o that_o hundred_o viz._n to_o the_o point_n p_o and_o for_o 2_o the_o last_o figure_n i_o count_v two_o unit_n of_o that_o ten_o which_o do_v i_o find_v the_o number_n give_v to_o be_v represent_v upon_o the_o three_o fraction_n at_o the_o point_n n._n so_o 2370._o be_v represent_v at_o the_o point_n p_o 2300._o upon_o the_o same_o fraction_n at_o the_o begin_n of_o the_o hundred_o sign_v by_o the_o figure_n 3_o and_o 2000_o 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x._o contrariwise_o by_o invert_v the_o rule_n of_o this_o proposition_n any_o point_n of_o the_o common_a line_n be_v give_v you_o may_v find_v the_o number_n represent_v by_o it_o so_o the_o point_n p_o n_o q_o r_o be_v give_v the_o number_n represent_v by_o they_o be_v 2370._o 2372._o 2080._o and_o 2008._o chap._n iii_o the_o description_n construction_n and_o use_v of_o the_o scale_n of_o number_n the_o scale_n of_o number_n be_v that_o describe_v abouc_fw-la the_o common_a line_n α_n ω_n viz._n in_o the_o three_o first_o space_n of_o the_o line_n of_o proportion_n which_o be_v first_o divide_v into_o nine_o proportional_a part_n distinguish_v by_o the_o great_a figure_n 0._o 1._o 2._o 3._o 4._o 5._o 6._o 7._o 8._o 9_o the_o first_o beginning_n at_o the_o begin_n of_o the_o line_n note_v by_o 1._o and_o end_v at_o the_o line_n that_o cross_v those_o three_o space_n upon_o the_o four_o fraction_n mark_v by_o the_o figure_n 2_o on_o the_o right_a hand_n and_o by_o a_o little_a 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which_o be_v hereafter_o term_v fift_n now_o the_o construction_n of_o this_o scale_n be_v in_o this_o manner_n repair_v unto_o mr._n brigges_n his_o table_n of_o logarithme_n and_o suppose_v 1000_o to_o be_v represent_v at_o the_o begin_n of_o the_o line_n of_o proportion_n find_v in_o those_o table_n the_o logarithme_n of_o 1001._o which_o be_v 3,00042_o 40774_o 7932_o whereof_o take_v only_o 3,0004_o the_o first_o five_o figure_n then_o cast_v away_o 3_o the_o characteristique_a his_o office_n be_v only_o to_o show_v of_o how_o many_o place_n the_o number_n unto_o which_o the_o logarithme_n belong_v consist_v as_o i_o have_v former_o show_v in_o the_o treatise_n abovementioned_n find_v by_o the_o proposition_n of_o the_o last_o chapter_n upon_o the_o scale_n of_o logarithme_n 0004_o the_o figure_n that_o remain_v which_o be_v represent_v upon_o the_o first_o fraction_n at_o the_o point_n x_o this_o do_v just_a against_o that_o point_n in_o the_o scale_n of_o number_n mark_v the_o point_n z_o which_o represent_v the_o number_n 1001_o upon_o that_o scale_n then_o take_v the_o logarithme_n of_o 1002._o do_v in_o like_a manner_n and_o so_o proceed_v till_o you_o have_v describe_v all_o the_o division_n of_o the_o scale_n of_o number_n upon_o the_o line_n the_o use_n of_o this_o scale_n appear_v in_o the_o resolution_n of_o the_o proposition_n follow_v proposition_n i._o a_o whole_a number_n be_v give_v to_o find_v it_o upon_o the_o scale_n of_o number_n find_v the_o first_o figure_n of_o the_o number_n give_v among_o the_o prime_n of_o that_o scale_n then_o find_v the_o second_o figure_n among_o the_o second_o of_o that_o prime_n 3._o for_o the_o three_o figure_n count_v so_o many_o three_o of_o that_o second_o 4._o for_o the_o four_o count_n so_o many_o fourth_n of_o that_o three_o and_o last_o if_o the_o number_n fall_v in_o the_o first_o second_o or_o three_o prime_n for_o the_o five_o figure_n count_v so_o many_o fift_n of_o the_o last_o four_o this_o do_v the_o point_n where_o the_o last_o figure_n fall_v upon_o the_o common_a line_n α_n ω_n be_v the_o point_n that_o represent_v the_o number_n give_v example_n 17268._o be_v give_v i_o demand_v the_o point_n upon_o the_o