of_o their_o intersection_n draw_v your_o line_n to_o the_o end_n of_o your_o give_a line_n and_o it_o shall_v be_v a_o perpendicular_a you_o see_v it_o be_v the_o same_o the_o other_o be_v prop._n iu._n to_o let_v fall_v a_o perpendicular_a from_o a_o point_n to_o a_o give_v line_n let_v the_o give_v line_n be_v d_o a_o the_o point_n from_o whence_o the_o perpendicular_a be_v to_o be_v let_v fall_n be_v at_o c._n from_o the_o point_n c_o draw_v a_o white_a line_n to_o the_o give_v line_n by_o guess_n as_o c_o a_o divide_v it_o into_o two_o equal_a part_n which_o be_v do_v at_o b_o then_o continue_v ½_n the_o line_n c_o a_o which_o be_v a_o b_o or_o c_o b_o in_o your_o compass_n and_o your_o compass_n fix_v one_o foot_n at_o b_o describe_v the_o arch_n c_o d_o and_o where_o it_o cut_v the_o give_v line_n there_o will_v your_o perpendicular_a fall_n from_o the_o give_v point_n for_o c_o d_o be_v perpendicular_a to_o the_o give_v line_n d_o a._n prop._n v._o to_o draw_v a_o line_n parallel_n to_o a_o line_n give_v let_v the_o give_v line_n be_v a_o s_n it_o be_v require_v to_o draw_v a_o line_n so_o that_o the_o two_o line_n may_v run_v at_o both_o end_n one_o by_o the_o other_o and_o never_o meet_v which_o be_v parallel_v one_o to_o the_o other_o open_v your_o compass_n to_o that_o extent_n as_o you_o will_v have_v the_o two_o line_n asunder_o and_o go_v towards_o one_o end_n of_o the_o give_v line_n as_o at_o s_n and_o describe_v the_o arch_n u._fw-mi and_o with_o the_o same_o distance_n come_v towards_o the_o other_o end_n as_o at_o a_o and_o describe_v another_o arch_n which_o be_v n_n and_o by_o the_o top_n of_o these_o two_o arch_n draw_v the_o line_n r_o o_fw-fr which_o be_v parallel_n to_o a_o s._n prop._n vi_o to_o draw_v a_o line_n parallel_n to_o a_o give_v line_n from_o any_o point_n assign_v let_v the_o give_v line_n be_v s_o l_o the_o point_n assign_v be_v a_o take_v the_o distance_n from_o s_n to_o a_o and_o carry_v it_o towards_o the_o other_o end_n of_o the_o give_v line_n as_o at_o l_o describe_v the_o arch_n n_z then_o take_v the_o distance_n from_o l_o to_o s_n and_o fix_v one_o foot_n of_o your_o compass_n in_o the_o give_v point_n a_o across_o the_o arch_n n_z with_o the_o arch_n o_o and_o by_o the_o place_n of_o their_o intersection_n and_o the_o point_n assign_v draw_v a_o line_n which_o shall_v be_v parallel_n to_o s_o l._n prop._n vii_o to_o make_v a_o square_a of_o a_o line_n give_v let_v the_o line_n give_v be_v a_o equal_a to_o which_o draw_v the_o side_n of_o the_o square_a b_o e_z and_o from_o one_o end_n of_o it_o raise_v a_o perpendicular_a and_o by_o it_o set_v off_o the_o length_n of_o a_o as_o here_o from_o e_z the_o perpendicular_a be_v raise_v and_o the_o length_n of_o a_o set_v upon_o it_o which_o be_v the_o side_n of_o the_o square_a e_o d_o continue_v the_o same_o distance_n in_o your_o compass_n and_o go_v to_o d_o and_o describe_v the_o arch_n 8_o carry_v the_o same_o distance_n to_o b_o and_o cross_v the_o arch_n 8_o with_o f_o and_o from_o the_o intersection_n of_o those_o two_o arch_n draw_v the_o side_n c_o d_o and_o c_o b_o which_o make_v the_o square_a bedc_n &_o this_o be_v a_o true_a square_n prop._n viii_o to_o make_v a_o square_a who_o length_n and_o breadth_n be_v give_v these_o sort_n of_o square_n be_v call_v geometrical_a square_n when_o but_o two_o side_n be_v equal_a namely_o the_o two_o long_a side_n or_o the_o two_o short_a side_n the_o angles_n be_v all_o equal_a namely_o right_a angle_n suppose_v the_o length_n of_o the_o square_n be_v a_o the_o breadth_n b_o i_o desire_v to_o make_v it_o first_o draw_v a_o line_n equal_a to_o a_o for_o the_o length_n of_o it_o as_o s_n v_n then_o from_o any_o end_n of_o that_o line_n raise_v a_o perpendicular_a as_o here_o from_o v._o and_o set_v off_o the_o line_n b_o for_o the_o breadth_n of_o it_o upon_o it_o which_o fall_v in_o the_o perpendicular_a line_n at_o l_o then_o take_v the_o length_n of_o a_o and_o describe_v the_o arch_n n_z fix_v your_o compass_n one_o foot_n in_o l_o then_o take_v b_o the_o breadth_n and_o cross_v that_o arch_n by_o another_o fix_v your_o compass_n one_o foot_n in_o s_n and_o draw_v l_o