measure_v off_o all_o the_o ark_n that_o can_v be_v find_v by_o calculation_n with_o the_o demonstration_n of_o all_o the_o former_a proportion_n from_o p._n 52_o to_o 59_o to_o determine_v what_o hour_n be_v proper_a to_o all_o plain_n p._n 60_o to_o 61_o another_o method_n of_o inscribe_v the_o hour-line_n in_o all_o plain_n by_o a_o parallelogram_n p._n 62_o to_o draw_v the_o tangent_fw-la scheme_n suit_v thereto_o p._n 63._o the_o hour-line_n so_o inscribe_v in_o a_o horizontal_a and_o south_n dial_n p._n 63_o 64_o as_o also_o in_o a_o upright_a decliner_n p._n 65_o with_o another_o tangent_fw-la scheme_n suit_v thereto_o for_o prick_v they_o down_o without_o the_o use_n of_o compass_n p._n 66_o 67_o a_o general_a method_n without_o proportional_a work_n for_o fit_v the_o parallelogram_n into_o oblique_a plain_n that_o have_v the_o requisite_n first_o place_v p._n 69_o 70_o by_o help_n of_o three_o shadow_n to_o find_v a_o meridian-line_n p._n 70_o 71_o another_o scheme_n suit_v 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line_n be_v project_v on_o any_o regular_a flat_a will_v become_v straight_a line_n dial_n be_v by_o clavius_n in_o the_o second_o chapter_n of_o his_o gnomonic_n distinguish_v into_o seventeen_o kind_n 1._o the_o horizontal_a be_v parallel_n to_o the_o horizon_n 2._o south_n and_o north_n direct_v by_o some_o call_v a_o vertical_a dial_n because_o parallel_v to_o the_o prime_a vertical_a or_o circle_n of_o east_n and_o west_n 3._o 4._o south_n direct_a recline_a or_o north_n incline_v less_o more_o than_o the_o pole_n 5._o 6._o south_n direct_a incline_v or_o north_n recline_v less_o more_o than_o the_o equinoctial_a 7._o south_n direct_v recline_v or_o north_n incline_v to_o the_o pole_n call_v a_o direct_a polar_a plain_n because_o parallel_v thereto_o 8._o south_n direct_v incline_v or_o north_n recline_v to_o the_o equator_n call_v a_o equinoctial_a plain_n because_o parallel_v thereto_o 9_o south_n and_o north_n decline_a east_n or_o west_n 10._o east_n and_o west_n direct_v 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small_a part_n 3._o to_o draw_v a_o horizontal_a and_o vertical_a line_n upon_o a_o plain_n the_o ready_a and_o most_o certain_a way_n to_o do_v this_o especial_o if_o the_o plain_a lean_a downward_o from_o the_o zenith_n will_v be_v by_o help_n of_o a_o thread_n and_o plummet_n hold_v steady_o to_o make_v two_o prick_n at_o a_o competent_a distance_n in_o the_o shadow_n of_o the_o thread_n on_o the_o plain_a project_v by_o the_o eye_n and_o a_o line_n draw_v through_o those_o or_o parallel_n to_o those_o two_o point_n shall_v be_v the_o plain_n perpendicular_a or_o vertical_a line_n and_o a_o line_n draw_v perpendicular_a to_o the_o say_a line_n shall_v be_v the_o plain_n horizontal_a line_n to_o find_v the_o reclination_n of_o a_o plain_n the_o reclination_n of_o a_o plain_n be_v the_o angle_n comprehend_v between_o the_o plain_n perpendicular_a and_o the_o
be_v illustrate_v by_o one_o or_o two_o example_n the_o distance_n between_o the_o hour_n line_n of_o two_o and_o a_o quarter_n and_o three_o and_o three_o quarter_n will_v be_v equal_a to_o twice_o the_o distance_n between_o the_o hour_n line_n of_o twelve_o and_o one_o and_o a_o half_a also_o the_o distance_n between_o four_o and_o a_o quarter_n and_o four_o and_o a_o half_a be_v equal_a to_o twice_o the_o distance_n between_o three_o and_o a_o half_a and_o three_o and_o three_o quarter_n or_o it_o be_v equal_a to_o the_o distance_n between_o one_o and_o three_o quarter_n and_o two_o and_o three_o quarter_n but_o the_o former_a be_v near_a the_o truth_n it_o will_v be_v inconvenient_a in_o such_o plain_n as_o also_o in_o direct_a east_n or_o west_n dial_n to_o express_v any_o hour_n line_n from_o the_o substile_n beyond_o 75d_o or_o 5_o hour_n to_o these_o i_o may_v add_v some_o other_o observation_n which_o be_v communicate_v by_o doctor_n richard_n sterne_n to_o mr._n sutton_n 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north_n recliner_n or_o south_n incliner_n who_o reinclination_n be_v equal_a to_o the_o latitude_n be_v parallel_n to_o the_o equinoctial_a circle_n and_o be_v therefore_o call_v equinoctial_a dial_n there_o be_v no_o difficulty_n in_o describe_v of_o these_o divide_v a_o circle_n into_o 24_o equal_a part_n for_o the_o whole_a hour_n and_o afterwards_o into_o half_n and_o quarter_n and_o place_v the_o meridian_n line_n in_o the_o plain_n perpendicular_a assume_v as_o many_o of_o the_o former_a hour_n as_o the_o sun_n can_v shine_v upon_o for_o either_o face_n and_o then_o place_v a_o round_a wire_n in_o the_o centre_n for_o the_o style_n perpendicular_a to_o the_o plain_n and_o the_o dial_n be_v finish_v a_o south_n plain_a decline_a 30d_o east_n latitude_n 51-32′_n page_n 13_o page_n 17_o a_o south_n dial_n declin_n 30d_o east_n latitude_n 51-32′_n to_o prick_v off_o the_o substile_n and_o stile_n height_n on_o upright_a decliner_n in_o their_o true_a coast_n and_o quantity_n on_o 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whence_o draw_v a_o line_n into_o the_o centre_n and_o the_o angle_z juk_n will_v show_v how_o much_o the_o stile_n be_v to_o be_v elevate_v above_o the_o substilar_a line_n in_o order_n to_o the_o demonstration_n hereof_o let_v it_o be_v observe_v that_o a_o angle_n may_v be_v prick_v off_o by_o sine_n or_o tangent_n in_o stead_n of_o chord_n to_o prick_v off_o a_o angle_n by_o sine_n or_o tangent_n as_o in_o the_o scheme_n annex_v let_v bc_o be_v radius_fw-la and_o let_v there_o be_v a_o arch_n prick_v off_o with_o chord_n as_o be_v i_o say_v if_o the_o tangent_fw-la of_o the_o say_a ark_n ba_o be_v take_v out_o of_o a_o line_n of_o tangent_n to_o the_o same_o radius_fw-la and_o be_v erect_v perpendicular_a to_o the_o end_n of_o the_o radius_fw-la as_o ba_o a_o line_n draw_v from_o a_o into_o the_o centre_n shall_v include_v the_o same_o angle_n as_o be_v prick_v off_o by_o chord_n as_o be_v evident_a from_o the_o definition_n of_o a_o tangent_fw-la in_o like_a manner_n a_o angle_n may_v be_v prick_v off_o by_o sine_n the_o near_a distance_n from_o e_o to_o bc_o be_v the_o sine_fw-la of_o the_o arch_n be_v so_o in_o like_o manner_n the_o near_a distance_n from_o b_o to_o aec_o be_v the_o sine_fw-la of_o the_o same_o arch_n wherefore_o if_o with_o the_o sine_fw-la of_o a_o arch_n from_o the_o end_n of_o the_o radius_fw-la be_v describe_v another_o ark_n as_o fletcher_o and_o from_o the_o centre_n or_o other_o extremity_n of_o the_o radius_fw-la a_o line_n draw_v just_a touch_v the_o same_o the_o angle_n include_v between_o the_o say_a line_n and_o the_o radius_fw-la shall_v be_v a_o ark_n equal_a to_o the_o ark_n belong_v to_o the_o say_v sine_fw-la and_o what_o be_v here_o do_v by_o a_o line_n of_o natural_a sine_n or_o tangent_n by_o help_n of_o a_o decimal_n line_n of_o equal_a part_n equal_a to_o the_o radius_fw-la radius_fw-la may_v be_v do_v by_o help_n of_o the_o natural_a table_n without_o they_o any_o proportion_n relate_v to_o the_o sixteen_o case_n of_o sphoerical_a triangle_n among_o which_o the_o radius_fw-la be_v always_o ingredient_n may_v be_v so_o vary_v that_o the_o radius_fw-la may_v be_v in_o the_o three_o place_n and_o a_o tangent_fw-la or_o a_fw-la sine_fw-la in_o the_o four_o place_n and_o then_o if_o the_o ark_n belong_v to_o the_o four_o proportional_a be_v know_v a_o angle_n equal_a thereto_o may_v be_v prick_v off_o by_o tangent_n or_o sine_n according_a to_o the_o nature_n of_o the_o four_o term_n as_o beforth_o if_o it_o be_v unknown_a notwithstanding_o a_o angle_n equal_a thereto_o may_v be_v prick_v off_o with_o sine_n or_o tangent_n according_a to_o the_o nature_n of_o the_o four_o term_n from_o the_o two_o first_o term_n of_o the_o proportion_n because_o they_o
