Selected quad for the lemma: hand_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
hand_n	left_a	right_a	turn_v	28,868	5	7.6673	4	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A52264	Institutio mathematica, or, A mathematical institution shewing the construction and use of the naturall and artificiall sines, tangents, and secants in decimal numbers, and also of the table of logarithms in the general solution of any triangle, whether plain or spherical, with their more particular application in astronomie, dialling, and navigation / by John Newton.	Newton, John, 1622-1678.	1654 (1654)	Wing N1061; ESTC R20441	128,229	417

zenith_n and_o therefore_o needless_a to_o put_v they_o on_o in_o the_o north_n recline_v more_o than_o the_o equator_fw-la the_o plane_n in_o our_o example_n must_v be_v elevate_v 120_o degr_n above_o the_o horizon_n and_o the_o style_n of_o both_o must_v point_v to_o the_o north_n pole_n last_o as_o all_o other_o plane_n have_v two_o face_n respect_v the_o contrary_a part_n of_o the_o heaven_n so_o these_o recliner_n have_v opposite_a side_n look_v downward_o the_o nadir_n as_o those_o do_v towards_o the_o zenith_n and_o may_v be_v therefore_o make_v by_o the_o same_o rule_n or_o if_o you_o will_v spare_v that_o labour_n and_o make_v the_o same_o dial_n serve_v for_o the_o opposite_a side_n turn_v the_o centre_n of_o the_o incliner_n downward_o which_o be_v upward_o in_o the_o recliner_n and_o those_o upward_o in_o the_o incliner_n which_o be_v downward_o in_o the_o recliner_n and_o after_o this_o conversion_n let_v the_o hour_n on_o the_o right_a hand_n of_o the_o meridian_n in_o the_o recliner_n become_v on_o the_o left_a hand_n in_o the_o incliner_n and_o contrary_o so_o have_v you_o do_v what_o you_o desire_v and_o this_o be_v a_o general_a rule_n for_o the_o opposite_a side_n of_o all_o plane_n probl._n 11._o to_o draw_v the_o hour-line_n upon_o a_o decline_a recline_a or_o decline_v incline_a plane_n decline_a recline_a plane_n have_v the_o same_o variety_n that_o be_v in_o the_o former_a recline_a north_n and_o south_n for_o either_o the_o declination_n may_v be_v such_o that_o the_o recline_a plane_n will_v fall_v just_a upon_o the_o pole_n and_o then_o it_o be_v call_v a_o decline_a equinoctial_a or_o it_o may_v fall_v above_o or_o under_o the_o pole_n and_o then_o it_o be_v call_v a_o south_n decline_v cast_v and_o west_n recliner_n on_o the_o other_o side_n the_o declination_n may_v be_v such_o that_o the_o recline_a plane_n shall_v fall_v just_a upon_o the_o intersection_n of_o the_o meridian_n and_o equator_fw-la and_o than_o it_o be_v call_v a_o decline_a polar_n or_o it_o may_v fall_v above_o or_o under_o the_o say_a intersection_n and_o then_o it_o be_v call_v a_o north_n decline_v east_n and_o west_n recliner_n the_o three_o variety_n of_o south_n recliner_n be_v represent_v by_o the_o three_o circle_n ahb_n fall_v between_o the_o pole_n of_o the_o world_n and_o the_o zenith_n agb_v just_a upon_o the_o pole_n and_o aeb_n between_o the_o pole_n and_o the_o horizon_n and_o the_o particular_a pole_n of_o each_o plane_n be_v so_o much_o elevate_v above_o the_o horizon_n upon_o the_o azimuth_n dzc_a cross_v the_o base_a at_o right_a angle_n as_o the_o plane_n itself_o recline_v from_o the_o zenith_n note_v in_o the_o scheme_n with_o i_o k_o