k_n and_o k_o s_o from_o the_o place_n of_o their_o intersection_n as_o you_o do_v in_o the_o other_o thus_o the_o two_o opposite_a side_n in_o this_o square_n be_v equal_a and_o the_o angle_n in_o both_o all_o equal_a for_o they_o be_v right_a angle_n prop._n ix_o to_o make_v a_o diamond_n figure_n of_o a_o line_n and_o a_o angle_n give_v a_o diamond_n figure_n be_v a_o figure_n of_o four_o equal_a side_n but_o the_o angle_n be_v two_o of_o they_o acute_a and_o two_o of_o they_o obtuse_a the_o acute_a angle_n be_v equal_a and_o the_o obtuse_a angle_n be_v equal_a one_o to_o another_o let_v a_o b_o be_v the_o give_v line_n c_o b_o the_o measure_n of_o the_o give_v angle_n a_o be_v the_o angular_a point_n first_o take_v the_o line_n ab_fw-la and_o draw_v a_o line_n of_o its_o length_n for_o one_o side_n of_o the_o figure_n namely_o 8_o 0_o then_o take_v the_o semidiameter_n of_o the_o arch_n c_o b_o which_o be_v c_z a_o and_o fix_v one_o foot_n of_o your_o compass_n in_o 8_o describe_z the_o arch_n s_o 0_o and_o take_v the_o arch_n b_o c_o and_o set_v it_o off_o from_o o_o to_o s_n then_o draw_v the_o line_n s_o 8_o equal_a to_o 8_o o_o this_o do_v keep_v the_o length_n of_o o_o 8_o in_o your_o compass_n and_o from_o s_n and_o o_o describe_v the_o arch_n n_o and_o it_o and_o draw_v the_o side_n o_o r_n and_o s_o r_n as_o you_o do_v in_o the_o other_o figure_n and_o thus_o s_n r_n be_v equal_a to_o 8_o o_o or_o r_o o_o be_v equal_a to_o 8_o s_n and_o the_o opposite_a angle_n also_o equal_a i_o forbear_v to_o show_v the_o reason_n of_o their_o be_v equal_a because_o it_o have_v be_v handle_v by_o other_o and_o indeed_o it_o be_v so_o plain_a that_o with_o a_o little_a consideration_n you_o may_v know_v it_o prop._n x._o to_o make_v a_o rhomboiade_n of_o two_o give_v side_n and_o a_o angle_n include_v a_o rhomboiades_n be_v a_o figure_n who_o opposite_a side_n and_o opposite_a angle_n be_v equal_a as_o a_o geometrical_a square_n be_v but_o in_o this_o they_o differ_v a_o rhomboiades_n have_v never_o a_o right_a angle_n but_o two_o obtuse_a and_o two_o acute_a whereas_o the_o other_o have_v all_o right_a angle_n it_o differ_v from_o the_o diamond_n figure_n also_o for_o in_o one_o the_o side_n be_v all_o equal_a and_o in_o this_o but_o two_o equal_a side_n i_o need_v not_o show_v the_o work_n of_o it_o because_o it_o differ_v not_o from_o a_o diamond_n figure_n only_o in_o take_v the_o two_o side_n apart_o to_o describe_v the_o arch_n at_o l_o ay_o suppose_v you_o may_v conceive_v how_o it_o be_v make_v by_o see_v this_o which_o be_v here_o make_v the_o give_v side_n and_o angle_n be_v s_o o_o n_n i_o have_v not_o set_v down_o the_o arch_n to_o measure_v the_o angle_n at_o o._n i_o suppose_v from_o what_o have_v be_v say_v you_o will_v conceive_v how_o that_o be_v prop._n xi_o to_o find_v the_o centre_n of_o a_o circle_n draw_v a_o line_n from_o side_n to_o side_n of_o the_o circle_n at_o a_o venture_n as_o a_o c_o and_o divide_v that_o line_n into_o two_o equal_a part_n by_o a_o perpendicular_a as_o be_v show_v before_o that_o perpendicular_a line_n draw_v through_o the_o circle_n from_o side_n to_o side_n as_o be_v u_z s_n and_o it_o shall_v be_v the_o diameter_n of_o the_o circle_n the_o half_a of_o which_o be_v the_o semidiameter_n or_o very_a centre_n ⊙_o it_o be_v possible_a to_o find_v the_o centre_n of_o a_o triangle_n after_o the_o same_o manner_n suppose_v the_o triangle_n who_o centre_n you_o will_v find_v be_v a_o c_o n_z divide_v any_o side_n into_o two_o equal_a part_n by_o a_o perpendicular_a and_o it_o will_v go_v through_o the_o centre_n of_o the_o triangle_n as_o the_o side_n a_o c_o be_v divide_v into_o two_o equal_a part_n by_o the_o perpendicular_a b_o s_n then_o i_o say_v b_o s_n go_v through_o the_o centre_n of_o this_o triangle_n but_o to_o find_v whereabouts_o in_o this_o perpendicular_a the_o centre_n of_o the_o triangle_n be_v i_o know_v by_o no_o other_o mean_n but_o by_o remove_v your_o compass_n in_o this_o line_n from_o place_n to_o place_n till_o you_o find_v it_o which_o be_v here_o find_v to_o be_v at_o u._n but_o this_o be_v but_o a_o botch_o way_n and_o with_o a_o little_a more_o labour_v you_o may_v find_v it_o at_o once_o therefore_o mind_v