and_o nk_o be_v go_o because_o kl_o be_v lg_o now_o in_o the_o equiangle_a triangle_n abh_o aog_n as_o bh_o ha_o ∷_o og_o ga_o and_o in_o the_o triangle_n ahe_o agk_o as_o ha_o he_o ∷_o ga_o to_o gk_o therefore_o ex_fw-la equo_fw-la it_o will_v be_v as_o bh_o he_o ∷_o og_o gk_o but_o as_o bh_o he_o ∷_o by_o im_o for_o ih_o be_v parallel_n to_o em_o therefore_o as_o by_o in_o ∷_o og_o gk_o but_o nk_o be_v og_o therefore_o by_o in_o ∷_o nk_o kg_o but_o i_o be_v half_a of_o im_o and_o kl_o half_a of_o kg_o therefore_o by_o i_o ∷_o nk_o kl_o therefore_o by_o composition_n of_o proportion_n as_o bc_o i_o ∷_o nl_o kl_o alternate_o as_o bc_o nl_o ∷_o i_o kl_o in_o the_o second_o figure_n bc_o be_v nl_o therefore_o i_o be_v kl_o wherefore_o because_o i_o be_v also_o parallel_a to_o kl_o ik_o be_v parallel_n to_o cl_o by_o like_o reason_n gm_o be_v also_o parallel_a to_o cl_o but_o in_o the_o three_o and_o four_o figure_n the_o side_n pc_o pl_z of_o the_o triangle_n pcl_n be_v cut_v proportional_o by_o the_o parallel_n bn_o and_o as_o before_o as_o bc_o nl_o ∷_o i_o kl_o and_o as_o pc_o pl_z ∷_o bc_o nl_o therefore_o pc_o pl_z ∷_o i_o kl_o wherefore_o ik_o be_v parallel_n to_o cl_o again_o as_o before_o bc_o nl_o ∷_o i_o kl_o but_o cm_o be_v i_o and_o ig_o be_v kl_o therefore_o as_o bc_o nl_o ∷_o cm_o lg_o but_o as_o have_v be_v say_v as_o bc_o nl_o ∷_o pc_o pl_z therefore_o as_o pc_o pl_z ∷_o cm_o lg_o wherefore_o because_o the_o side_n pm_o pg_o of_o the_o triangle_n pmg_o be_v cut_v proportional_o in_fw-la c_o and_o l_o cl_o and_o mg_o be_v parallel_v one_o to_o another_o now_o in_o all_o the_o three_o figure_n it_o have_v be_v prove_v that_o ik_o and_o mg_o be_v either_o of_o they_o parallel_n to_o cl_o therefore_o they_o be_v parallel_v one_o to_o another_o wherefore_o the_o angle_n gmi_fw-la be_v equal_a to_o the_o angle_n kib_o which_o before_o be_v prove_v to_o be_v equal_a to_o amf_n therefore_o the_o angle_n gmi_fw-la and_o amf_n be_v equal_a one_o to_o another_o wherefore_o fm_o and_o mg_o be_v one_o and_o the_o same_o right_a line_n which_o be_v to_o be_v prove_v to_o draw_v a_o east_n or_o west_n dial_n let_v the_o hour_n line_n of_o six_o which_o be_v also_o the_o substilar_a line_n make_v a_o angle_n equal_a to_o the_o latitude_n of_o the_o place_n with_o the_o plain_n horizontal_a line_n above_o that_o end_n of_o it_o that_o point_v to_o the_o coast_n of_o the_o elevate_a pole_n then_o draw_v a_o line_n perpendicular_a to_o the_o substilar_a line_n which_o some_o call_v a_o contingent_a line_n and_o with_o such_o a_o radius_fw-la as_o you_o determine_v the_o style_n shall_v have_v parallel_a height_n above_o the_o substile_n divide_v a_o tangent_fw-la line_n of_o hour_n and_o quarter_n according_a to_o the_o direction_n for_o direct_v polar_a recliner_n then_o through_o those_o division_n draw_v line_n parallel_v to_o the_o substile_n and_o they_o shall_v be_v the_o hour_n line_n require_v thus_o the_o hour_n of_o 5_o and_o 7_o be_v each_o of_o they_o tangent_n of_o 15d_o from_o six_o and_o so_o for_o the_o rest_n let_v they_o be_v number_v on_o each_o side_n of_o six_o be_v the_o substile_n for_o a_o east_n dial_n with_o the_o morning_n hour_n for_o a_o west_n dial_n with_o the_o afternoon_n hour_n the_o one_o be_v the_o compliment_n of_o the_o other_o to_o 12_o hour_n and_o therefore_o we_o have_v but_o one_o example_n namely_o a_o east_n dial_n for_o the_o latitude_n of_o london_n how_o to_o fill_v this_o or_o any_o other_o plain_a with_o any_o determine_a number_n of_o hour_n shall_v afterwards_o be_v handle_v a_o west_n dial_n recline_v 50d_o latitude_n 51-32′_n a_o east_n dial_n incline_v 40d_o latitude_n 50d_o to_o prick_v off_o the_o requisite_n of_o a_o east_n or_o west_n recline_v or_o incline_v dial_n in_o their_o true_a situation_n and_o quantity_n in_o these_o plain_n first_o draw_v the_o plain_n perpendicular_a and_o cross_v it_o with_o a_o horizontal_a line_n which_o be_v also_o the_o meridian_n line_n in_o any_o one_o of_o the_o quarter_n make_v a_o geometrical_a square_n as_o before_o direct_v for_o upright_a decliner_n and_o from_o the_o plain_n vertical_a in_o the_o say_a quarter_n count_v off_o the_o latitude_n to_o l_o and_o from_o the_o horizontal_a line_n in_o the_o say_a quarter_n count_v off_o the_o reclination_n or_o inclination_n to_o r_o and_o from_o l_o and_o r_o draw_v line_n into_o the_o centre_n and_o where_o the_o latitude_n line_n intersect_v the_o side_n of_o the_o square_a set_v f._n 1._o to_o prick_v off_o the_o substile_n for_o latitude_n above_o 45d_o place_n bf_o from_o v_o the_o centre_n in_o the_o horizontal_a line_n for_o recliner_n northwards_o for_o incliner_n southwards_o and_o thereto_o set_v o_o and_o through_o the_o say_a point_n draw_v a_o line_n parallel_n to_o the_o vertical_a and_o place_v the_o near_a distance_n from_o a_o to_o rv_o on_o the_o former_a parallel_n from_o o_o for_o recliner_n upward_o to_o i_o but_o for_o incliner_n downward_o and_o a_o line_n thence_o draw_v into_o the_o centre_n shall_v be_v the_o substile_n 2._o the_o style_v height_n place_v the_o near_a distance_n from_o b_o to_o rv_o on_o a_o perdendicular_a raise_v from_o the_o point_n i_o in_o the_o substile_n and_o it_o find_v the_o point_n k_o whence_o a_o line_n draw_v into_o the_o centre_n shall_v represent_v the_o style_n in_o the_o other_o hemisphere_n the_o word_n northwards_o and_o southwards_o must_v be_v mutual_o change_v for_o latitude_n under_o 45′_n the_o side_n of_o the_o square_n must_v be_v place_v from_o five_o to_o o_o and_o af_o must_v be_v place_v on_o the_o line_n of_o reclination_n or_o inclination_n from_o five_o to_o c_o the_o near_a distance_n from_o c_o to_o va_o place_v on_o the_o perpendicular_a pass_v through_o o_o for_o recliner_n upward_o but_o for_o incliner_n downward_o find_v the_o point_n i_o through_o which_o the_o substile_n be_v to_o pass_v and_o the_o near_a distance_n from_o c_o to_o ub_z raise_v perpendicular_o on_o the_o point_n i_o in_o the_o substile_n find_v the_o point_n k_o for_o the_o style_n as_o before_o demonstration_n 1._o for_o the_o substile_n in_o these_o plain_n ub_z or_o va_o being_n radius_fw-la fb_fw-la be_v the_o contangent_fw-la of_o the_o latitude_n and_o the_o near_a distance_n from_o a_o to_o rv_o be_v the_o cousin_n of_o reclination_n or_o inclination_n to_o the_o same_o radius_fw-la and_o the_o near_a distance_n from_o b_o to_o rv_o the_o sine_fw-la thereof_o this_o for_o latitude_n above_o 45_o but_o for_o lesser_a latitude_n af_o the_o tangent_fw-la of_o the_o latitude_n be_v make_v radius_fw-la and_o thereupon_o the_o radius_fw-la or_o side_n of_o the_o square_a be-became_v the_o cotangent_fw-la of_o the_o latitude_n and_o the_o near_a distance_n from_o c_o to_o va_o be_v the_o cousin_n and_o from_o c_o to_o ub_z the_o sine_fw-la of_o the_o reclination_n or_o inclination_n so_o that_o the_o prescribe_a construction_n in_o both_o case_n erect_v the_o cousin_n of_o the_o reinclination_n on_o the_o cotangent_fw-la of_o the_o latitude_n which_o be_v make_v good_a from_o this_o proportion_n