and_o l._n 1._o of_o the_o equinoctial_a decline_a and_o recline_a plane_n this_o plane_n represent_v by_o the_o circle_n agb_n have_v his_o base_a azb_n decline_v 30_o degree_n from_o the_o east_n and_o west_n line_n ezw_n equal_a to_o the_o declination_n of_o the_o south_n pole_n thereof_o 30_o degree_n from_o saint_n the_o south_n part_n of_o the_o meridian_n easterly_n unto_o d_o recline_v from_o the_o zenith_n upon_o the_o azimuth_n czd_v the_o quantity_n zg_v 34_o degree_n 53_o min._n and_o pass_v through_o the_o pole_n at_o p._n set_v off_o the_o reclination_n zg_v from_o d_o to_o k_o and_o k_o shall_v represent_v the_o pole_n of_o the_o recline_a plane_n so_o much_o elevate_v above_o the_o horizon_n at_o d_o as_o the_o circle_n agb_n represent_v the_o plane_n decline_v from_o the_o zenith_n z_o from_o p_o the_o pole_n of_o the_o world_n to_o k_o the_o pole_n of_o the_o plane_n draw_v a_o arch_n of_o a_o great_a circle_n pk_n thereby_o the_o better_a to_o inform_v the_o fancy_n in_o the_o rest_n of_o the_o work_n and_o if_o any_o be_v desirous_a to_o any_o declination_n give_v to_o fit_v a_o plane_n recline_v just_a to_o the_o pole_n or_o any_o reclination_n be_v give_v to_o find_v the_o declination_n proper_a to_o it_o this_o diagram_n will_v satisfy_v they_o therein_o for_o in_o the_o triangle_n zgp_n we_o have_v limit_v first_o the_o hypothenusal_a pz_n 38_o degree_n 47_o min._n second_o the_o angle_n at_o the_o base_a pzg_n the_o plane_n declination_n 30_o degree_n hence_o to_o find_v the_o base_a gz_n by_o the_o seven_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o cousin_a of_o gzp_n 30_o 9.937531_o so_o the_o tangent_fw-la of_o pz_n 38.47_o 9.900138_o to_o the_o tangent_fw-la of_o gz_n 34.53_o 9.837669_o the_o reclination_n require_v if_o the_o declination_n be_v require_v to_o a_o reclination_n give_v then_o by_o the_o 13_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o tangent_fw-la of_o zg_v 34.53_o 9.837669_o so_o the_o co-tangent_a of_o pz_n 38.47_o 10.099861_o o_o the_o cousin_a of_o gzp_n 39_o 9.937530_o and_o now_o to_o calculate_v the_o hour-line_n of_o this_o dial_n you_o be_v to_o find_v two_o thing_n first_o the_o arch_n of_o the_o plane_n or_o distance_n of_o the_o meridian_n and_o substile_a from_o the_o horizontal_a line_n which_o in_o this_o scheme_n be_v pb_n the_o intersection_n of_o the_o recline_a plane_n with_o the_o horizon_n be_v at_o b._n and_o second_o the_o distance_n of_o the_o meridian_n of_o the_o place_n szpn_n from_o the_o meridian_n of_o the_o plane_n pk_n which_o be_v have_v the_o dial_n be_v easy_o make_v wherefore_o in_o the_o triangle_n zgp_n right_o angle_v at_o g_o you_o have_v the_o angle_n gzp_v give_v 30_o degree_n the_o declination_n and_o zp_v 38_o degr_n 47_o min._n the_o compliment_n of_o the_o pole_n to_o find_v gp_n and_o therefore_o by_o the_o eight_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o sine_fw-la of_o zp_n 38.47_o 9.793863_o so_o be_v the_o sine_fw-la of_o gzp_n 30_o 9.698970_o to_o the_o sine_fw-la of_o gp_n 18.12_o 9.492833_o who_o be_v compliment_n 71_o deg_n 88_o min._n be_v the_o arch_a pb_n desire_v the_o second_o thing_n to_o be_v find_v be_v the_o distance_n of_o the_o meridian_n of_o the_o place_n which_o be_v the_o hour_n of_o 12_o from_o the_o substile_a or_o meridian_n of_o the_o plane_n