common_a line_n where_o it_o be_v represent_v 1._o the_o first_o figure_n direct_v i_o to_o the_o first_o prime_n and_o 7._o to_o the_o seven_o second_v of_o that_o prime_n place_v upon_o the_o three_o fraction_n at_o the_o little_a figure_n 71._o then_o for_o 2_o i_o count_v two_o three_o of_o that_o second_o viz._n to_o the_o point_n μ_n and_o for_o 6_o i_o count_v six_o fourth_n of_o that_o three_o that_o be_v to_o the_o point_n ν_fw-gr and_o last_o of_o all_o for_o 8_o the_o last_o figure_n i_o take_v eight_o fift_n of_o that_o four_o so_o that_o i_o find_v 17268._o the_o number_n give_v to_o be_v represent_v at_o the_o point_n ε_n upon_o the_o three_o fraction_n so_o 1726._o or_o 17260._o be_v find_v at_o the_o point_n ν_fw-gr 172_o 1720_o or_o 17200._o at_o the_o point_n μ_n 17_o 170_o 1700_o or_o 17000_o at_o the_o seven_o second_v of_o the_o first_o prime_n 1._o 10._o 100_o 1000_o etc._n etc._n at_o the_o begin_n of_o the_o line_n and_o 2._o 20._o 200._o 2000_o etc._n etc._n at_o the_o begin_n of_o the_o second_o prime_n in_o like_a manner_n 2040._o be_v represent_v at_o the_o point_n φ_n upon_o the_o four_o fraction_n the_o cypher_n in_o the_o second_o place_n signify_v that_o no_o second_o and_o the_o other_o in_o the_o four_o or_o last_o place_n show_v that_o no_o fourth_n be_v to_o be_v take_v in_o find_v out_o that_o number_n upon_o the_o scale_n so_o likewise_o 2008_o be_v represent_v upon_o the_o same_o fraction_n at_o the_o point_n ψ_n the_o cypher_n in_o the_o second_o and_o three_o place_n show_v that_o no_o second_n or_o three_o be_v to_o be_v take_v in_o the_o discovery_n of_o that_o number_n contrariwise_o by_o change_v the_o rule_n of_o this_o proposition_n any_o point_n of_o the_o common_a line_n be_v give_v you_o may_v find_v the_o number_n represent_v by_o it_o so_o the_o point_n ε_n upon_o the_o three_o fraction_n and_o ψ_n upon_o the_o four_o represent_v the_o number_n 17268._o &_o 2008._o from_o the_o premise_n arisethese_fw-mi corrollary_n 1._o a_o number_n that_o consist_v of_o more_o figure_n than_o five_o and_o fall_v in_o the_o first_o second_o or_o three_o prime_n be_v represent_v at_o the_o point_n where_o the_o five_o figure_n fall_v so_o 17268347._o be_v represent_v upon_o the_o three_o fraction_n at_o the_o point_n and_o 20080372._o upon_o the_o four_o at_o the_o pâ—â—nt_a ψ._n 2._o a_o number_n that_o consist_v of_o more_o figure_n than_o four_o and_o fall_v between_o the_o beginning_n of_o the_o four_o prime_n and_o the_o end_n of_o the_o line_n be_v represent_v at_o the_o point_n where_o the_o four_o figure_n fall_v so_o 4236_o and_o 4236873._o be_v both_o represent_v upon_o the_o seven_o fraction_n at_o the_o point_n θ._n 3._o a_o point_n of_o the_o common_a line_n in_o the_o first_o second_o or_o three_o prime_n always_o give_v you_o a_o number_n that_o consist_v of_o fine_a place_n so_o the_o point_n ε_n ν_fw-gr &_o μ_n be_v give_v the_o number_n represent_v by_o they_o be_v 17268_o 17260_o &_o 17200._o 4._o a_o point_n of_o the_o common_a line_n between_o the_o beginning_n of_o the_o four_o prime_n and_o the_o end_n of_o the_o line_n always_o yield_v you_o a_o number_n compose_v of_o four_o place_n so_o θ_o and_o χ_n upon_o the_o seven_o fraction_n represent_v 4236_o and_o 4230._o prop_n 2._o a_o break_a number_n be_v give_v to_o find_v it_o upon_o the_o scale_n of_o number_n prefix_v the_o whole_a part_n of_o the_o number_n give_v before_o the_o numerator_n of_o the_o fraction_n and_o thereby_o make_v