as_o the_o cotangent_fw-la of_o the_o latitude_n be_v to_o the_o cousin_n of_o the_o reinclination_n from_o the_o zenith_n so_o be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o substilar_n from_o the_o meridian_n 2._o for_o the_o style_n a_o proportion_n that_o will_v serve_v to_o prick_v it_o off_o be_v as_o the_o cosecant_v of_o the_o latitude_n be_v to_o the_o sine_fw-la of_o the_o reinclination_n so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o style_n height_n page_n 25_o a_o west_n dial_n recline_v 50d_o latitude_n 51_o 32′_o a_o south_n plain_a decline_a 40d_o east_n incline_v 15_o deg_n a_o west_n plain_n recline_v 50_o degree_n latitude_n 51_o degree_n 32_o minute_n the_o point_n sol_fw-la substile_n style_n and_o hour-line_n be_v all_o find_v after_o the_o same_o manner_n as_o in_o a_o upright_a decliner_n the_o substile_n be_v set_v off_o from_o the_o meridian_n line_n here_o after_o the_o same_o manner_n as_o it_o be_v from_o the_o meridian_n there_o a_o south_n plain_a decline_a east_n 40_o degree_n incline_v 20_o degree_n latitude_n 51_o degree_n 32_o minute_n to_o prick_v off_o the_o requisite_n in_o all_o decline_a recline_a and_o incli_v plain_n in_o their_o true_a coast_n and_o quantity_n now_o follow_v those_o direction_n enlarge_v which_o as_o i_o say_v in_o the_o epistle_n i_o receive_v from_o mr._n thomas_n rice_n upon_o any_o plain_n first_o draw_v a_o true_a horizontal_a line_n at_o the_o east_n end_n thereof_o set_v e_o and_o at_o the_o west_n end_n w_o across_o the_o say_a line_n with_o a_o perpendicular_a which_o may_v be_v call_v
the_o plain_n perpendicular_a by_o some_o term_v the_o plain_n vertical_a line_n or_o line_n of_o reclination_n at_o the_o intersection_n of_o these_o two_o line_n set_v v_o and_o upon_o it_o as_o a_o centre_n describe_z a_o circle_n the_o radius_fw-la whereof_o may_v be_v equal_a to_o 60d_o of_o a_o line_n of_o chord_n at_o the_o upper_a end_n of_o the_o vertical_a line_n set_v s_o and_o at_o the_o low_a end_n n._n as_o the_o declination_n be_v set_v it_o off_o with_o chord_n from_o s_o or_o n_o towards_o the_o true_a coast_n and_o at_o it_o set_v d_o from_o whence_o draw_v a_o line_n through_o the_o centre_n also_o set_v off_o the_o latitude_n of_o the_o place_n the_o same_o way_n and_o in_o the_o same_o quarter_n and_o at_o it_o set_v l_o from_o which_o draw_v a_o line_n to_o the_o centre_n from_o that_o end_n of_o the_o horizontal_a line_n towards_o which_o the_o declination_n be_v count_v set_v the_o inclination_n which_o as_o well_o as_o the_o reclinais_n reckon_v from_o the_o zenith_n the_o former_a be_v the_o denomination_n of_o the_o under_o the_o latter_a of_o the_o upper_a face_n upward_o towards_o s_o and_o the_o reclination_n downward_o towards_o n_o and_o at_o it_o set_v r_o from_o whence_o draw_v a_o line_n through_o the_o centre_n to_o the_o other_o side_n of_o the_o circle_n in_o the_o same_o quarter_n of_o declination_n draw_v ha_o parallel_n to_o the_o horizontal_a line_n and_o fg_o parallel_n to_o the_o vertical_a line_n in_o a_o geometrical_a square_n of_o like_a and_o of_o any_o convenient_a distance_n from_o the_o centre_n at_o pleasure_n and_o where_o the_o latitude_n line_n intersect_v the_o side_n of_o the_o square_a let_v the_o letter_n f_o be_v place_v on_o the_o line_n of_o declination_n beyond_o the_o centre_n make_v vo_o equal_a to_o fg_o and_o draw_v ob_o parallel_n to_o the_o horizontal_a line_n continue_v till_o it_o meet_v with_o the_o side_n of_o the_o square_a fg_o produce_v at_o the_o point_n of_o concurrence_n set_v b_o and_o where_o it_o intersect_v the_o plain_n perpendicular_a set_v p_o and_o draw_v oc_o parallel_n to_o the_o vertical_a line_n cut_v we_o at_o c_o and_o make_v ha_o equal_a to_o oc_o or_o bg_o now_o by_o help_n of_o the_o three_o point_n a_o b_o c_o thus_o find_v the_o requisite_n will_v be_v easy_o prick_v off_o 1._o the_o substilar_a line_n the_o near_a distance_n from_o a_o to_o rv_o set_v on_o the_o line_n co_z produce_v if_o need_v be_v from_o c_o to_o i_o the_o same_o way_n the_o distance_n be_v take_v from_o a_o that_o be_v if_o downward_o or_o upward_o the_o other_o must_v be_v so_o too_o will_v show_v where_o vi_o the_o substilar_a line_n be_v to_o be_v draw_v 2._o the_o style_v height_n the_o least_o distance_n from_o b_o to_o rv_o set_v on_o a_o perpendicular_a raise_v upon_o the_o substile_n from_o the_o point_n i_o will_v find_v the_o point_n k_o from_o whence_o draw_v a_o line_n into_o the_o centre_n at_o v_o and_o the_o angle_z juk_n will_v show_v how_o much_o the_o style_n be_v elevate_v above_o the_o substile_n and_o if_o the_o work_n be_v true_a vk_o and_o vf_o will_v be_v equal_a whence_o it_o follow_v that_o the_o trouble_n of_o raise_v the_o mention_v perpendicular_a may_v be_v shun_v 3._o the_o meridian_n line_n the_o least_o distance_n from_o c_o to_o rv_o set_v upon_o the_o line_n opb_n from_o p_o on_o that_o side_n which_o be_v far_a from_o the_o line_n rv_o will_v find_v the_o point_n m_o from_o whence_o a_o line_n draw_v into_o the_o centre_n shall_v be_v the_o meridian_n line_n and_o i_o add_v that_o on_o all_o north_n recliner_n in_o the_o northern_a hemisphere_n the_o meridian_n line_n must_v be_v draw_v through_o the_o centre_n on_o the_o other_o side_n and_o than_o the_o construction_n of_o the_o scheme_n will_v place_v it_o below_o the_o plain_n horizontal_a line_n which_o be_v its_o proper_a situation_n for_o the_o say_a upper_a face_n and_o for_o the_o under_o face_n the_o scheme_n place_v it_o true_a without_o caution_n 4._o a_o polar_a plain_n how_o know_v if_o the_o line_n rv_o fall_v just_a into_o b_o the_o plain_n be_v a_o polar_a plain_n in_o such_o a_o plain_a the_o style_n have_v no_o height_n but_o be_v parallel_n to_o the_o axis_n in_o this_o case_n the_o inclination_n of_o meridian_n must_v be_v know_v direction_n for_o such_o plain_n must_v afterwards_o follow_v but_o if_o the_o line_n rv_o fall_v between_o the_o point_n b_o and_o p_o then_o must_v the_o substile_n style_n and_o meridian_n be_v all_o draw_v through_o the_o centre_n and_o stand_v beyond_o on_o the_o other_o side_n annotation_n on_o the_o former_a scheme_n 1._o that_o for_o latitude_n under_o 45d_o this_o construction_n of_o the_o scheme_n suppose_v the_o side_n of_o the_o square_a produce_v which_o will_v therefore_o be_v liable_a to_o large_a excursion_n or_o other_o inconvenience_n wherefore_o for_o such_o latitude_n i_o shall_v somewhat_o vary_v from_o the_o construction_n prescribe_v 2._o in_o find_v the_o substilar_a line_n in_o stead_n of_o erect_v ci_o upon_o vc_o you_o may_v prick_v the_o same_o on_o the_o vertical_a line_n vn_fw-fr and_o thereto_o erect_v vc_o and_o get_v the_o point_n i_o possible_o with_o more_o certainty_n by_o find_v the_o intersection_n of_o two_o ark_n where_o the_o say_a point_n be_v to_o pass_v 3._o in_o prick_v off_o the_o meridian_n line_n the_o distance_n of_o c_o from_o the_o centre_n may_v be_v double_v or_o triple_a but_o so_o must_v likewise_o up_o and_o the_o near_a distance_n from_o c_o to_o rv_o erect_v on_o a_o line_n draw_v parallel_n to_o we_o pass_v through_o the_o point_n p_o so_o find_v and_o in_o stead_n of_o draw_v such_o a_o parallel_n the_o point_n m_o may_v be_v find_v by_o the_o intersection_n of_o two_o arks_n 4._o that_o this_o scheme_n place_v the_o requisite_n of_o all_o dial_n in_o their_o true_a coast_n and_o quantity_n yet_o