represent_v by_o the_o angle_n zpg_n which_o may_v be_v find_v by_o the_o 11_o case_n of_o right_n angle_v spherical_a triangle_n for_o as_o the_o radius_fw-la 90_o 10.000000_o be_v to_o the_o sine_fw-la of_o gp_n 18.12_o 9.492833_o so_o be_v the_o co-tang_n of_o gz_n 34.53_o 10.162379_o to_o the_o co-tang_n of_o gpz_n 65.68_o 9.655212_o who_o be_v compliment_n be_v zpk_v 24_o deg_n 32_o min._n the_o arch_n desire_v now_o because_o 24_o deg_n 32_o min._n be_v more_o than_o 15_o deg_n one_o hour_n distance_n from_o the_o meridian_n and_o less_o than_o 30_o deg_n two_o hour_n distance_n i_o conclude_v that_o the_o stile_n shall_v fall_v between_o 10_o and_o 11_o of_o the_o clock_n on_o the_o west_n side_n of_o the_o meridian_n because_o the_o plain_n decline_v east_n if_o then_o you_o take_v 15_o deg_n from_o 24_o deg_n 32_o min._n there_o shall_v remain_v 9_o deg_n 32_o min._n for_o the_o equinoctial_a distance_n of_o the_o 11_o a_o clock_n hour_n line_n from_o the_o substile_a and_o take_v 24_o deg_n 32_o min._n out_o of_o 30_o deg_n there_o shall_v remain_v 5_o deg_n 68_o min._n for_o the_o distance_n of_o the_o hour_n of_o 10_o from_o the_o substile_a the_o rest_n of_o the_o hour_n distance_n be_v easy_o find_v by_o continual_a addition_n of_o 15_o deg_n unto_o these_o hour_n distance_n join_v the_o natural_a tangent_n as_o in_o the_o east_n and_o west_n dial_n which_o will_v give_v you_o the_o true_a distance_n of_o each_o hour_n from_o the_o substile_a the_o plane_n be_v project_v as_o in_o the_o 5_o pro._n for_o the_o east_n &_o west_n dial_n or_o as_o in_o the_o 8_o prob._n for_o the_o equinoctial_a according_a to_o which_o rule_v you_o may_v proportion_v the_o length_n of_o the_o stile_n also_o which_o be_v erect_v over_o the_o substile_a and_o the_o dial_n place_v according_a to_o the_o declination_n 30_o deg_n easterly_a and_o the_o whole_a plain_n raise_v to_o a_o angle_n of_o 55_o deg_n 47_o min._n the_o compliment_n of_o the_o reclination_n the_o shadow_n of_o the_o stile_n shall_v give_v the_o hour_n of_o the_o day_n desire_v 2._o to_o draw_v the_o hour_n line_n upon_o a_o south_n recline_v plain_a decline_a east_n or_o west_n which_o pass_v between_o the_o zenith_n and_o the_o pole_n in_o these_o kind_n of_o decline_a recline_a plain_n the_o south_n pole_n be_v elevate_v above_o the_o plane_n as_o be_v clear_a by_o the_o circle_n ahb_n represent_v the_o same_o which_o fall_v between_o the_o zenith_n and_o the_o north_n pole_n and_o therefore_o hide_v

fraction_n be_v in_o the_o calculation_n very_o tedious_a beside_o here_o no_o fraction_n almost_o be_v exquisite_o true_a therefore_o the_o radius_fw-la for_o the_o make_n of_o rhese_fw-mi table_n be_v to_o be_v take_v so_o much_o the_o more_o that_o there_o may_v be_v no_o error_n in_o so_o many_o of_o the_o figure_n towards_o the_o left_a hand_n as_o you_o will_v have_v place_v in_o the_o table_n and_o as_o for_o the_o number_n superfluous_a they_o be_v to_o be_v cut_v off_o from_o the_o right_a hand_n towards_o the_o left_a after_o the_o end_n of_o the_o supputation_n thus_o to_o find_v the_o number_n answer_v to_o each_o degree_n and_o minute_n of_o the_o quadrant_n to_o the_o radius_fw-la of_o 10000000_o or_o ten_o million_o i_o add_v eight_o cipher_n more_o and_o then_o my_o radius_fw-la do_v consist_v of_o sixteen_o