to_o good_a reason_n that_o be_v the_o angle_n in_o this_o triangle_n be_v the_o same_o with_o the_o angle_n in_o the_o question_n therefore_o as_o the_o sum_n of_o these_o side_n be_v in_o proportion_n to_o the_o sum_n of_o the_o side_n there_o so_o be_v the_o sum_n of_o the_o side_n 30_o here_o to_o the_o sum_n of_o the_o side_n that_o be_v correspondent_a to_o it_o there_o or_o run_v upon_o the_o same_o course_n to_o find_v the_o ship_n distance_n that_o sail_v s_o w_n correspondent_a to_o c_z n_z 30_o league_n as_o sum_n of_o the_o side_n of_o the_o triangle_n c_o n_o b_o 95_o le_fw-fr co_fw-la be_v 7,021363_o be_v to_o the_o sum_n of_o the_o side_n in_o the_o quest_n 148_o league_n 3,170261_o so_o be_v the_o side_n c_o n_o 30_o league_n 2,477121_o to_o the_o ship_n distance_n that_o sail_v s_o w_n 46_o league_n 6_o 19_o 2,668745_o this_o side_n be_v thus_o find_v we_o to_o abbreviate_v the_o work_n will_v lay_v it_o down_o in_o the_o triangle_n before_o who_o angle_n be_v equal_a to_o to_o the_o angle_n here_o set_v 46_o 6_o 10_o league_n for_o the_o length_n of_o the_o side_n the_o westermost_a ship_n run_v which_o be_v n_n r._n draw_z r_z s_z parallel_n to_o c_o b_o and_o extend_v n_z b_o till_o it_o intersect_v r_o s_o which_o it_o do_v in_o we_o this_o be_v as_o much_o as_o protract_v of_o it_o anew_o for_o the_o angle_n at_o r_n at_z s_z and_o at_o n_n be_v the_o same_o that_o they_o be_v give_v to_o be_v in_o the_o question_n and_o the_o side_n r_o n_n 47_o league_n as_o it_o be_v find_v you_o may_v measure_v the_o side_n unknown_a by_o the_o plain_a scale_n and_o set_v they_o down_o if_o you_o will_v or_o you_o may_v work_v by_o the_o rule_n of_o three_o use_v this_o proportion_n as_o n_n c_o 30_o league_n be_v to_o n_n r_n 46_o 6_o 10_o leag_fw-mi so_o be_v n_z b_o 41_o 6_o 10_o to_o n_n s_o multiply_v and_o divide_v and_o it_o make_v 64_o 6_o 10_o league_n 14_o 57_o of_o a_o ten_o by_o the_o table_n for_o n_n be_v as_o n_n c_o 30_o league_n comp_n arith_n 7,52287_o to_o n_o r_n 46_o 6_o 10_o league_n  _fw-fr 2,66839_o so_o be_v n_z b_o 41_o 6_o 10_o  _fw-fr 2,61909_o to_z n_o s_z 64_o 6_o 10_o league_n  _fw-fr 2,81035_o we_o may_v have_v do_v this_o by_o the_o table_n put_v the_o angle_n and_o side_n in_o proportion_n as_o we_o have_v do_v all_o along_o in_o the_o other_o triangle_n but_o i_o suppose_v your_o own_o reason_n will_v give_v you_o that_o this_o have_v the_o same_o give_v in_o it_o that_o the_o angle_n c_o n_o b_o have_v and_o be_v wrought_v so_o the_o fraction_n here_o and_o the_o other_o agree_v only_o this_o be_v not_o so_o true_a altogether_o because_o here_o we_o do_v it_o but_o to_o ten_o there_o in_o small_a fraction_n but_o this_o be_v within_o a_o small_a part_n of_o one_o ten_o of_o a_o unite_v for_o the_o side_n r_o s_o as_o sine_fw-la n_o s_o r_n 45_o deg_fw-la 00_o min._n comp_n arith_n 0,15051_o be_v to_o r_o n_n 46_o league_n 6_o 10_o  _fw-fr 2,66839_o so_o be_v sine_fw-la n_n 33_o deg_fw-la 45_o min._n  _fw-fr 9,74473_o to_z r_o s_z 36_o 6_o 10_o league_n  _fw-fr 2,56363_o if_o you_o add_v the_o side_n now_o find_v all_o together_o it_o will_v come_v to_o 2_o 10_o of_o a_o unite_v less_o than_o the_o give_v side_n together_o in_o the_o question_n be_v now_o that_o arise_v by_o neglect_v the_o take_n of_o the_o absolute_a number_n answer_v to_o the_o logarithm_n that_o come_v out_o but_o this_o be_v sufficient_a in_o case_n of_o this_o kind_n if_o you_o have_v set_v it_o in_o pace_n and_o wrought_v to_o the_o ten_o part_n you_o will_v have_v be_v out_o but_o 2_o 10_o of_o a_o pace_n therefore_o use_v your_o mind_n in_o such_o case_n if_o you_o have_v wrought_v to_o centism_n you_o will_v have_v be_v near_a but_o this_o i_o count_v near_o enough_o for_o sail_v and_o i_o be_o sure_a be_v more_o proper_a to_o be_v use_v than_o such_o small_a fraction_n because_o no_o long_a distance_n can_v be_v guess_v to_o a_o mile_n or_o a_o league_n you_o may_v have_v wrought_v your_o last_o proportion_n by_o the_o rule_n of_o three_o as_o you_o do_v the_o other_o three_o side_n of_o a_o triangle_n give_v to_o find_v the_o centre_n