notwithstanding_o if_o this_o scheme_n be_v hold_v before_o a_o looking-glass_n the_o effigy_n thereof_o in_o the_o glass_n show_v how_o the_o scheme_n will_v happen_v and_o place_v the_o requisite_n namely_o the_o style_n substile_n and_o meridian_n for_o a_o plain_n of_o the_o same_o denomination_n but_o decline_v to_o the_o contrary_a coast_n and_o if_o the_o face_n of_o the_o say_a scheme_n be_v lay_v upon_o a_o window_n and_o the_o substile_n style_n and_o meridian_n be_v continue_v through_o the_o centre_n on_o the_o backside_n thereof_o it_o show_v you_o how_o these_o requisite_n be_v to_o be_v place_v on_o the_o opposite_a side_n of_o the_o plain_a which_o be_v do_v may_v be_v hold_v before_o a_o looking-glass_n as_o before_o and_o will_v be_v represent_v for_o the_o contrary_a declination_n of_o that_o opposite_a face_n the_o truth_n of_o all_o which_o will_v be_v confirm_v from_o the_o scheme_n itself_o this_o scheme_n for_o decline_a recline_a or_o incline_v plain_n use_v a_o new_a method_n of_o calculation_n derive_v from_o a_o oblique_a triangle_n in_o the_o sphere_n wherein_o there_o be_v two_o side_n with_o the_o angle_n comprehend_v give_v to_o find_v both_o the_o other_o angle_n which_o be_v reduce_v by_o a_o perpendicular_a to_o two_o right_a angle_a triangle_n from_o which_o the_o follow_a proportion_n be_v derive_v i_o shall_v therefore_o first_o deliver_v the_o say_a method_n then_o demonstrate_v that_o the_o say_a proportion_n be_v carry_v on_o in_o the_o scheme_n and_o last_o from_o the_o sphere_n show_v how_o those_o proportion_n do_v arise_v 1._o to_o find_v a_o polar_a plain_n reclination_n or_o inclination_n as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o plain_n declination_n so_o be_v the_o cotangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o reclination_n or_o inclination_n seek_v 2._o to_o find_v the_o distance_n of_o the_o substile_n or_o meridian_n line_n from_o the_o plain_n perpendicular_a for_o a_o polar_a plain_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o a_o polar_a plain_n reclination_n so_o be_v the_o tangent_fw-la of_o the_o declination_n to_o the_o tangent_fw-la of_o the_o substilar_a line_n from_o the_o plain_n perpendicular_a 3._o the_o inclination_n of_o meridian_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o latitude_n so_o be_v the_o tangent_fw-la of_o the_o declination_n to_o the_o tangent_fw-la of_o the_o inclination_n of_o meridian_n affection_n of_o a_o polar_a plain_n the_o substilar_n on_o the_o upper_a face_n lie_v above_o that_o end_n of_o the_o horizontal_a line_n towards_o the_o coast_n of_o declination_n and_o the_o meridian_n lie_v parallel_n to_o the_o substile_n beyond_o it_o towards_o that_o end_n of_o the_o horizontal_a line_n that_o be_v towards_o the_o coast_n of_o declination_n for_o decline_a recline_a or_o incline_v plain_n first_o find_v a_o polar_a plain_n reclination_n for_o the_o same_o
declination_n then_o for_o south_n recliner_n and_o north_n incliner_n get_v the_o difference_n but_o for_o north_n recliner_n and_o south_n incliner_n the_o sum_n of_o a_o polar_a plain_n reclination_n and_o of_o the_o re_z inclination_n of_o the_o plain_a propose_v and_o then_o it_o hold_v 1._o for_o the_o substile_n as_o the_o cousin_n of_o the_o say_a ark_n of_o difference_n or_o sum_n according_a as_o the_o plain_n lean_v northward_o or_o southwards_o be_v to_o the_o sine_fw-la of_o the_o polar_a plain_n reclination_n so_o be_v the_o tangent_fw-la of_o the_o declination_n to_o the_o tangent_fw-la of_o the_o substilar_n from_o the_o plain_n perpendicular_a 2._o for_o the_o style_n height_n as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o substile_n distance_n from_o the_o plain_n perpendicular_a so_o be_v the_o tangent_fw-la of_o the_o sum_n or_o difference_n of_o reclination_n as_o before_o limit_v to_o the_o tangent_fw-la of_o the_o style_n height_n 3._o meridian_n distance_n from_o the_o plain_n perpendicular_a as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o re_z inclination_n so_o be_v the_o tangent_fw-la of_o the_o declination_n to_o the_o tangent_fw-la of_o the_o meridian_n from_o the_o plain_n perpendicular_a 4._o inclination_n of_o meridian_n as_o the_o sine_fw-la of_o the_o style_n height_n be_v to_o the_o tangent_fw-la of_o the_o distance_n between_o the_o meridian_n and_o substile_n so_o be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o inclination_n of_o meridian_n for_o south_n recliner_n or_o north_n incliner_n the_o difference_n between_o the_o substile_n distance_n from_o the_o plain_n perpendicular_a and_o the_o meridian_n distance_n therefrom_o be_v equal_a to_o the_o distance_n between_o the_o meridian_n and_o substile_n the_o like_a for_o such_o north_n recliner_n or_o south_n incliner_n as_o recline_v or_o incline_v more_o than_o a_o equinoctial_a plain_n have_v the_o same_o declination_n but_o if_o they_o lean_v above_o it_o or_o have_v a_o lesser_a reclination_n the_o sum_n be_v the_o distance_n between_o the_o meridian_n and_o the_o substile_n the_o three_o first_o proportion_n beside_o the_o find_n of_o a_o polar_a reclination_n be_v use_v in_o the_o scheme_n for_o the_o place_n of_o the_o requisite_n and_o the_o latter_a proportion_n in_o the_o circular_a scheme_n for_o draw_v the_o hour_n another_o proportion_n for_o find_v the_o inclination_n of_o meridian_n by_o calculation_n be_v as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o sine_fw-la of_o the_o substile_n distance_n from_o the_o plain_n perpendicular_a so_o be_v the_o cousin_n of_o the_o re_z inclination_n of_o the_o plain_a to_o the_o sine_fw-la of_o the_o inclination_n of_o meridian_n the_o reclination_n of_o a_o equinoctial_a plain_n to_o any_o assign_a declination_n be_v necessary_a for_o the_o determine_n of_o divers_a affection_n the_o proportion_n to_o find_v it_o be_v as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o plain_n declination_n so_o be_v the_o tangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o reclination_n seek_v the_o upper_a face_n of_o a_o equinoctial_a plain_n be_v call_v a_o north_n recliner_n the_o meridian_n descend_v from_o the_o end_n of_o the_o horizontal_a line_n opposite_a to_o the_o coast_n of_o declination_n the_o substilar_a line_n be_v the_o hour-line_n of_o six_o and_o make_v right_a angle_n with_o the_o meridian_n line_n direction_n for_o the_o true_a scituate_n of_o the_o meridian_n and_o substile_n suit_v to_o the_o former_a method_n of_o calculation_n 1._o for_o plain_n lean_v northwards_o if_o a_o south_n plain_n recline_v more_o than_o a_o polar_a plain_n have_v the_o same_o declination_n the_o plain_n pass_v beneath_o the_o pole_n of_o the_o world_n the_o north_n pole_n be_v elevate_v upon_o the_o upper_a face_n the_o substile_n and_o meridian_n line_n lie_v above_o that_o end_n of_o the_o plain_n horizontal_a line_n towards_o the_o coast_n of_o declination_n the_o substilar_a line_n be_v next_o the_o plain_n perpendicular_a for_o the_o under_o face_n be_v a_o north_n incliner_n the_o south_n pole_n be_v elevate_v the_o line_n lie_v in_o the_o same_o position_n below_o the_o plain_n horizontal_a line_n and_o on_o the_o contrary_a side_n of_o the_o plain_n perpendicular_a if_o a_o south_n plain_n reclination_n be_v less_o than_o the_o polar_a plain_n reclination_n the_o plain_n pass_v above_o the_o pole_n and_o the_o north_n pole_n be_v elevate_v on_o the_o