place_n this_o do_v you_o must_v next_o find_v out_o the_o right_a sin_n of_o all_o the_o arch_n less_o than_o a_o quadant_n in_o the_o same_o part_n as_o the_o radius_fw-la be_v take_v of_o whatsoever_o bigness_n it_o be_v and_o from_o those_o right_a sin_n the_o tangent_n and_o secant_v must_v be_v find_v out_o 21._o the_o right_a sin_n in_o make_v of_o the_o table_n be_v either_o primary_n or_o secondary_a the_o primarie_a sin_n be_v those_o by_o which_o the_o rest_n be_v find_v and_o thus_o the_o radius_fw-la or_o whole_a sine_fw-la be_v the_o first_o primary_n sine_fw-la the_o which_o how_o great_a or_o little_o soever_o be_v equal_a to_o the_o side_n of_o a_o sixangled_n figure_n inscribe_v in_o a_o circle_n that_o be_v to_o the_o subtense_n of_o 60_o degree_n the_o which_o be_v thus_o demonstrate_v out_o of_o the_o radius_fw-la or_o subtense_n of_o 60_o degree_n the_o sine_fw-la of_o 30_o degree_n be_v easy_o find_v the_o half_a of_o the_o subtense_n be_v the_o measure_n of_o a_o angle_n at_o the_o circumference_n opposite_a thereunto_o by_o the_o 19_o of_o the_o second_o if_o therefore_o your_o radius_fw-la consist_v of_o 16_o place_n be_v 1000.0000.0000.0000_o the_o sine_fw-la of_o 30_o degree_n will_v be_v the_o one_o half_a thereof_o to_o wit_n 500.0000.0000.0000_o 22._o the_o other_o primary_n sin_n be_v the_o sin_n of_o 60_o 45_o 36_o and_o of_o 18_o degree_n be_v the_o half_a of_o the_o subtense_n of_o 120_o 90_o 72_o and_o of_o 36_o degree_n 23._o the_o subtense_n of_o 120_o degree_n be_v the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v find_v the_o rule_n subtract_v the_o square_a of_o the_o subtense_n of_o 60_o degree_n from_o the_o square_n of_o the_o diameter_n the_o square_a root_n of_o what_o remain_v be_v the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circle●_n or_o the_o subtense_n of_o 120_o degree_n the_o reason_n of_o the_o rule_n the_o subtense_n of_o a_o arch_n with_o the_o subtense_n of_o the_o compliment_n thereof_o to_o 180_o with_o the_o diameter_n make_v in_o the_o meeting_n of_o the_o two_o subtense_n a_o right_a angle_a triangle_n as_o the_o subtense_n ab_fw-la 60_o degree_n with_o the_o subtense_n ac_fw-la 120_o degree_n and_o the_o diameter_n cb_n make_v the_o right_a angle_a triangle_n abc_n right_o angle_v at_o a_o by_o the_o 19_o of_o the_o second_o and_o therefore_o the_o side_n include_v the_o right_a angle_n be_v equal_a in_o power_n to_o the_o three_o side_n by_o the_o 〈◊〉_d of_o the_o second_o therefore_o the_o square_a of_o ab_fw-la be_v take_v from_o the_o square_n of_o cb_n there_o remain_v the_o square_a of_o ac_fw-la who_o squar_fw-la root_n be_v the_o subtense_n of_o 〈◊〉_d degree_n or_o the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circle_n example_n let_v the_o diameter_n cb_n be_v 2000.0000_o 0000.0000_o the_o square_a thereof_o be_v 400000._o 00000.00000.00000.00000.00000_o the_o subtense_n of_o ab_fw-la be_v 100000.00000.00000_o the_o square_a thereof_o be_v 100000.00000.00000_o 00000.00000.00000_o which_o be_v substract_v from_o the_o square_n of_o cb_n the_o remainder_n be_v 300000.00000.00000.00000.00000.00000_o who_o square_a root_n 173205.08075.68877_o the_o subtense_n of_o 120_o degree_n consectary_n hence_o it_o follow_v that_o the_o subtense_n of_o a_o arch_n less_o than_o a_o semicircle_n