they_o as_o it_o be_v one_o entire_a number_n then_o by_o the_o proposition_n aforego_v find_v the_o point_n which_o represent_v that_o number_n which_o also_o will_v be_v the_o point_n that_o represent_v the_o break_a number_n propound_v example_n 172_o 68_o 100_o be_v give_v 172_o be_v prefix_v before_o 68_o the_o numerator_n of_o the_o fraction_n constitute_v the_o whole_a number_n 17268_o which_o by_o the_o proposition_n aforego_v be_v represent_v upon_o the_o three_o fraction_n at_o the_o point_n â—_o so_o 17.26_o that_o be_v 17_o 26_o 100_o and_o 1._o 726_o viz._n 1_o 726_o 1000_o be_v both_o represent_v upon_o the_o same_o fraction_n at_o the_o point_n ν_fw-gr in_o like_a manner_n 20.40_o and_o 20.08_o be_v find_v upon_o the_o four_o fraction_n at_o the_o point_n φ_n and_o ψ._n but_o here_o it_o be_v to_o be_v observe_v that_o the_o fraction_n of_o the_o break_a number_n propound_v to_o be_v find_v upon_o this_o scale_n must_v always_o have_v for_o their_o denominator_fw-la a_o number_n consist_v of_o a_o unit_fw-la in_o the_o first_o place_n towards_o the_o left_a hand_n and_o nothing_o but_o cypher_n towards_o the_o right_n such_o as_o be_v 10._o 100_o 1000_o 10000_o etc._n etc._n and_o if_o the_o fraction_n of_o the_o break_a number_n give_v be_v not_o such_o they_o ought_v to_o be_v reduce_v to_o fraction_n of_o that_o kind_n now_o other_o fraction_n be_v reduce_v to_o fraction_n of_o that_o kind_n for_o the_o most_o part_n upon_o view_n as_o if_o the_o number_n give_v be_v 12._o foot_n and_o 9_o inch_n that_o number_n be_v reduce_v be_v 12.75_o viz._n 12_o 75_o 100_o and_o 12._o pound_n 14_o shilling_n after_o reduction_n be_v 12.7_o that_o be_v 12_o 1_o 10._o but_o when_o you_o meet_v with_o a_o break_a number_n who_o fraction_n be_v not_o reduccable_a upon_o view_n it_o may_v be_v reduce_v by_o the_o rule_n of_o three_o for_o as_o the_o denominator_fw-la of_o the_o fraction_n give_v be_v to_o 10._o 100_o or_o 1000_o etc._n etc._n so_o be_v the_o numerator_n of_o the_o same_o fraction_n to_o the_o numerator_n of_o the_o fraction_n require_v so_o 17_o 98_o 305_o that_o be_v 17_o year_n and_o 98_o day_n be_v give_v the_o proportion_n will_v be_v as_o 365_o to_o 1000_o so_o 98_o to_o 268._o so_o that_o 1000_o be_v the_o denominator_fw-la and_o 268_o the_o numerator_n of_o the_o fraction_n require_v your_o number_n after_o reduction_n will_v stand_v thus_o 17_o 268_o 1000_o or_o thus_o 17.268_o now_o to_o find_v 268._o the_o four_o proportional_a by_o the_o help_n of_o the_o logarithme_n i_o refer_v you_o to_o the_o three_o problem_n of_o the_o five_o chapter_n of_o my_o book_n abovementioned_n but_o in_o this_o case_n let_v the_o denominator_fw-la of_o the_o fraction_n require_v always_o exceed_v the_o denominator_fw-la of_o the_o fraction_n give_v as_o in_o the_o example_n aforego_v 1000_o exceed_v 365._o chap._n four_o the_o joint_n use_v of_o the_o scale_n of_o number_n and_o the_o scale_n of_o logarithme_n together_o prop._n i._n a_o whole_a number_n be_v give_v to_o find_v the_o logarithme_n thereof_o find_v upon_o the_o scale_n of_o number_n by_o the_o first_o proposition_n of_o the_o last_o chapter_n the_o point_n that_o represent_v the_o number_n give_v then_o by_o the_o proposition_n of_o the_o second_o chapter_n observe_v upon_o the_o scale_n of_o logarithme_n the_o number_n represent_v by_o that_o point_n this_o do_v if_o you_o prefix_v before_o that_o number_n his_o proper_a characteristique_a that_o entire_a number_n be_v the_o logarithme_n require_v now_o the_o characteristique_a be_v the_o first_o figure_n of_o the_o logarithme_n consist_v of_o as_o many_o unity_n within_o one_o as_o the_o number_n unto_o which_o that_o logarithme_n belong_v consist_v of_o place_n so_o the_o characteristique_a of_o the_o number_n betwixt_o 1_o and_z 10_o be_v 0_o betwixt_o 10_o and_o 100_o be_v 1_o betwixt_o 100_o and_o 1000_o be_v 2_o betwixt_o 1000_o and_o 10000_o be_v 3_o etc._n etc._n