under_o face_n be_v the_o incline_a side_n the_o substile_n and_o meridian_n lie_v above_o that_o end_n of_o the_o plain_n horizontal_a line_n that_o be_v opposite_a to_o the_o coast_n of_o declination_n the_o meridian_n be_v near_a the_o plain_n perpendicular_a for_o the_o upper_a face_n be_v a_o south_n recliner_n the_o south_n pole_n be_v elevate_v and_o the_o line_n lie_v in_o the_o same_o position_n below_o the_o plain_n horizontal_a line_n but_o on_o the_o contrary_a side_n of_o the_o plain_n perpendicular_a descend_n below_o that_o end_n of_o the_o horizontal_a line_n opposite_a to_o the_o coast_n of_o declination_n 2._o for_o plain_n lean_v southwards_o general_o on_o the_o upper_a face_n the_o north_n pole_n be_v elevate_v on_o the_o under_o face_n the_o south_n pole_n to_o place_v the_o substile_n such_o north_n recliner_n who_o reclination_n be_v less_o than_o the_o compliment_n of_o a_o polar_a plain_n reclination_n the_o substile_n be_v elevate_v above_o the_o end_n of_o the_o horizontal_a line_n contrary_a to_o the_o coast_n of_o declination_n and_o on_o the_o under_o face_n be_v a_o south_n incliner_n the_o substilar_n be_v depress_v below_o the_o end_n of_o the_o horizontal_a line_n opposite_a to_o the_o coast_n of_o declination_n but_o when_o the_o reclination_n be_v more_o than_o the_o compliment_n of_o the_o reclination_n of_o a_o polar_a plain_n the_o substile_n be_v to_o lie_v below_o the_o plain_n horizontal_a line_n from_o that_o end_n opposite_a to_o the_o coast_n of_o declination_n but_o for_o south_n incliner_n be_v the_o under_o face_n the_o substile_n be_v elevate_v above_o the_o end_n of_o the_o horizontal_a line_n opposite_a to_o the_o coast_n of_o declination_n to_o place_v the_o meridian_n on_o all_o north_n recliner_n the_o meridian_n lie_v below_o the_o horizontal_a line_n from_o that_o end_n thereof_o opposite_a to_o the_o coast_n of_o declination_n because_o at_o noon_n the_o sun_n be_v south_n cast_v the_o shadow_n of_o the_o style_n to_o the_o northwards_o on_o the_o under_o face_n be_v a_o south_n incliner_n it_o must_v always_o be_v place_v below_o the_o horizontal_a line_n below_o that_o end_n of_o it_o towards_o the_o coast_n of_o declination_n these_o direction_n suppose_v the_o declination_n to_o be_v denominate_v from_o the_o situation_n of_o that_o face_n of_o the_o plain_a on_o which_o the_o dial_n be_v to_o be_v make_v and_o the_o horizontal_a line_n for_o all_o dial_n that_o have_v centre_n be_v suppose_v to_o pass_v through_o the_o same_o now_o to_o the_o demonstration_n of_o the_o former_a scheme_n 1._o it_o be_v assert_v that_o if_o rv_o fall_v into_o the_o point_n b_o the_o plain_a be_v a_o polar_a plain_n in_o which_o case_n the_o style_n be_v parallel_n to_o the_o axis_n of_o the_o world_n demonstration_n every_o decline_a plain_n may_v have_v such_o a_o reclination_n find_v thereto_o as_o shall_v make_v the_o say_a plain_n become_v a_o polar_a plain_n and_o the_o proportion_n to_o find_v it_o may_v be_v thus_o as_o the_o tangent_fw-la of_o the_o latitude_n be_v to_o the_o cousin_n of_o the_o declination_n so_o be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o reclination_n seek_v in_o the_o former_a scheme_n if_o we_o make_v fg_o the_o cotangent_fw-la of_o the_o latitude_n radius_fw-la the_o side_n of_o the_o square_n will_v be_v the_o tangent_fw-la of_o the_o latitude_n now_o vo_o equal_a to_o fg_o be_v radius_fw-la oc_o equal_a to_o gb_o be_v the_o cousin_n of_o the_o declination_n wherefore_o a_o line_n draw_v into_o the_o centre_n from_o b_o shall_v include_v the_o angle_n of_o a_o polar_a plain_n reclination_n agreeable_a to_o the_o two_o first_o term_n of_o the_o proportion_n and_o to_o the_o direction_n for_o prick_v off_o a_o angle_n by_o tangent_n 2._o that_o the_o substile_n be_v true_a prick_v off_o upon_o v_o as_o a_o centre_n with_o the_o radius_fw-la ub_z imagine_v or_o describe_v a_o circle_n than_o be_v bg_o equal_a to_o up_o the_o sine_fw-la of_o a_o polar_a plain_n reclination_n which_o be_v equal_a to_o ha_o and_o the_o ark_n comprehend_v between_o a_o and_o b_o will_v be_v a_o quadrant_n but_o in_o a_o quadrant_n any_o line_n be_v draw_v from_o the_o limb_n pass_v through_o the_o centre_n the_o near_a distance_n from_o the_o end_n of_o one_o of_o the_o radij_fw-la will_v be_v the_o sine_fw-la of_o the_o ark_n thence_o count_v and_o the_o near_a distance_n from_o the_o other_o radius_fw-la thence_o count_v
half_a the_o declination_n to_o the_o tangent_fw-la of_o a_o four_o arch._n again_o as_o the_o cousin_n of_o half_a the_o sum_n of_o the_o former_a compliment_n be_v to_o the_o cousin_n of_o half_a their_o difference_n so_o be_v the_o contangent_fw-la of_o half_a the_o declination_n to_o the_o tangent_fw-la of_o a_o seven_o arch._n get_v the_o sum_n and_o difference_n of_o the_o four_o and_o seven_o arch_n then_o if_o the_o colatitude_n be_v great_a than_o the_o compliment_n of_o the_o reclination_n the_o sum_n be_v the_o substile_n distance_n from_o the_o plain_n perpendicular_a and_o the_o difference_n the_o inclination_n of_o meridian_n but_o if_o it_o be_v less_o the_o difference_n be_v the_o substile_n distance_n from_o the_o plain_n perpendicular_a and_o the_o sum_n the_o inclination_n of_o meridian_n to_o place_v the_o substile_n for_o plain_n lean_v southwards_o when_o the_o angle_n of_o the_o substile_n from_o the_o plain_n perpendicul_fw-la be_v less_o than_o a_o quadrant_n it_o will_v on_o the_o upperface_n lie_v above_o that_o end_n of_o the_o horizontal_a line_n that_o be_v opposite_a to_o the_o coast_n of_o declination_n and_o on_o the_o under_o face_n lie_v beneath_o it_o but_o when_o it_o be_v great_a it_o will_v lie_v below_o the_o say_a end_n on_o the_o upper_a face_n and_o above_o it_o on_o the_o under_o face_n but_o this_o will_v not_o be_v till_o the_o reclination_n be_v more_o than_o the_o compliment_n of_o the_o reclination_n of_o a_o polar_a plain_n that_o have_v the_o same_o declination_n for_o plain_n lean_v northwards_o the_o direction_n of_o the_o first_o method_n suffice_v to_o place_v the_o meridian_n either_o calculate_v it_o and_o place_n it_o according_a to_o the_o direction_n of_o the_o first_o and_o second_o method_n or_o else_o calculate_v it_o by_o this_o proportion_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o style_n height_n so_o be_v the_o tangent_fw-la of_o the_o inclination_n of_o meridian_n when_o it_o be_v obtuse_a take_v its_o compliment_n to_o a_o semicircle_n to_o the_o tangent_fw-la of_o the_o meridian_n line_n from_o the_o substilar_a for_o plain_n lean_v northward_o the_o first_o direction_n must_v serve_v but_o for_o southern_a plain_n the_o second_o because_o the_o distance_n of_o the_o meridian_n be_v calculate_v from_o the_o substile_n suppose_a to_o be_v place_v and_o here_o the_o work_n be_v converse_v to_o that_o for_o in_o that_o we_o suppose_v the_o meridian_n place_v and_o not_o the_o substile_n for_o the_o style_n height_n as_o the_o sine_fw-la of_o the_o four_o arch_n be_v to_o the_o sine_fw-la of_o the_o seven_o arch_n so_o be_v the_o tangent_fw-la of_o half_a the_o difference_n of_o the_o compliment_n both_o of_o the_o latitude_n and_o reclination_n to_o the_o tangent_fw-la of_o a_o arch_n seek_v how_o much_o the_o say_a ark_n be_v double_v want_n or_o exceed_v 90d_o be_v the_o style_n height_n in_o south_n recliner_n if_o the_o say_a ark_n be_v double_v be_v less_o than_o 