be_v give_v the_o subtense_n of_o the_o compliment_n of_o that_o arch_n to_o a_o semicircle_n be_v also_o give_v 24._o the_o subtense_n of_o 90_o degree_n be_v the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v find_v the_o rule_n multiply_v the_o diameter_n in_o itself_o and_o the_o square_a root_n of_o half_a the_o product_n be_v the_o subtense_n of_o 90_o degree_n or_o the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n the_o reason_n of_o this_o rule_n the_o diagonal_a line_n of_o a_o square_n inscribe_v in_o a_o circle_n be_v two_o diameter_n and_o the_o right_a angle_a figure_n make_v of_o the_o diagonal_o be_v equal_a to_o the_o right_a angle_a figure_n make_v of_o the_o opposite_a side_n by_o the_o 20_o the_o of_o the_o second_o now_o because_o the_o diagonal_a line_n ab_fw-la and_o cd_o be_v equal_a it_o be_v all_o one_o whether_o i_o multiply_v ac_fw-la by_o itself_o or_o by_o the_o other_o diagonal_a cd_o the_o product_n will_v be_v still_o the_o same_o then_o because_o the_o side_n ab_fw-la ac_fw-la and_o bc_n do_v make_v a_o right_a angle_a triangle_n right_o angle_v at_o c_o by_o the_o 〈◊〉_d of_o the_o second_o &_o that_o the_o 〈◊〉_d ac_fw-la and_o ●b_n be_v equal_a by_o the_o work_n the_o half_a of_o the_o square_n of_o ab_fw-la must_v needs_o be_v the_o square_n of_o ac_fw-la or_o cb_n by_o the_o 17_o the_o of_o the_o second_o who_o square_a root_n the_o subtense_n of_o cb_n the_o side_n of_o a_o square_a or_o 90_o degree_n example_n let_v the_o diameter_n ab_fw-la be_v 200000.00000_o 00000_o the_o square_a thereof_o be_v 400000.00000_o 00000.00000.00000.00000_o the_o half_a whereof_o be_v 200000.00000.00000.00000.00000_o 00000._o who_o square_a root_n 14142●_n 356●3_n 73095._o be_v the_o subtense_n of_o 90_o degree_n or_o the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n 25._o the_o subtense_n of_o 36_o degree_n be_v the_o side_n of_o a_o decangle_n and_o may_v thus_o be_v find_v the_o rule_n divide_v the_o radius_fw-la by_o two_o then_o multiply_v the_o radius_fw-la by_o itself_o and_o the_o half_a thereof_o by_o itself_o and_o from_o the_o square_a root_n of_o the_o sum_n of_o these_o two_o product_n subtract_v the_o half_a of_o radius_fw-la what_o remain_v be_v the_o side_n of_o a_o decangle_n or_o the_o subtense_n of_o 36_o degree_n the_o reason_n of_o the_o rule_n for_o example_n let_v the_o radius_fw-la ebb_n be_v 100000.00000.00000_o then_o be_v bh_n or_o the_o half_a thereof_o 500000._o 00000.00000_o the_o square_a of_o ebb_n be_v 100000_o 00000.00000.00000.00000.00000_o and_o the_o square_a of_o bh_n 250000.00000.00000.00000_o 00000.00000.00000_o the_o sum_n of_o these_o two_o square_n viz_o 125000.00000.00000_o 00000_o 00000._o 00000_o be_v the_o square_a of_o he_o or_o hk_n who_o square_a root_n be_v 1118033●_n 887●9895_n from_o which_o deduct_v the_o half_a radius_fw-la bh_n 500000000000000_o and_o there_o remain_v 618033988749895_o the_o right_a line_n kb_n which_o be_v the_o side_n of_o a_o decangle_n or_o the_o subtense_n of_o 36_o degree_n 26_o the_o subtense_n of_o 72_o degree_n be_v the_o side_n of_o a_o pentagon_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v sound_a the_o rule_n subtract_v the_o side_n of_o a_o decangle_n from_o the_o diameter_n the_o remainder_n multiply_v by_o the_o radius_fw-la shall_v be_v the_o square_n