question_n ix_o there_o be_v three_o ship_n bind_v to_o one_o place_n the_o eastermost_a be_v distant_a from_o the_o middlemost_a 40_o league_n and_o bear_v s_n e_o b_o e_o the_o middlemost_a be_v distant_a from_o the_o westermost_a 50_o league_n and_o bear_v n_o e_z now_o they_o every_o one_o know_v as_o much_o as_o i_o have_v write_v they_o also_o know_v they_o be_v of_o a_o like_a distance_n from_o this_o port._n i_o desire_v to_o know_v what_o distance_n the_o port_n be_v from_o they_o and_o how_o it_o bear_v from_o each_o ship_n i_o have_v apply_v this_o question_n to_o sail_v here_o but_o before_o in_o this_o book_n i_o have_v use_v it_o in_o a_o thing_n which_o be_v very_o proper_a for_o it_o and_o indeed_o it_o may_v be_v of_o use_v many_o time_n which_o make_v i_o give_v it_o place_n here_o i_o shall_v say_v nothing_o of_o this_o question_n by_o the_o plain_a scale_n for_o it_o be_v do_v as_o before_o you_o see_v but_o we_o will_v do_v it_o by_o the_o table_n of_o tangent_n and_o logarithm_n here_o i_o have_v give_v the_o side_n m_o er_fw-mi 40_o league_n the_o side_n m_o w_n 50_o league_n and_o the_o angle_n at_o m_n namely_o its_o compliment_n to_o 180_o deg_fw-la r_o m_o o_o 78_o deg_fw-la 45_o min._n to_o find_v the_o angle_n m_o ere_fw-we w_n and_o m_o w_n o_z as_o w_n m_o add_v m_o o_o 90_o league_n comp_n arith_n 7,04575_o be_v to_o their_o difference_n 10_o league_n  _fw-fr 2_o so_o be_v the_o tan._n of_o ½_n m_o er_fw-mi w_o and_o m_o w_n oer_o 39_o d._n 22_o m._n 9,91404_o to_o tan._n of_o their_o difference_n 5_o deg_fw-la 12_o min._n 8,95980_o subtract_v from_o 39_o deg_fw-la 22_o min._n and_o the_o remainder_n be_v m_o w_n oer_o 34_o deg_fw-la 50_o min._n add_v to_o it_o make_v m_n oer_o w_n 44_o deg_fw-la 34_o min._n for_o the_o side_n er_fw-mi w._n as_o sine_fw-la m_o o_o w_n 44_o deg_fw-la 34_o min._n comp_n arith_n 0,153824_o to_o w_n m_o 50_o league_n  _fw-fr 2,698970_o so_o be_v sine_fw-la r_o m_o o_o 78_o d._n 45_o m._n its_o comp_n log_n 180_o d._n 9,991573_o to_o w_o o_z 69_o 9_o 10_o league_n  _fw-fr 2,844367_o but_o now_o you_o will_v demand_v of_o i_o how_o we_o shall_v find_v any_o thing_n in_o the_o triangle_n w_n f_o er_fw-mi be_v there_o be_v but_o one_o thing_n know_v in_o it_o namely_o the_o distance_n of_o the_o eastermost_a ship_n from_o the_o westermost_a er_fw-mi w_n i_o will_v tell_v you_o it_o mr._n euclid_n prove_v this_o which_o make_v i_o forbear_v it_o mr._n euclid_n book_n 3_o prop._n 20._o say_v that_o in_o a_o circle_n a_o angle_n at_o the_o centre_n be_v double_a to_o a_o angle_n at_o the_o circumference_n provide_v that_o both_o the_o angle_n have_v to_o their_o base_a one_o and_o the_o same_o part_n of_o the_o circumference_n the_o centre_n be_v f_o here_o the_o triangle_n w_n f_o o_z and_o w_n m_o ere_fw-we have_v w_o o_z to_o both_o their_o base_n then_o must_v b_o f_o a_o be_v double_a to_o i_o m_o o_o or_o c_o b_o a_o double_a to_o d_o m_o i_o that_o be_v to_o the_o arch_n i_o o_o d_o so_o that_o if_o you_o double_a d_o o_fw-fr i_o 101_o deg_fw-la 15_o min._n it_o make_v c_o b_o a_o 202_o deg_fw-la 30_o min._n that_o subtract_v from_o a_o circle_n 360_o deg_fw-la leave_v the_o triangle_n w_n f_o o_z it_o be_v evident_a than_o that_o w_n f_o ere_fw-we be_v double_a to_o r_o m_o i_o because_o r_n m_o i_o be_v what_o i_o o_o d_o want_v of_o a_o semicircle_n as_o w_n f_o ere_fw-we be_v what_o c_o b_o a_o want_v of_o a_o whole_a circle_n which_o be_v double_a to_o it_o example_n  _fw-fr deg_fw-la min._n w_n m_o ere_fw-we be_v 101_o deg_fw-la 30_o min._n double_v it_o be_v c_o b_o a_o 202_o 30_o that_o subtract_v from_o a_o circle_n 360_o 00_o the_o remainder_n be_v w_n f_o oer_o 157_o 30_o and_o this_o subtract_v from_o 180_o 00_o the_o remainder_n be_v f_o w_n o_z and_o f_o oer_o w_n for_o the_o angle_n of_o a_o right_a angle_a triangle_n be_v 180_o d._n 22_o 30_o the_o half_a of_o which_o be_v f_o w_n o_z or_o f_o ere_fw-we w_o 11_o 15_o for_o they_o must_v be_v equal_a because_o the_o side_n opposite_a to_o they_o be_v equal_a namely_o f_o o_z and_o f_o w._n and_o thus_o we_o have_v the_o three_o angle_n of_o the_o triangle_n w_n f_o er_fw-mi and_z the_o side_z w_o o_z to_o