90d_o its_o compliment_n be_v the_o elevation_n of_o the_o north_n pole_n and_o the_o plain_n fall_v below_o the_o pole_n but_o if_o the_o say_a arch_n exceed_v 90d_o the_o plain_n pass_v above_o the_o pole_n and_o the_o excess_n be_v the_o elevation_n of_o the_o north_n pole_n on_o the_o under_o face_n of_o a_o south_n recliner_n call_v a_o north_n incliner_n and_o the_o affection_n be_v determine_v in_o the_o first_o method_n where_o the_o declination_n have_v its_o denomination_n from_o that_o coast_n of_o the_o meridian_n to_o which_o the_o plain_n look_v these_o method_n of_o calculation_n may_v not_o precise_o agree_v one_o with_o another_o though_o all_o true_a unless_o the_o part_n proportional_a be_v exact_o calculate_v from_o large_a table_n in_o every_o operation_n which_o to_o do_v as_o to_o the_o example_n in_o this_o book_n my_o leisure_n will_v not_o permit_v this_o last_o method_n be_v derive_v also_o from_o the_o former_a oblique_a triangle_n the_o proportion_n here_o apply_v be_v demonstrate_v in_o trigonometria_fw-la brittanica_fw-la by_o mr._n newton_n the_o demonstration_n of_o the_o former_a proportion_n in_o project_v the_o sphere_n it_o be_v frequent_o require_v to_o draw_v a_o arch_n through_o any_o two_o different_a point_n within_o a_o circle_n that_o shall_v divide_v the_o say_a circle_n into_o two_o equal_a semicircle_n construction_n 1._o draw_v a_o line_n from_o one_o of_o the_o give_a point_n through_o the_o centre_n for_o conveniency_n through_o that_o point_n which_o be_v most_o remote_a 2._o from_o the_o centre_n raise_v a_o radius_fw-la perpendicular_a to_o that_o line_n 3._o and_o from_o the_o say_a point_n draw_v a_o line_n to_o the_o end_n of_o the_o radius_fw-la 4._o from_o the_o end_n of_o the_o radius_fw-la raise_v a_o line_n perpendicular_a to_o the_o line_n last_o draw_v and_o where_o it_o intersect_v the_o former_a line_n draw_v through_o the_o first_o point_n and_o centre_n be_v a_o three_o point_n give_v describe_v a_o circle_n through_o these_o three_o point_n and_o the_o proposition_n will_v be_v effect_v example_n let_v it_o be_v require_v to_o draw_v the_o arch_n of_o a_o circle_n through_o the_o two_o point_n e_o and_o f_o that_o shall_v divide_v the_o circle_n bd_o into_o two_o equal_a part_n operation_n from_o e_o draw_v eg_o through_o the_o centre_n a_o make_v ad_fw-la perpendicular_a thereto_o join_v ed_o and_o make_v dg_o perpendicular_a to_o ed_o cut_v eg_o in_o g_o through_o e_o f_o and_o g_o draw_v the_o arch_n of_o a_o circle_n which_o will_v divide_v the_o circumference_n bdc_o into_o two_o equal_a part_n in_o b_o and_o c_o that_o be_v if_o ca_o be_v draw_v it_o will_v pass_v through_o b_o if_o not_o let_v it_o pass_v above_o or_o below_o as_o let_v it_o pass_v below_o and_o cut_v bfe_n in_o h._n demonstration_n by_o construction_n edg_n and_o dag_o be_v right_a angle_n therefore_o â–¡_o ad_fw-la â–_o eag_o by_o 13_o prop._n 6_o euclid_n because_o edg_n be_v a_o right_a angle_n ad_fw-la be_v a_o mean_a proportional_a between_o ea_fw-la and_o ag_o but_o â–_o eag_o shall_v be_v â–_o cah_o by_o 35._o prop._n of_o 3_o euclid_n therefore_o â–_o cah_o â–¡_o ad_fw-la but_o â–¡_o ad_fw-la â–¡_o a_o that_o be_v â–_o cai_o therefore_o â–_o cah_o â–_o cai_o which_o be_v absurd_a therefore_o ci_o can_v pass_v below_o b_o the_o same_o absurdity_n will_v follow_v if_o it_o be_v think_v to_o pass_v above_o it_o therefore_o ca_o produce_v will_v fall_v in_o the_o point_n b_o wherefore_o bdc_o be_v a_o semicircle_n which_o be_v to_o be_v prove_v and_o hereof_o i_o acknowledge_v i_o have_v see_v a_o demonstration_n by_o the_o learned_a teacher_n of_o the_o mathematics_n mr._n john_n leak_n to_o this_o effect_n to_o project_v the_o sphere_n and_o measure_n off_o the_o ark_n of_o a_o upright_a decliner_n upon_o z_o as_o a_o centre_n describe_v the_o arch_n of_o a_o circle_n and_o cross_v it_o with_o two_o diameter_n at_o right_a angle_n in_o the_o centre_n whereto_o set_v nesw_n to_o represent_v the_o north_n east_n south_n and_o west_n prick_v off_o the_o latitude_n from_o n_o to_o l_o and_o lay_v a_o ruler_n to_o it_o from_o e_o and_o where_o it_o cut_v nz_o set_v p_o to_o represent_v the_o pole_n prick_v off_o the_o declination_n of_o the_o plain_a from_o e_o to_o a_o and_o from_o s_o to_o d_o and_o draw_v the_o diameter_n azb_o which_o represent_v the_o plain_a and_o dzc_a which_o represent_v the_o pole_n thereof_o through_o the_o three_o point_n cpd_n draw_v the_o arch_n of_o a_o circle_n and_o there_o will_v be_v frame_v aright_o angle_v triangle_n zhp_n right_o angle_v at_o h_o in_o which_o there_o will_v be_v give_v the_o side_n zp_o the_o compliment_n of_o the_o latitude_n with_o the_o angle_n pzh_o the_o compliment_n of_o the_o declination_n whereby_o may_v be_v find_v the_o style_n height_n represent_v by_o the_o side_n ph_o the_o substile_n distance_n from_o the_o plain_n perpendicular_a represent_v by_o zh_o and_o the_o angle_n between_o that_o meridian_n which_o make_v right_a angle_n with_o the_o plain_a and_o the_o meridian_n of_o the_o place_n represent_v by_o the_o angle_n zph_o show_v the_o arch_n of_o time_n between_o the_o substile_n and_o meridian_n call_v the_o inclination_n of_o meridian_n from_o which_o triangle_n be_v educe_v those_o proportion_n deliver_v for_o upright_a decliner_n to_o measure_v off_o these_o arks_n 1._o the_o substile_n a_o ruler_n lay_v from_o d_o to_o h_o find_v the_o point_n fletcher_o in_o the_o limb_n and_o the_o arch_n cf_o be_v the_o measure_n of_o the_o substile_n distance_n from_o the_o meridian_n to_o wit_n 21d_o 41′_n 2._o the_o style_v height_n set_a off_o a_o quadrant_n from_o fletcher_o to_o g_o lay_v a_o ruler_n from_o g_o to_o d_o and_o where_o it_o intersect_v bz_o set_a ☉_o which_o be_v the_o pole_n of_o the_o circle_n cpd_n lie_v a_o ruler_n from_o ☉_o to_o
a_o horizontal_a dial_n for_o the_o latitude_n of_o london_n 51_o degree_n 32_o minute_n cross_n the_o same_o with_o a_o perpendicular_a to_o wit_v the_o line_n of_o six_o and_o draw_v a_o oblong_a or_o square_a figure_n as_o here_o be_v do_v wherein_o to_o write_v the_o hour_n from_o m_o to_o c_o set_v off_o such_o a_o radius_fw-la as_o you_o intend_v to_o draw_v a_o circle_n withal_o which_o for_o convenience_n may_v be_v as_o big_a as_o the_o plain_n will_v admit_v and_o therewith_o upon_o c_o as_o a_o centre_n describe_z a_o circle_n and_o set_v off_o the_o radius_fw-la of_o the_o say_a circle_n from_o fletcher_o to_o d_o from_o m_o to_o k_o and_o draw_v the_o right_a line_n fd_o then_o prick_v the_o pole_n height_n or_o latitude_n of_o the_o place_n from_o f_z to_o l_o and_o from_o l_o to_o s_o and_o through_o the_o point_n s_o draw_v the_o line_n ms_n and_o it_o shall_v represent_v the_o cock_n or_o style_n height_n from_o the_o point_n l_o take_v the_o near_a distance_n to_o fm_o and_o prick_v that_o extent_n from_o fletcher_o to_o n_o on_o the_o line_n fd_o than_o a_o ruler_n lay_v from_o n_o to_o k_o where_o it_o intersect_v the_o meridian_n be_v the_o regulate_a point_n or_o point_v ☉_o then_o from_o m_o divide_v the_o circle_n into_o twelve_o equal_a part_n for_o the_o whole_a hour_n set_v the_o letter_n of_o the_o alphabet_n thereto_o and_o lay_v a_o ruler_n from_o the_o regulate_a point_n to_o each_o of_o those_o division_n and_o it_o will_v intersect_v the_o circle_n on_o the_o opposite_a side_n from_o which_o intersection_n line_n draw_v into_o the_o