of_o one_o side_n of_o a_o pentagon_n who_o square_a root_n shall_v be_v the_o side_n itself_o or_o subtense_n of_o 72_o degree_n the_o reason_n of_o the_o rule_n in_o the_o follow_a diagram_n let_v ac_fw-la be_v the_o side_n of_o a_o decangle_n equal_a to_o cx_o in_o the_o diameter_n and_o let_v the_o rest_n of_o the_o semicircle_n be_v bisect_v in_o the_o point_n e_o then_o shall_v either_o of_o the_o right_a line_n ae_n or_o ebb_n represent_v the_o side_n of_o a_o equilateral_a pentagon_n for_o ac_fw-la the_o side_n of_o a_o decangle_n subtend_v a_o arch_n of_o 36_o degree_n the_o ten_o part_n of_o a_o circle_n and_o therefore_o aeb_fw-mi the_o remain_a arch_n of_o a_o semicircle_n be_v 144_o degree_n the_o half_a whereof_o ae_n or_o ebb_n be_v 72_o degree_n the_o five_o part_n of_o a_o circle_n or_o side_n of_o a_o equilateral_a pentagon_n the_o square_a whereof_o be_v equal_a to_o the_o oblong_v make_v of_o db_n and_o bx_n demonstration_n draw_v the_o right_a line_n exit_fw-la ed_z and_o aec_fw-la then_o will_v the_o side_n of_o the_o angle_n ace_n and_o ecx_n be_v equal_a because_o cx_o be_v make_v equal_a to_o ac_fw-la and_o aec_fw-la common_a to_o both_o and_o the_o angle_n themselves_o be_v equal_a because_o they_o be_v in_o equal_a segment_n

appear_v by_o that_o which_o follow_v 6._o §_o when_o of_o four_o number_n give_v the_o second_o exceed_v the_o first_o as_o much_o as_o the_o four_o exceed_v the_o three_o the_o sum_n of_o the_o first_o and_o four_o be_v equal_a to_o the_o sum_n of_o the_o second_o and_o three_o and_o contrary_o as_o 8_o 5_o 6_o 3._o here_o 8_o exceed_v 5_o as_o much_o as_o 6_o exceed_v 3_o therefore_o the_o sum_n of_o the_o first_o and_o four_o namely_o of_o 8_o and_o 3_o be_v equal_a to_o the_o sum_n of_o the_o second_o and_o three_o namely_o of_o 5_o and_o 6_o from_o whence_o necessary_o follow_v this_o corollary_n when_o four_o number_n be_v proportional_a the_o sum_n of_o the_o logarithme_n of_o the_o mean_a number_n be_v equal_a to_o the_o sum_n of_o the_o logarithme_n of_o the_o extreme_n example_n let_v the_o four_o proportional_a number_n be_v those_o express_v in_o the_o first_o column_n of_o the_o first_o table_n in_o this_o chapter_n viz._n 4_o 16_o 32_o 128_o in_o which_o table_n the_o logarithme_n of_o 4_o under_o the_o letter_n a_o be_v 3_o the_o logarithme_n of_o 16_o 5_o the_o logarithme_n of_o 32_o 6_o and_o the_o logarithme_n of_o 128_o be_v 8._o now_o as_o the_o sum_n of_o 5_o and_o 6_o the_o logarithme_n of_o the_o mean_a number_n do_v make_v 11_o so_o the_o sum_n of_o 3_o and_o 8_o the_o logarithme_n of_o the_o extreme_n do_v make_v 11_o also_o 7._o §_o when_o four_o number_n be_v proportional_a the_o logarithme_n of_o the_o first_o substract_v from_o the_o sum_n of_o the_o logarithme_n of_o the_o second_o and_o three_o leave_v the_o logarithme_n of_o the_o four_o example_n let_v the_o proportion_n be_v as_o 128_o to_o 32_o so_o be_v 16_o to_o a_o four_o number_n here_o add_v 5_o and_o 6_o the_o logarithme_n of_o the_o second_o and_o three_o the_o sum_n be_v 11_o from_o which_o substract_v 8_o the_o logarithme_n of_o 128_o the_o first_o proportional_a the_o remainder_n be_v 3_o the_o logarithm_n of_o 4_o the_o four_o proportional_a 8._o §_o if_o instead_o of_o substract_v the_o aforesaid_a logarithme_n of_o the_o first_o we_o