find_v f_o o_z or_o f_o w_n or_o f_o m_o the_o distance_n of_o any_o of_o the_o ship_n from_o the_o port._n for_o the_o distance_n of_o the_o ship_n from_o the_o port._n as_o sine_fw-la w_n f_o oer_o 22_o deg_fw-la 30_o min._n comp_n arith_n 0,417160_o be_v to_o w_o o_z 69_o 9_o 10_o league_n  _fw-fr 2,844477_o so_o be_v sine_fw-la f_o w_n o_z 11_o deg_fw-la 15_o min._n  _fw-fr 9,290235_o to_o any_o of_o the_o ship_n distance_n from_o the_o port_n which_o i_o find_v to_o be_v 35_o 6_o 10_o league_n 2,551872_o i_o need_v not_o count_v the_o triangle_n for_o the_o point_n for_o i_o have_v show_v that_o sufficient_o so_o that_o you_o i_o suppose_v can_v tell_v how_o to_o find_v the_o ship_n bear_v from_o the_o port_n for_o it_o differ_v nothing_o from_o the_o way_n you_o count_v in_o other_o question_n question_n x._o there_o be_v two_o ship_n set_v from_o one_o port_n one_o sail_a n_z w_o b_o n_z 6_o league_n a_o watch_n the_o other_o sail_v n_o e_z 8_o league_n a_o watch_n these_o ship_n arrive_v at_o two_o several_a port_n in_o one_o instant_a of_o time_n the_o port_n be_v 12_o league_n asunder_o i_o demand_v the_o distance_n run_v of_o each_o ship_n and_o the_o port_n bear_v one_o from_o the_o other_o for_o the_o do_n of_o this_o question_n first_o draw_v a_o n_n w_n b_o n_n line_n and_o set_v off_o 6_o league_n upon_o it_o which_o be_v a_o i_o then_o draw_v a_o n_n e_o line_n and_o set_v 8_o league_n off_o upon_o it_o which_o be_v a_o d_o draw_v the_o side_n i_o d_o then_o consider_v that_o as_o a_o i_o be_v to_o a_o d_o so_o be_v the_o westermost_a ship_n run_v to_o the_o eastermost_a so_o that_o the_o side_n a_o d_o and_o a_o i_o extend_v so_o far_o that_o a_o line_n of_o 12_o league_n long_o draw_v parallel_n to_o d_o i_o will_v just_o touch_v the_o extend_a side_n which_o must_v cut_v they_o in_o the_o place_n be_v the_o port_n be_v namely_o a_o b_o in_o b_o and_o a_o i_o in_o c_o measure_v the_o side_n c_o a_n and_o a_o b_o and_o also_o the_o angle_n of_o the_o port_n bear_v a_o b_o c_o or_o a_o c_o b_o and_n count_v they_o as_o have_v be_v show_v before_o i_o find_v the_o side_n a_o b_o the_o eastermost_a ship_n run_v to_o be_v 10_o leag_n 7_o 10._o i_o find_v a_o c_o the_o westermost_a ship_n run_v to_o be_v 8_o league_n the_o westermost_a port_n bear_v from_o the_o eastermost_a west_n 4_o deg_fw-la 14_o min._n southerly_a by_o the_o table_n first_o for_o the_o port_n bear_v one_o from_o the_o other_o a_o d_o i_o equal_a to_o a_o b_o c_o or_o a_o i_o d_o equal_a to_o a_o c_o b_o they_o be_v equal_a because_o i_o d_o and_o c_o b_o be_v parallel_a and_o i_o a_o d_o be_v a_o angle_n opposite_a to_o they_o both_o as_o a_o d_o and_o a_o i_o 14_o league_n comp_n arith_n 6,853871_o be_v to_o their_o difference_n 2_o league_n  _fw-fr 2,301030_o so_o be_v tan._n ½_n a_o i_o d_o and_o a_o d_o i_o 50_o deg_fw-la 37_o min._n 30_o sec_fw-la 10,085698_o to_o tan._n their_o diff._n 9_o deg_fw-la 51_o min._n  _fw-fr 9,240599_o which_o add_v to_o the_o half_a sum_n be_v a_o i_o d_o 60_o deg_fw-la 28_o min._n a_o c_o b_o equal_a to_o a_o d_o i_o or_o a_o b_o c_o the_o bearing_n subtract_v from_o the_o half_a sum_n be_v 40_o deg_fw-la 46_o min._n for_o i_o d._n as_o sine_fw-la d_o 40_o deg_fw-la 46_o min._n 0,185100_o to_o i_o a_o 6_o league_n 0,778151_o so_o be_v sine_fw-la i_o a_o d_o 78_o deg_fw-la 45_o min._n 9,991573_o to_o d_o i_o 9_o league_n 0,954824_o for_o c_o a._n as_o i_o d_o 9_o leag_fw-mi co_fw-la be_v 8,04575_o be_v to_o c_o b_o 12_o league_n 2,07918_o so_o be_v i_o a_o 6_o league_n 1,77815_o to_o a_o c_o 8_o league_n 1,90309_o for_o a_o b._n as_o i_o d_o 9_o leag_fw-mi co_fw-la be_v 8,04575_o be_v to_o c_o b_o 12_o league_n 2,07928_o so_o be_v a_o d_o 8_o league_n 1,90309_o to_o a_o b_o 10_o 7_o 10_o league_n 2,02802_o the_o reason_n that_o the_o proportion_n stand_v thus_o be_v because_o the_o angles_n be_v equal_a in_o each_o triangle_n and_o therefore_o the_o side_n must_v be_v proportional_a thus_o the_o eastermost_a ship_n sail_v 10_o 7_o 10_o league_n the_o westermost_a 8_o league_n and_o the_o eastermost_a port_n bear_v from_o the_o westermost_a east_n 4_o deg_fw-la 14_o min._n northerly_a