centre_n at_o m_o shall_v be_v the_o hour_n line_n require_v to_o be_v produce_v beyond_o the_o centre_n as_o many_o as_o be_v needful_a which_o shall_v be_v the_o hour_n before_o or_o after_o 6_o in_o the_o summer_n half_a year_n the_o half_n and_o quarter_n be_v after_o the_o same_o manner_n to_o be_v inscribe_v by_o divide_v each_o equal_a division_n of_o the_o circle_n into_o half_n and_o quarter_n this_o circular_a dial_v be_v in_o effect_n publish_v and_o invent_v by_o mr._n foster_n in_o his_o book_n of_o a_o quadrant_n in_o anno_fw-la 1638._o for_o the_o demonstration_n of_o this_o work_n will_v demonstrate_v the_o truth_n of_o those_o circular_a performance_n which_o he_o operate_v on_o the_o back_n of_o that_o quadrant_n but_o be_v more_o express_o hint_v in_o his_o posthuma_fw-la the_o demonstration_n whereof_o shall_v follow_v those_o that_o have_v plain_a table_n with_o a_o frame_n may_v have_v tangent_fw-la line_n put_v on_o the_o side_n of_o their_o frame_n and_o then_o if_o a_o centre_n be_v find_v upon_o the_o paper_n under_o the_o frame_n by_o the_o intersection_n of_o a_o ruler_n lay_v over_o those_o tangent_n the_o requisite_a division_n of_o a_o circle_n to_o any_o radius_fw-la that_o can_v be_v describe_v upon_o the_o paper_n will_v be_v most_o ready_o give_v without_o divide_v any_o circle_n or_o set_v of_o mark_n or_o letter_n thereto_o the_o frame_n keep_v fast_o that_o paper_n on_o which_o the_o draught_n of_o the_o dial_n be_v make_v which_o may_v also_o be_v supply_v from_o a_o circle_n divide_v on_o pasteboard_n cut_v out_o which_o be_v to_o be_v lace_v upon_o a_o board_n over_o the_o paper_n whereon_o the_o draught_n be_v to_o be_v make_v a_o upright_a full_a south_n dial_n this_o be_v no_o other_o than_o a_o horizontal_a dial_n in_o that_o latitude_n which_o be_v equal_a to_o the_o compliment_n of_o the_o latitude_n of_o the_o place_n you_o be_v in_o only_o the_o hour_n must_v not_o be_v continue_v beyond_o the_o centre_n and_o the_o delineation_n require_v no_o other_o direction_n than_o the_o former_a on_o a_o direct_a north_n dial_n in_o this_o latitude_n there_o will_v not_o be_v above_o two_o hour_n above_o and_o two_o hour_n beneath_o each_o end_n of_o the_o horizontal_a line_n to_o be_v express_v and_o the_o style_n will_v have_v the_o same_o elevation_n and_o point_n upward_o horizontal_a dial_n a_o upright_o south_n dial_n in_o all_o upright_a north_n and_o south_n plain_n the_o meridian_n or_o hour_n line_n of_o 12_o be_v perpendicular_a to_o the_o plain_n horizontal_a line_n if_o the_o plain_n be_v direct_v the_o height_n of_o the_o style_n above_o the_o substile_n be_v equal_a to_o the_o compliment_n of_o the_o latitude_n if_o the_o plain_n be_v south_n recline_v or_o north_n incline_v the_o height_n of_o the_o style_n above_o the_o substile_n be_v equal_a to_o the_o ark_n of_o difference_n between_o the_o compliment_n of_o the_o latitude_n and_o the_o ark_n of_o reinclination_n and_o if_o this_o latter_a ark_n be_v great_a than_o the_o former_a the_o contrary_a pole_n be_v elevate_v if_o the_o plain_n be_v south_n incline_v or_o north_n recline_v the_o height_n of_o the_o style_n above_o the_o substile_n be_v equal_a to_o the_o sum_n of_o the_o colatitude_n and_o of_o the_o reinclination_n and_o when_o this_o latter_a ark_n be_v great_a than_o the_o latitude_n the_o style_n height_n will_v be_v great_a than_o a_o quadrant_n such_o horizontal_a or_o direct_a south_n dial_n both_o upright_a and_o lean_v whereon_o the_o stile_n have_v but_o small_a elevation_n be_v to_o be_v draw_v with_o a_o double_a tangent_fw-la line_n without_o a_o centre_n wherein_o the_o follow_a direction_n for_o direct_v polar_a plain_n and_o those_o for_o other_o oblique_a plain_n whereon_o the_o stile_n have_v but_o small_a elevation_n will_v full_o direct_v you_o all_o plain_n that_o cut_v the_o axis_n of_o the_o world_n have_v a_o centre_n but_o if_o they_o be_v parallel_v thereto_o the_o stile_n have_v no_o elevation_n such_o direct_a south_n recline_v or_o north_n incline_v plain_n who_o arch_n of_o reclination_n be_v equal_a to_o the_o compliment_n of_o the_o latitude_n be_v parallel_n to_o the_o axis_n and_o be_v call_v polar_a plain_n in_o these_o the_o hour_n line_n will_v be_v tangent_fw-la line_n of_o any_o assume_v radius_fw-la and_o the_o parallel_a height_n of_o the_o stile_n above_o the_o plain_a must_v be_v make_v equal_a to_o the_o radius_fw-la of_o the_o tangent_fw-la by_o which_o the_o hour_n be_v set_v off_o first_o draw_v the_o tangent_fw-la line_n cg_o and_o perpendicular_a thereto_o acb_o upon_o c_o as_o a_o centre_n with_o any_o radius_fw-la describe_v the_o quadrant_n of_o a_o circle_n hb_o and_o prick_v off_o the_o radius_fw-la from_o c_o to_o a_o from_o h_o to_o e_o from_o b_o to_o d_o from_o d_o to_o fletcher_o from_o fletcher_o twice_o to_o g._n a_o ruler_n lay_v from_o a_o to_o e_o find_v the_o point_n 1_o on_o the_o tangent_fw-la cg_o for_o the_o hour_n line_n of_o one_o and_o lay_v from_o a_o to_o d_o find_v the_o point_n 2_o for_o the_o hour_n line_n of_o two_o the_o hour_n line_n of_o three_o be_v at_o the_o point_n h_o of_o four_o at_o the_o point_n fletcher_o of_o five_o at_o the_o point_n g_o thus_o be_v the_o whole_a hour_n easy_o insert_v now_o to_o insert_v the_o half_n and_o quarter_n divide_v the_o arch_n be_v ed_z dh_o into_o half_n and_o a_o ruler_n lay_v from_o a_o to_o those_o division_n will_v find_v point_n upon_o ch_o where_o all_o the_o half_a hour_n under_o 45d_o be_v to_o be_v graduate_v and_o if_o a_o ruler_n be_v lay_v from_o b_o to_o those_o respective_a division_n of_o the_o quadrant_n bh_o it_o will_v find_v point_n on_o the_o tangent_fw-la cg_o where_o the_o whole_a hour_n and_o half_n be_v to_o be_v graduate_v above_o 45d_o after_o the_o same_o manner_n be_v the_o quarter_n to_o be_v insert_v but_o in_o regard_n the_o half_n and_o quarter_n above_o 45d_o will_n by_o this_o direction_n be_v find_v with_o much_o uncertainty_n i_o have_v add_v the_o help_n follow_v first_o divide_v the_o half_n and_o quarter_n under_o 45d_o as_o now_o direct_v then_o for_o those_o above_o make_v use_n of_o this_o table_n in_o a_o direct_a south_n polar_a dial_n the_o distance_n of_o these_o hour_n line_n be_v equal_a to_o the_o distance_n of_o these_o hour_n line_n 3_o and_o 3_o ¼_o 3_o ½_o and_o 4_o 2_o ¼_o and_o 3_o ¾_o 4_o and_o 4_o ¼_o 4_o ¼_o and_o 4_o ½_o 4_o ¼_o and_o 4_o ¼_o ¾_o and_o 1_o ¼_o 1_o ¼_o and_o 2_o ½_o 1_o ½_o and_o double_a ¾_o and_o 1_o ¾_o 3_o ½_o and_o 3_o ¾_o double_v otherwise_o 1_o ¾_o and_o 2_o ¾_o 2_o and_o 3_o ¾_o this_o i_o do_v not_o assert_v to_o be_v absolute_o true_a but_o so_o near_o the_o truth_n that_o there_o will_v not_o arrive_v above_o one_o thousand_o part_n of_o the_o radius_fw-la difference_n in_o the_o great_a dial_n that_o be_v make_v and_o will_v be_v more_o certain_a than_o any_o sector_n though_o of_o a_o vast_a radius_fw-la or_o then_o they_o can_v with_o convenience_n be_v prick_v down_o the_o common_a way_n by_o a_o contingent_a line_n the_o meaning_n of_o the_o table_n will_v