add_v his_o compliment_n arithmetical_a to_o any_o number_n the_o total_a abate_v that_o number_n be_v as_o much_o as_o the_o remainder_n will_v have_v be_v the_o compliment_n arithmetical_a of_o one_o number_n to_o another_o as_o here_o we_o take_v it_o be_v that_o which_o make_v that_o first_o number_n equal_a to_o the_o other_o thus_o the_o compliment_n arithmetical_a of_o 8_o to_o 10_o be_v 2_o because_o 8_o and_o 2_o be_v 10._o now_o than_o whereas_o in_o the_o example_n of_o the_o last_o proposition_n substract_v 8_o from_o 11_o there_o remain_v 3_o if_o instead_o of_o substract_v 8_o we_o add_v his_o compliment_n arithmetical_a to_o 10_o which_o be_v 2_o the_o total_a be_v 13_o from_o which_o abate_v 10_o there_o remain_v 3_o as_o before_z both_o the_o operation_n stand_v thus_o as_o 128_o be_v to_o 32_o so_o be_v 16_o logar_a 8_o compl_a arithmetical_a 2_o 6_o  _fw-fr 6_o 5_o  _fw-fr 5_o the_o aggreg_n of_o 1.2_o  _fw-fr 11_o their_a aggregate_v be_v 13_o to_o 4_o  _fw-fr 3_o  _fw-fr  _fw-fr from_o which_o abate_v 10_o there_o remain_v 3_o and_o the_o like_a be_v to_o be_v understand_v of_o any_o other_o the_o reason_n be_v manifest_a for_o whereas_o we_o shall_v have_v abate_v 8_o out_o of_o 11_o we_o do_v not_o only_o not_o abate_v it_o but_o add_v moreover_o his_o compliment_n to_o 10_o which_o be_v 2_o wherefore_o the_o total_a be_v more_o than_o if_o shall_v be_v by_o 8_o &_o 2_o that_o be_v by_o 10_o wherefore_o abate_v 10_o from_o it_o we_o have_v the_o logarithme_n desire_v which_o rule_n although_o it_o be_v general_a yet_o we_o shall_v seldom_o have_v occasion_n to_o use_v any_o other_o compliment_n than_o such_o as_o be_v the_o compliment_n of_o the_o logarithme_n give_v either_o to_o 10,000000_o or_o to_o 20,000000_o the_o ●_o compliment_n arithmetical_a of_o any_o logarithme_n to_o either_o of_o these_o number_n be_v that_o which_o make_v the_o logarithme_n give_v equal_a to_o either_o of_o they_o thus_o the_o compliment_n arithmetical_a of_o the_o logarithme_n of_o 2_o viz._n 0301030_o be_v 9698970_o because_o these_o two_o number_n add_v together_o do_v make_v 10.000000_o and_o thus_o the_o compliment_n thereof_o to_o 20_o 000000is_fw-la 19698970_o if_o therefore_o 0301030_o be_v substract_v from_o 10.000000_o the_o remainder_n be_v his_o compliment_n arithmetical_a but_o to_o find_v it_o ready_o you_o may_v instead_o of_o substract_v the_o logarithme_n give_v from_o 10.000000_o write_v the_o compliment_n of_o every_o figure_n thereof_o unto_o 9_o begin_v with_o the_o first_o figure_n towards_o the_o left_a hand_n and_o so_o on_o till_o you_o come_v to_o the_o last_o figure_n towards_o the_o right_a hand_n and_o thereof_o set_v down_o the_o residue_n unto_o 10._o thus_o for_o the_o compliment_n arithmetical_a of_o the_o aforesaid_a logarithme_n 0301030_o i_o write_v for_o 0_o 9_o for_o 3_o 6_o for_o 0_o 9_o for_o 1_o 8_o for_o 0_o 9_o for_o 3_o again_o i_o shall_v write_v 6_o but_o because_o the_o last_o place_n of_o the_o logarithme_n be_v a_o cipher_n and_o that_o i_o must_v write_v the_o compliment_n thereof_o to_o 10_o instead_o of_o 6_o i_o write_v 7_o and_o for_o 0_o 0_o and_o so_o have_v i_o this_o number_n 9698970_o which_o be_v the_o compliment_n arithmetical_a of_o 0301030_o as_o before_o 9_o §_o every_o logarithme_n have_v his_o proper_a characteristic_a