there_o be_v no_o distance_n that_o a_o man_n sail_n but_o if_o he_o return_v the_o same_o way_n that_o he_o go_v out_o he_o shall_v find_v the_o same_o distance_n backward_o as_o he_o do_v out_o but_o now_o the_o world_n be_v round_o and_o if_o a_o man_n go_v out_o one_o way_n and_z returns_z another_z as_o it_o be_v common_a he_o will_v find_v a_o great_a deal_n of_o difference_n in_o long_a run_v for_o to_o demonstrate_v the_o reason_n of_o this_o i_o shall_v not_o because_o my_o father_n and_o other_o have_v do_v that_o sufficient_o beside_o it_o will_v take_v up_o too_o much_o room_n in_o this_o little_a book_n these_o question_n of_o sail_v may_v be_v understand_v in_o such_o case_n as_o this_o be_v namely_o when_o a_o man_n sail_v the_o same_o way_n home_o as_o he_o do_v out_o but_o because_o the_o wind_n will_v not_o suffer_v a_o man_n to_o do_v so_o it_o be_v necessary_a to_o know_v some_o better_a way_n to_o sail_v by_o let_v a_o man_n return_v never_o so_o contrary_a to_o what_o he_o sail_v outward_o yet_o at_o all_o time_n to_o know_v how_o near_o or_o how_o far_o he_o may_v be_v off_o now_o there_o be_v no_o way_n better_o than_o mercator_n way_n of_o sail_v it_o be_v very_o excellent_a and_o have_v be_v treat_v of_o by_o many_o and_o that_o make_v i_o presume_v to_o think_v that_o i_o may_v treat_v of_o it_o as_o well_o as_o other_o and_o first_o here_o be_v the_o table_n of_o meridional_a part_n by_o which_o you_o be_v to_o work_v a_o table_n of_o meridional_a part_n to_o every_o three_o minute_n latitud_n mer._n part_n d._n m._n 0_o 3_o 3_o  _fw-fr 6_o 6_o  _fw-fr 9_o 9_o  _fw-fr 12_o 12_o  _fw-fr 15_o 15_o  _fw-fr 18_o 18_o  _fw-fr 21_o 21_o  _fw-fr 24_o 24_o  _fw-fr 27_o 27_o  _fw-fr 30_o 30_o  _fw-fr 33_o 33_o  _fw-fr 36_o 36_o  _fw-fr 39_o 39_o  _fw-fr 42_o 42_o  _fw-fr 45_o 45_o  _fw-fr 48_o 48_o  _fw-fr 51_o 51_o  _fw-fr 54_o 54_o  _fw-fr 57_o 57_o 1_o 0_o 60_o  _fw-fr 3_o 63_o  _fw-fr 6_o 66_o  _fw-fr 9_o 69_o  _fw-fr 12_o 72_o  _fw-fr 15_o 75_o  _fw-fr 18_o 78_o  _fw-fr 21_o 81_o  _fw-fr 24_o 84_o  _fw-fr 27_o 87_o  _fw-fr 30_o 90_o  _fw-fr 33_o 93_o  _fw-fr 36_o 96_o  _fw-fr 39_o 99_o  _fw-fr 42_o 102_o  _fw-fr 45_o 105_o  _fw-fr 48_o 108_o  _fw-fr 51_o 111_o  _fw-fr 54_o 114_o  _fw-fr 57_o 117_o 2_o 0_o 120_o  _fw-fr 3_o 123_o  _fw-fr 6_o 126_o  _fw-fr 9_o 129_o  _fw-fr 12_o 132_o  _fw-fr 15_o 135_o  _fw-fr 18_o 138_o  _fw-fr 21_o 141_o  _fw-fr 24_o 144_o  _fw-fr 27_o 147_o  _fw-fr 30_o 150_o  _fw-fr 33_o 153_o  _fw-fr 36_o 156_o  _fw-fr 39_o 159_o  _fw-fr 42_o 162_o  _fw-fr 45_o 165_o  _fw-fr 48_o 168_o  _fw-fr 51_o 171_o  _fw-fr 54_o 174_o  _fw-fr 57_o 177_o 3_o 0_o 180_o  _fw-fr 3_o 183_o  _fw-fr 6_o 186_o  _fw-fr 9_o 189_o  _fw-fr 12_o 192_o  _fw-fr 15_o 195_o  _fw-fr 18_o 198_o  _fw-fr 21_o 201_o  _fw-fr 24_o 204_o  _fw-fr 27_o 207_o  _fw-fr 30_o 210_o  _fw-fr 33_o 213_o  _fw-fr 36_o 216_o  _fw-fr 39_o 219_o  _fw-fr 42_o 222_o  _fw-fr 45_o 225_o  _fw-fr 48_o 228_o  _fw-fr 51_o 231_o  _fw-fr 54_o 234_o  _fw-fr 57_o 237_o 4_o 0_o 240_o  _fw-fr 3_o 243_o  _fw-fr 6_o 246_o  _fw-fr 9_o 249_o  _fw-fr 12_o 252_o  _fw-fr 15_o 255_o  _fw-fr 18_o 258_o  _fw-fr 21_o 262_o  _fw-fr 24_o 264_o  _fw-fr 27_o 267_o  _fw-fr 30_o 270_o  _fw-fr 33_o 273_o  _fw-fr 36_o 276_o  _fw-fr 39_o 279_o  _fw-fr 42_o 282_o  _fw-fr 45_o 285_o  _fw-fr 48_o 288_o  _fw-fr 51_o 291_o  _fw-fr 54_o 294_o  _fw-fr 57_o 297_o 5_o 0_o 300_o  _fw-fr 3_o 303_o  _fw-fr 6_o 306_o  _fw-fr 9_o 309_o  _fw-fr 12_o 312_o  _fw-fr 15_o 315_o  _fw-fr 18_o 318_o  _fw-fr 21_o 321_o  _fw-fr 24_o 324_o  _fw-fr 27_o 327_o  _fw-fr 30_o 330_o  _fw-fr 33_o 333_o  _fw-fr 36_o 337_o  _fw-fr 39_o 340_o  _fw-fr 42_o 343_o  _fw-fr 45_o 346_o  _fw-fr 48_o 349_o  _fw-fr 51_o 353_o  _fw-fr 54_o 355_o  _fw-fr 57_o 358_o 6_o 0_o 361_o  _fw-fr 3_o 364_o  _fw-fr 6_o 367_o  _fw-fr 9_o 370_o  _fw-fr 12_o 373_o  _fw-fr 15_o 376_o  _fw-fr 18_o 379_o  _fw-fr 21_o 382_o  _fw-fr 24_o 385_o  _fw-fr 27_o 388_o  _fw-fr 30_o 391_o  _fw-fr 33_o 394_o  _fw-fr 36_o 397_o  _fw-fr 39_o 400_o  _fw-fr 42_o 403_o  _fw-fr 45_o 406_o  _fw-fr 48_o 409_o  _fw-fr 51_o 412_o  _fw-fr 54_o 415_o  _fw-fr 57_o 418_o