the_o style_n height_n the_o inclination_n of_o meridian_n be_v find_v in_o the_o oblique_a spherical_a triangle_n by_o the_o proportion_n of_o opposite_a side_n and_o angle_n last_o in_o the_o right_a angle_a triangle_n zrc_o there_o be_v give_v rc_o and_o rcz_n to_o find_v rz_n the_o distance_n of_o the_o meridian_n from_o the_o plain_n perpendicular_a to_o measure_v the_o respective_a ark_n abovesaid_a 1._o the_o new_a latitude_n a_o ruler_n lay_v from_o w_o to_o z_o and_o i_o will_v give_v you_o the_o arch_n mo_o in_o the_o limb_n the_o compliment_n of_o the_o new_a latitude_n to_o wit_n 27d_o 40′_n 2._o the_o new_a declination_n the_o ruler_n lay_v from_o the_o plain_n zenith_n at_o z_o to_o its_o pole_n at_o ☉_o find_v the_o point_n q_o in_o the_o limb_n and_o the_o arch_n sq_fw-la be_v the_o new_a declination_n to_o wit_n 18d_o 56′_n 3._o the_o substile_n distance_n from_o the_o plain_n perpendicular_a a_o ruler_n lay_v from_o ☉_o to_o h_o find_v the_o point_n t_o in_o the_o limb_n and_o the_o arch_a ct_o be_v 26d_o 26′_n be_v the_o substile_n distance_n from_o the_o plain_n perpendicular_a 4._o the_o meridian_n distance_n from_o the_o plain_n perpendicular_a a_o ruler_n lay_v from_o ☉_o to_o z_o find_v the_o point_n v_o in_o the_o limb_n and_o the_o arch_n dv_o be_v 35d_o 56′_n be_v the_o meridian_n distance_n from_o the_o plain_n perpendicular_a 5._o the_o style_v height_n set_a off_o a_o quadrant_n from_o g_o to_o x_o and_o draw_v xc_o where_o it_o intersect_v the_o plain_a as_o at_o y_o be_v the_o pole_n of_o the_o arch_a fg_o then_o lay_v a_o ruler_n from_o y_o to_o h_o and_o p_o you_o shall_v find_v the_o style_n height_n in_o the_o limb_n to_o be_v the_o arch_n 2_o 3_o namely_o 26d_o 6′_n a_o south_n plain_a declin_n 40d_o east_n recline_v 60d_o lat_z 51-32′_n page_n 55_o a_o south_n plain_a declin_n 30d_o east_n recline_v 25d_o lat_n 51_o 32′_o 6._o the_o inclination_n of_o meridian_n a_o ruler_n lay_v from_o p_o to_o y_o intersect_v the_o limb_n at_o the_o arch_n w_o be_v 20d_o 58′_n and_o so_o much_o be_v the_o inclination_n of_o meridian_n the_o polar_a reclination_n cs_o be_v 31d_o 19′_n the_o scheme_n determine_v all_o the_o affection_n of_o the_o plain_a 1._o it_o show_v that_o h_o the_o point_n of_o the_o substilar_n lie_v on_o that_o side_n the_o plain_n perpendicular_a that_o be_v towards_o the_o coast_n of_o declination_n 2._o that_o z_o the_o point_n for_o the_o place_n of_o the_o meridian_n lie_v towards_o the_o same_o coast_n as_o before_o but_o below_o the_o substilar_a line_n 3._o the_o arch_n ph_o show_v you_o that_o the_o north_n pole_n be_v elevate_v above_o the_o upper_a or_o recline_a face_n after_o the_o same_o manner_n may_v all_o the_o requisite_a ark_n be_v measure_v and_o affection_n determine_v for_o all_o plain_n whatsoever_o a_o south_n plain_a decline_a 30_o degree_n eastward_o recline_v 25_o degree_n or_o a_o north_n plain_a decline_a 30_o degree_n westward_o incline_v â—â—_z degree_n in_o this_o scheme_n we_o have_v the_o same_o oblique_a triangle_n pc_o ☉_o reduce_v to_o two_o right_a angle_a triangle_n psc_o and_o ps_o ☉_o sc_o be_v the_o inclination_n of_o a_o polar_a plain_n and_o rc_o the_o inclination_n of_o the_o plain_a propose_v the_o difference_n be_v sir_o and_o the_o compliment_n of_o it_o be_v the_o compliment_n of_o s_o ☉_o to_o a_o semicircle_n because_o s_o ☉_o be_v great_a than_o a_o quadrant_n and_o the_o proportion_n be_v whole_o the_o same_o though_o the_o triangle_n have_v side_n great_a than_o a_o quadrant_n the_o north_n pole_n be_v elevate_v on_o the_o incline_a face_n the_o meridian_n z_o lie_v from_o the_o plain_n perpendicular_a towards_o that_o end_n of_o the_o horizontal_a line_n opposite_a to_o the_o coast_n of_o declination_n the_o same_o way_n and_o beneath_o it_o lie_v the_o substilar_n the_o compliment_n of_o the_o new_a latitude_n zp_o be_v 10d_o 10′_n the_o new_a declination_n viz._n the_o compliment_n of_o nza_o be_v 26d_o 57′_n the_o 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nza_o be_v 38d_o 23′_n the_o meridian_n distance_n from_o the_o plain_n perpendicular_a zr_o 12d_o 33′_n the_o substile_n distance_n from_o the_o plain_n perpendicular_a rh_o 32d_o 16′_n the_o style_n height_n ph_o be_v 41d_o 30′_n the_o inclination_n of_o meridian_n ☉_o pc_o rather_o the_o acute_a angle_z ☉_o pn_o be_v 55d_o 58′_n the_o meridian_n z_o lie_v from_o the_o plain_n perpendicular_a towards_o the_o coast_n of_o declination_n on_o the_o recline_a side_n but_o must_v be_v draw_v through_o the_o centre_n because_o the_o sun_n at_o noon_n cast_v his_o shadow_n northwards_o unless_o in_o the_o torrid_a or_o freeze_a zone_n and_o the_o substile_n hlyes_n on_o the_o other_o side_n the_o plain_n perpendicular_a a_o north_n plain_a decline_a 40_o degree_n eastward_o recline_v 75_o degree_n in_o this_o plain_a likewise_o the_o oblique_a triangle_n c_o ☉_o p_o be_v reduce_v to_o two_o right_a angle_a triangle_n ps_o ☉_o and_o psc_fw-fr by_o the_o perpendicular_a ps_o which_o be_v part_n of_o the_o arch_n of_o a_o polar_a plain_a here_o cr_o more_o cs_o be_v equal_a to_o the_o sum_n of_o the_o plain_n reclination_n propose_v and_o of_o the_o polar_a plain_n reclination_n which_o be_v great_a than_o a_o quadrant_n for_o the_o arch_n r_o ☉_o be_v a_o quadrant_n now_o the_o cousin_n of_o a_o ark_n great_a than_o a_o quadrant_n be_v the_o sine_fw-la of_o that_o ark_n excess_n above_o a_o quadrant_n wherefore_o the_o sine_fw-la of_o s_o ☉_o be_v the_o cousin_n of_o the_o sum_n of_o both_o the_o reclination_n and_o the_o case_n the_o same_o as_o before_o a_o south_n plain_a declin_n 40d_o east_n incline_v 15d_o lat_n 51d_o 72′_n page_n 56_o a_o north_n plain_a declin_n 40d_o east_n reclin_n 75d_o lat_fw-la 51d_o 32′_o prick_v off_o a_o quadrant_n from_o g_o to_o x_o and_o draw_v xc_o it_o cut_v the_o plain_a at_o y_o a_o ruler_n lay_v from_o y_o to_o h_o and_o p_o find_v the_o point_n 2_o 3_o in_o the_o limb_n and_o the_o arch_n 2_o 3_o be_v 61d_o 31′_n be_v the_o style_n height_n orther_z compliment_n thereof_o to_o a_o semicircle_n may_v be_v find_v by_o measure_v the_o arch_n pf_o a_o ruler_n lay_v from_o ☉_o to_o z_o find_v the_o point_n v_o in_o the_o limb_n for_o the_o meridian_n line_n from_o which_o draw_v a_o line_n through_o the_o centre_n on_o the_o other_o side_n and_o it_o will_v be_v place_v in_o its_o true_a coast_n and_o quantity_n from_o the_o plain_n perpendicular_a at_o a_o to_o wit_n 39d_o 2′_n the_o inclination_n of_o meridian_n to_o wit_n the_o angle_n cp_o ☉_o be_v 20d_o 30′_n the_o new_a latitude_n zae_o 26d_o 38′_n the_o new_a declination_n be_v 9d_o 35′_n to_o wit_n the_o compliment_n of_o krb._n the_o polar_a reclination_n cs_o be_v 31d_o 19′_n the_o truth_n of_o this_o stereographick_a projection_n be_v full_o handle_v by_o aguilonius_fw-la in_o his_o optic_n and_o how_o to_o determine_v the_o affection_n of_o any_o angle_n of_o a_o oblique_a spherical_a triangle_n i_o have_v full_o show_v in_o a_o treatise_n call_v the_o sector_n on_o a_o quadrant_n for_o the_o resolution_n of_o spherical_a proposition_n delineation_n from_o proportion_n or_o the_o analemma_n will_v be_v more_o speedy_a and_o certain_a though_o they_o may_v also_o be_v thus_o resolve_v which_o i_o have_v handle_v at_o large_a in_o the_o mariner_n plain_a scale_n new_o plain_v to_o determine_v what_o hour_n be_v proper_a to_o all_o kind_n of_o plain_n to_o do_v this_o it_o will_v be_v necessary_a to_o project_v upon_o the_o plain_a of_o the_o horizon_n the_o summer_n and_o winter_n tropic_n get_v the_o sum_n and_o difference_n of_o the_o colatitude_n and_o of_o the_o sun_n great_a declination_n so_o we_o shall_v obtain_v his_o great_a and_o least_o meridian_n altitude_n the_o depression_n of_o the_o tropic_a of_o