and_o the_o character_n or_o characteristical_a root_n of_o every_o logarithme_n be_v the_o first_o figure_n or_o figure_n towards_o the_o left_a hand_n distinguish_v from_o the_o rest_n by_o a_o point_n or_o comma_n thus_o the_o character_n of_o the_o logarithme_n of_o every_o number_n less_o than_o 10_o be_v 0_o but_o the_o character_n of_o the_o logarithme_n of_o 10_o be_v 1_o and_o so_o of_o all_o other_o number_n to_o 100_o but_o the_o character_n of_o the_o logarithme_n of_o 100_o be_v 2_o and_o so_o of_o the_o rest_n to_o 1000_o and_o the_o character_n of_o the_o logarithme_n of_o 1000_o be_v 3_o and_o so_o of_o the_o rest_n to_o 10000_o in_o brief_a the_o characteristic_a of_o any_o logarithme_n must_v consist_v of_o a_o unite_v less_o than_o the_o give_v number_n consist_v of_o digit_n or_o place_n and_o therefore_o by_o the_o character_n of_o a_o logarithme_n you_o may_v know_v of_o how_o many_o place_n the_o absolute_a number_n answer_v to_o that_o logarithme_n do_v consist_v 10._o §_o if_o one_o number_n multiply_v another_o the_o sum_n of_o their_o logarithme_n be_v equal_a to_o the_o logarithme_n of_o the_o product_n as_o let_v the_o two_o number_n multiply_v together_o be_v 2_o and_o 2_o the_o product_n be_v 4_o i_o say_v then_o that_o the_o sum_n of_o the_o logarithme_n of_o 2_o and_o 2_o or_o the_o logarithme_n of_o 2_o double_v be_v equal_a to_o the_o logarithme_n of_o 4_o as_o here_o you_o may_v see_v 2._o 0.301030_o 2._o 0.301030_o  _fw-fr  _fw-fr 4._o 0.602060_o again_o let_v the_o two_o number_n multiply_v together_o be_v 2_o and_o 4_o the_o product_n be_v 8_o i_o say_v then_o that_o the_o sum_n of_o the_o logarithme_n of_o 2_o and_o 4_o be_v equal_a to_o the_o logarithme_n of_o 8_o as_o here_o you_o may_v also_o see_v 2._o 0.301030_o 4._o 0.602060_o  _fw-fr  _fw-fr 8._o 0.903090_o and_o so_o for_o any_o other_o the_o reason_n be_v for_o that_o by_o the_o ground_n of_o multiplication_n as_o unit_fw-la be_v in_o proportion_n to_o the_o multiplier_n so_o be_v the_o multiplicand_a to_o the_o product_n therefore_o by_o the_o six_o of_o this_o chapter_n the_o sum_n of_o the_o logarithme_n of_o a_o unit_fw-la and_o of_o the_o product_n be_v equal_a to_o the_o sum_n of_o the_o logarithme_n of_o the_o multiplier_n and_o multiplicand_a but_o the_o logarithme_n of_o a_o unit_fw-la be_v 0_o therefore_o the_o logarithme_n of_o the_o product_n alone_o be_v equal_a to_o the_o sum_n of_o the_o logarithme_n of_o the_o multiplier_n and_o multiplicand_a and_o by_o the_o like_a reason_n it_o three_o or_o more_o number_n be_v multiply_v together_o the_o sum_n of_o all_o their_o logarithme_n be_v equal_a to_o the_o logarithme_n of_o the_o product_n of_o they_o all_o 11._o §_o if_o one_o number_n divide_v another_o the_o logarithme_n of_o the_o divisor_n be_v substract_v from_o the_o logari●hme_n of_o the_o dividend_n leave_v the_o logarithme_n of_o the_o quotient_a as_o let_v 10_o be_v divide_v by_o 2_o the_o quotient_n be_v 5._o i_o say_v then_o if_o the_o logarithme_n of_o 2_o be_v substract_v from_o the_o logarithme_n of_o 10_o there_o will_v remain_v the_o logarithme_n of_o 5_o as_o here_o be_v to_o be_v see_v 10._o 1.000000_o 2._o 0.301030_o  _fw-fr  _fw-fr 5._o 0.698970_o for_o see_v that_o the_o quotient_a multiply_v by_o the_o divisor_n produce_v the_o dividend_n therefore_o