example_n 17268_o be_v give_v i_o demand_v his_o logarithme_n by_o the_o first_o proposition_n of_o the_o last_o chapter_n i_o find_v 17268_o upon_o the_o three_o fraction_n at_o the_o point_n â—_o which_o give_v i_o upon_o the_o scale_n of_o logarithme_n by_o the_o proposition_n of_o the_o second_o chapter_n the_o number_n 2372_o before_o which_o because_o the_o number_n give_v consist_v of_o five_o place_n i_o prefix_v four_o so_o that_o the_o entire_a logarithme_n of_o 17268_o the_o number_n give_v be_v 42372_o so_o the_o logarithme_n of_o 2040_o be_v 33096_o and_o the_o logarithme_n of_o 2008_o be_v 33028_o etc._n etc._n prop._n 2._o a_o break_a number_n be_v give_v to_o find_v the_o logarithme_n thereof_o find_v upon_o the_o scale_n of_o number_n by_o the_o last_o proposition_n of_o the_o last_o chapter_n the_o point_n that_o represent_v the_o number_n give_v then_o by_o the_o proposition_n of_o the_o second_o chapter_n take_v upon_o the_o scale_n of_o logarithme_n the_o number_n represent_v by_o that_o point_n this_o do_v if_o you_o place_n before_o that_o number_n his_o proper_a characteristique_a that_o be_v a_o figure_n consist_v of_o so_o many_o unity_n save_o one_o as_o the_o whole_a part_n of_o the_o number_n give_v consist_v of_o place_n that_o entire_a number_n be_v that_o you_o look_v for_o example_n 172._o 68_o be_v give_v i_o demand_v his_o logarisme_n that_o number_n be_v find_v by_o the_o last_o proposition_n of_o the_o last_o chapter_n upon_o the_o three_o fraction_n at_o the_o point_n â—_o which_o yield_v you_o upon_o the_o scale_n of_o logarithme_n by_o the_o proposition_n of_o the_o second_o chapter_n the_o number_n 2372_o and_o now_o because_o 172_o be_v the_o whole_a part_n of_o the_o number_n give_v consist_v of_o three_o place_n prefix_v before_o 2372_o the_o charactcristique_a 2_o which_o do_v the_o entire_a logarithme_n of_o 172.68_o will_v be_v find_v 22372_o so_o the_o logarithme_n of_o 17.26_o be_v 12370_o and_o the_o logarithme_n of_o 1.726_o be_v 02370._o prop._n 3._o a_o logarithme_n be_v give_v to_o find_v the_o number_n unto_o which_o it_o belong_v neglect_v the_o characteristique_a of_o the_o logarithme_n give_v find_v by_o the_o proposition_n of_o the_o second_o chapter_n the_o point_n where_o the_o other_o figure_n thereof_o be_v represent_v upon_o the_o scale_n of_o logarithme_n then_o by_o the_o first_o proposition_n of_o the_o last_o chapter_n take_v off_o upon_o the_o scale_n of_o number_n the_o number_n represent_v by_o that_o point_n this_o do_v observe_v of_o how_o many_o unity_n the_o characteristique_a of_o the_o logarithme_n give_v consist_v take_v one_o more_o of_o the_o first_o figure_n that_o the_o number_n take_v upon_o the_o scale_n of_o number_n have_v towards_o the_o left_a hand_n as_o if_o the_o characteristique_a be_v 0_o take_v one_o of_o those_o figure_n if_o it_o be_v 1_o take_v two_o if_o 2_o take_z three_z etc._n etc._n which_o figure_n will_v be_v the_o whole_a part_n of_o the_o number_n require_v and_o if_o there_o remain_v any_o figure_n towards_o the_o right_a hand_n they_o be_v the_o numerator_n of_o a_o fraction_n who_o denominator_fw-la be_v a_o number_n consist_v of_o a_o unity_n in_o the_o first_o place_n towards_o the_o left_a hand_n and_o of_o so_o many_o cypher_n towards_o the_o right_n as_o there_o be_v figure_n remain_v which_o fraction_n be_v the_o break_a part_n of_o the_o number_n demand_v example_n the_o logarithme_n 42372_o be_v give_v i_o demand_v the_o number_n unto_o which_o it_o belong_v 2372_o the_o other_o figure_v beside_o 4_o the_o characteristique_a i_o find_v by_o the_o prop_n of_o the_o 2._o chap._n to_o be_v represent_v in_o the_o scale_n of_o logarithme_n upon_o the_o three_o