of_o the_o compass_n it_o be_v to_o be_v set_v off_o in_o as_o also_o whether_o it_o be_v near_o the_o east_n or_o west_n than_o the_o north_n or_o south_n for_o if_o it_o be_v near_o the_o east_n or_o west_n than_o the_o north_n or_o south_n you_o must_v do_v the_o same_o from_o a_o parallel_n as_o here_o you_o do_v from_o a_o meridian_n for_o the_o side_n of_o a_o square_n be_v but_o the_o measure_n of_o four_o point_n which_o be_v but_o half_a the_o point_n between_o a_o meridian_n and_o a_o parallel_n this_o way_n of_o work_v may_v seem_v hard_a and_o tedious_a at_o first_o but_o you_o will_v soon_o find_v that_o it_o be_v free_a from_o mistake_n and_o both_o exact_a and_o easy_a if_o you_o practise_v it_o place_v this_o between_o page_n 112_o and_o 113._o of_o oblique_a triangle_n two_o side_n with_o a_o angle_n opposite_a to_o one_o of_o they_o give_v to_o find_v the_o other_o angles_n and_o side_n question_n i._n two_o ship_n set_v sail_n from_o the_o rock_n of_o lisbon_n one_o sail_v w_n s_o w_n the_o other_o sail_v n_o w_n b_o w_n 38_o league_n and_o at_o the_o end_n of_o their_o sail_v they_o be_v 58_o league_n asunder_o i_o demand_v the_o southermost_a ship_n distance_n run_v and_o how_o the_o ship_n bear_v one_o from_o the_o other_o first_o draw_v a_o north_n and_o south_n line_v white_a and_o then_o from_o that_o set_v off_o the_o northermost_a ship_n course_n make_v the_o rock_n or_o place_n set_v out_o the_o place_n in_o the_o north_n and_o south_n line_n that_o you_o draw_v that_o course_n from_o which_o be_v c_o upon_o this_o course_n set_v off_o 38_o league_n because_o the_o question_n say_v you_o sail_v 38_o league_n upon_o it_o and_o extend_v the_o side_n c_o a_o to_o the_o arch_n of_o 60_o deg_fw-la at_o it_o and_o set_v off_o five_o point_n from_o it_o to_o s_n and_o draw_v s_o c_o a_o white_a line_n which_o be_v w_n s_o w_n course_n for_o it_o be_v five_o point_n between_o the_o w_n s_o w_n and_o the_o n_z w_o b_o w_o this_o do_v take_v 58_o league_n from_o the_o scale_n of_o equal_a part_n and_o fix_v one_o foot_n of_o your_o compass_n in_o a_o and_o where_o the_o other_o intersect_v the_o course_n b_o c_o which_o it_o do_v in_o b_o there_o be_v the_o other_o ship_n then_o to_o measure_v the_o angle_n of_o the_o ship_n bear_v one_o from_o the_o other_o it_o be_v b_o and_z b_o c_z be_v an_z e_o n_o e_o line_n extend_v b_o a_o to_o the_o arch_n of_o 60_o deg_fw-la who_o centre_n be_v at_o b_o and_o see_v how_o many_o degree_n or_o point_n it_o be_v more_o northerly_a than_o a_o e_o n_o e_o line_n and_o so_o the_o other_o ship_n namely_o the_o ship_n at_o a_o bear_v from_o the_o ship_n at_o b_o then_o take_v the_o length_n of_o b_o c_o and_o apply_v it_o to_o your_o scale_n and_o see_v how_o many_o league_n or_o mile_n it_o be_v i_o have_v wrought_v this_o in_o league_n but_o i_o will_v work_v the_o rest_n in_o mile_n because_o it_o be_v more_o exact_a i_o find_v that_o the_o ship_n at_o a_o bear_v from_o b_o n_o e_o b_o n_o 45_o min._n easterly_n the_o distance_n run_v of_o the_o southermost_a ship_n be_v 70_o league_n by_o the_o table_n the_o proportion_n of_o this_o and_o all_o other_o of_o this_o kind_n be_v the_o same_o that_o hold_v in_o right_a angle_a triangle_n namely_o that_o the_o sine_fw-la of_o every_o angle_n be_v proportional_a to_o its_o opposite_a side_n or_o every_o side_n be_v proportional_a to_o the_o sine_fw-la of_o its_o opposite_a angle_n here_o we_o have_v give_v the_o side_n c_o a_o 38_o league_n the_o side_n a_o b_o 58_o league_n and_o the_o angle_n at_o c_z which_o be_v 5_o point_n or_o 56_o deg_fw-la 15_o min._n say_v then_o for_o the_o angle_n at_o b._n as_o a_o b_o 58_o league_n 580_o tem_n comp_n arith_n 7,2365719_o be_v to_o a_o c_o b_o 56_o deg_fw-la 15_o min._n sine_fw-la  _fw-fr 9,9198464_o so_o be_v a_o c_o 38_o league_n 380_o tem_n  _fw-fr 2,5797836_o to_o a_o b_o c_o sine_fw-la 33_o deg_fw-la 0_o min._n  _fw-fr 9,7362019_o which_o subtract_v from_o 6_o point_n or_o 67_o d._n 30_o m._n  _fw-fr 33_o 00_o remainder_n be_v the_o course_n from_o north_n 34_o d._n 30_o m._n which_o the_o ship_n a_o bear_v from_o the_o ship_n b_o which_o be_v n_n e_o b_o n_o 45_o min._n easterly_n for_o the_o distance_n run_v of_o the_o ship_n at_o b_o the_o side_n b_o c._n as_o sine_fw-la c_o 56_o deg_fw-la 15_o min._n comp_n arith_n 0,080153_o be_v to_o a_o b_o 58_o league_n  _fw-fr 2,763428_o so_o be_v sine_fw-la b_o a_o c_o 90_o deg_fw-la 45_o min._n take_v the_o sine_fw-la of_o the_o acute_a angle_n b_o a_o t_o 89_o d._n 15_o m._n 9,999961_o to_o b_o c_z 68_o 8_o 10_o  _fw-fr 2,843542_o the_o reason_n why_o you_o take_v the_o acute_a angle_n be_v because_o the_o table_n go_v no_o further_o than_o 90_o deg_fw-la neither_o indeed_o be_v any_o sine_fw-la beyond_o 90_o deg_fw-la but_o as_o my_o father_n say_v in_o his_o trigonometry_n p._n 2_o the_o sine_fw-la of_o a_o arch_n less_o than_o a_o quadrant_n be_v also_o the_o right_a sine_fw-la of_o a_o arch_n as_o much_o great_a than_o a_o quadrant_n so_o than_o the_o right_a sine_fw-la of_o the_o arch_n 90_o deg_fw-la 45_o min._n which_o be_v 45_o min._n great_a than_o a_o quadrant_n must_v be_v the_o right_n sine_fw-la of_o the_o arch_n 89_o deg_fw-la 15_o min._n which_o be_v as_o much_o less_o namely_o 45_o min._n the_o geometrical_a demonstration_n of_o it_o be_v there_o lay_v down_o you_o may_v ask_v how_o i_o come_v to_o know_v the_o angle_n b_o a_o c_o which_o be_v thus_o i_o find_v one_o of_o the_o other_o two_o angle_n namely_o b_o and_o the_o other_o i_o have_v give_v i_o i_o add_v they_o both_o together_o and_o that_o sum_n i_o subtract_v from_o 180_o deg_fw-la the_o remainder_n must_v then_o be_v the_o angle_n at_o a_o because_o 180_o deg_fw-la be_v the_o sum_n of_o the_o three_o angle_n of_o any_o right_n line_v triangle_n and_o if_o i_o subtract_v two_o of_o they_o from_o three_o of_o they_o there_o will_v remain_v one_o of_o they_o which_o be_v 90_o deg_fw-la 45_o min._n the_o three_o angle_n of_o a_o triangle_n give_v with_o one_o of_o the_o side_n to_o find_v the_o other_o two_o side_n question_n ii_o admit_v i_o set_v from_o a_o head-land_n lie_v in_o the_o latitude_n of_o 50_o deg_fw-la 00_o min._n north_n latitude_n and_o sail_v w_o s_o w_n 38_o league_n and_o then_o meet_v with_o a_o ship_n that_o come_v from_o a_o place_n which_o lie_v s_n s_o w_n from_o the_o head-land_n now_o this_o ship_n have_v sail_v n_o w._n i_o demand_v the_o distance_n of_o that_o place_n from_o the_o other_o and_o also_o the_o distance_n the_o ship_n have_v sail_v that_o come_v from_o the_o southermost_a place_n the_o distance_n between_o the_o two_o place_n be_v a_o b_o mile_n 58_o mile_n the_o distance_n that_o the_o ship_n sail_v be_v a_o o_o mile_n 44_o 4_o 10_o mile_n if_o you_o have_v occasion_n to_o find_v the_o latitude_n the_o place_n at_o a_o lie_v in_o let_v fall_v the_o perpendicular_a a_o n_n upon_o the_o south_n line_n that_o come_v from_o b_o and_o it_o cut_v it_o in_o n_n and_o leave_v the_o difference_n of_o latitude_n b_o n_n see_v how_o many_o mile_n it_o be_v and_o subtract_v it_o from_o 50_o deg_fw-la and_o you_o have_v your_o desire_n measure_v a_o n_n and_o you_o have_v the_o westing_n that_o a_o lie_v from_o the_o head-land_n b._n by_o the_o table_n here_o the_o angle_n at_o b_o be_v a_o angle_n of_o 4_o point_n or_o 45_o d._n 00_o m._n the_o angle_n at_o a_o be_v a_o angle_n of_o 6_o point_n or_o 67_o d._n 30_o m._n the_o angle_n at_o o_o be_v a_o angle_n of_o 6_o point_n or_o 67_o d._n 30_o m._n the_o side_n b_o o_fw-fr be_v 58_o mile_n  _fw-fr  _fw-fr first_n for_o the_o side_n a_o b._n the_o general_a rule_n say_v that_o the_o sine_fw-la of_o every_o angle_n be_v proportional_a to_o its_o opposite_a side_n then_o from_o this_o i_o conclude_v that_o b_o a_o shall_v be_v equal_a to_o b_o o_o because_o the_o angel_n opposite_a to_o they_o be_v equal_a and_o so_o you_o will_v find_v they_o for_o as_o sine_fw-la a_o 67_o deg_fw-la 30_o min._n comp_n arith_n 0,034384_o be_v to_o o_o b_o 58_o mile_n  _fw-fr 1,763428_o so_o be_v sine_fw-la o_fw-la 67_o deg_fw-la 30_o min._n  _fw-fr 9,965615_o to_o a_o b_o 58_o mile_n the_o two_o place_n distance_n 1,763428_o you_o see_v it_o be_v so_o exact_o for_o these_o three_o number_n add_v together_o and_o radius_fw-la cast_v away_o produce_v the_o same_o logarithm_n that_o 58_o take_v out_o of_o the_o book_n do_v and_o this_o question_n i_o do_v on_o