fraction_n at_o the_o point_n n_o at_o which_o point_n upon_o the_o scale_n of_o number_n i_o find_v by_o the_o 1_o prop._n of_o the_o last_o ch_z to_o be_v represent_v the_o number_n 17268_o and_o now_o because_o the_o characteristique_a of_o the_o logarithme_n give_v be_v 4_o the_o whole_a number_n 17268_o be_v the_o number_n unto_o which_o the_o logarithme_n give_v appertain_v but_o if_o the_o logar_n give_v be_v 22372_o his_o number_n will_v be_v 172.68_o the_o charact._n 2_o show_v that_o 172_o the_o three_o first_o figure_n of_o the_o number_n find_v aught_o to_o be_v take_v for_o the_o whole_a part_n and_o 68_o for_o the_o fraction_n of_o the_o number_n unto_o which_o that_o logarithme_n belong_v from_o this_o proposition_n arise_v these_o corrallary_n 1._o when_o a_o logarithme_n who_o charact._n exceed_v 4_o falls_z within_z the_o first_o second_o or_o three_o prime_n the_o first_o five_o figure_n of_o the_o number_n unto_o which_o it_o belong_v can_v only_o be_v know_v so_o if_o the_o logarithine_n give_v be_v 72372_o the_o five_o first_o figure_n of_o the_o number_n unto_o which_o it_o belong_v be_v 17268._o 2._o when_o a_o logarithme_n who_o charact._n exceed_v 3_o happen_v to_o fall_v betwixt_o the_o beginning_n of_o the_o four_o prime_a and_o the_o end_n of_o the_o line_n the_o first_o four_o figure_n of_o the_o number_n unto_o which_o it_o belong_v be_v only_o discoverable_a upon_o the_o line_n so_o the_o logarithme_n 76270_o be_v give_v the_o four_o first_o figure_n of_o the_o number_n unto_o which_o it_o belong_v be_v 4236_o which_o you_o shall_v find_v represent_v upon_o the_o seven_o fraction_n at_o the_o point_n θ._n but_o now_o in_o take_v the_o number_n upon_o either_o of_o the_o scale_n observe_v this_o rule_n when_o you_o have_v direct_v your_o eye_n unto_o a_o point_n upon_o the_o common_a line_n in_o take_v a_o nâ—mber_n upon_o either_o of_o the_o scale_n first_o take_v the_o least_o part_n represent_v by_o that_o point_n and_o then_o the_o rest_n in_o the_o same_o order_n as_o in_o the_o example_n of_o the_o last_o proposition_n the_o logarithme_n 42372_o be_v propound_v your_o eye_n be_v direct_v by_o it_o upon_o the_o scale_n of_o logarithme_n unto_o the_o point_n n_o and_o therefore_o in_o remove_v your_o view_n for_o take_v upon_o the_o scale_n of_o number_n the_o number_n unto_o which_o that_o logarithme_n belong_v first_o take_v the_o filt_n viz._n 8_o then_z 6_o the_o fourth_n and_o so_o the_o rest_n in_o order_n which_o do_v carry_v in_o you_o mind_n eight_o six_o two_o seven_o one_o and_o beginning_n with_o 8_o first_o set_v they_o down_o thus_o 17268_o as_o before_o in_o like_a manner_n in_o the_o example_n of_o the_o 1._o prop._n of_o this_o ch_z the_o number_n 17268_o be_v give_v your_o eye_n be_v direct_v upon_o the_o scale_n of_o number_n unto_o the_o point_n â—_o and_o therefore_o in_o remove_v your_o view_n for_o take_v upon_o the_o scale_n of_o logarithme_n the_o logarithme_n of_o that_o number_n first_o observe_v the_o unit_n viz._n 2_o then_z 7_o the_o ten_o and_o so_o the_o rest_n in_o order_n this_o do_v keep_v in_o your_o mind_n the_o figure_n so_o take_v set_v they_o down_o as_o before_o thus_o 2372._o and_o in_o observe_v this_o rule_n after_o a_o little_a practice_n you_o shall_v find_v much_o ease_n and_o readiness_n thus_o have_v show_v you_o how_o to_o find_v upon_o the_o line_n of_o proportion_n the_o number_n of_o any_o logarithme_n and_o the_o logarithme_n of_o any_o number_n propound_v under_o the_o several_a limitation_n of_o the_o rule_n aforego_v for_o the_o use_n of_o the_o logarithme_n be_v find_v i_o refer_v you_o to_o the_o treatise_n mention_v at_o the_o begin_n of_o this_o discourse_n *_o âŽ_o *_o finis_fw-la