Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	east_n	north_n	west_n	5,069	5	9.1516	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A33999	Geometrical dyalling, or, Dyalling performed by a line of chords onely, or by the plain scale wherein is contained two several methods of inscribing the hour-lines in all plains, with the substile, stile and meridian, in their proper coasts and quantities : being a full explication and demonstration of divers difficulties in the works of learned Mr. Samuel Foster deceased ... : whereto is added four new methods of calculation, for finding the requisites in all leaning plains ... : also how by projecting the sphere, to measure off all the arks found by calculation ... : lastly, the making of dyals from three shadows of a gnomon ... / written by John Collins ...	Collins, John, 1625-1683.	1659 (1659)	Wing C5373; ESTC R17003	56,110	109

calculate_v all_o the_o requisite_n in_o all_o upright_a plain_n the_o meridian_n lie_v in_o the_o plain_n perpendicular_a and_o if_o they_o decline_v from_o the_o south_n in_o this_o hemisphere_n it_o be_v to_o descend_v or_o run_v downward_o if_o from_o the_o north_n it_o ascend_v and_o the_o substile_n lie_v on_o that_o side_n thereof_o opposite_a to_o the_o coast_n of_o declination_n in_o east_n or_o west_n reincliner_n it_o lie_v in_o the_o plain_n horizontal_a line_n on_o the_o incline_a side_n the_o south_n pole_n be_v elevate_v but_o on_o the_o upper_a side_n the_o north_n pole_n and_o the_o substile_n lie_v above_o or_o below_o that_o end_n of_o the_o meridian_n line_n which_o point_v to_o the_o pole_n elevate_v above_o the_o plain_n on_o all_o plain_n whatsoever_o to_o calculate_v the_o hour_n distance_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 above_o the_o substile_n so_o be_v the_o tangent_fw-la of_o the_o angle_n at_o the_o pole_n to_o the_o tangent_fw-la of_o the_o hour-line_n distance_n from_o the_o substilar_a line_n by_o the_o angle_n at_o the_o pole_n be_v mean_v the_o ark_n of_o difference_n between_o the_o ark_n call_v the_o inclination_n of_o meridian_n and_o the_o distance_n of_o any_o hour_n from_o the_o meridian_n for_o all_o hour_n on_o the_o same_o side_n of_o the_o meridian_n the_o substile_n fall_v and_o the_o sum_n of_o these_o two_o ark_n for_o all_o hour_n on_o the_o other_o side_n the_o meridian_n all_o hour_n on_o any_o plain_a go_v to_o the_o contrary_a coast_n of_o their_o situation_n in_o the_o sphere_n thus_o all_o the_o morning_n or_o eastern_a hour_n go_v to_o the_o western_a coast_n of_o the_o plain_a and_o all_o the_o evening_n or_o western_a hour_n go_v to_o the_o eastern_a coast_n of_o the_o plain_n a_o three_o method_n of_o calculation_n for_o lean_v plain_n that_o be_v for_o all_o sort_n of_o plain_n that_o do_v both_o decline_v and_o also_o incline_v or_o recline_v they_o may_v be_v refer_v to_o a_o new_a latitude_n in_o which_o they_o shall_v stand_v as_o upright_a plain_n and_o then_o they_o will_v have_v a_o new_a declination_n in_o that_o new_a latitude_n which_o two_o thing_n be_v find_v the_o former_a proportion_n for_o upright_a decliner_n will_v serve_v to_o calculate_v all_o the_o ark_n require_v how_o this_o may_v be_v do_v on_o a_o globe_n be_v not_o difficult_a to_o apprehend_v have_v set_v the_o globe_n to_o your_o latitude_n let_v one_o of_o the_o meridian_n of_o the_o ecliptic_a or_o longitude_n in_o the_o heaven_n represent_v a_o decline_a recline_a plain_n this_o circle_n intersect_v the_o meridian_n of_o the_o place_n in_o two_o point_n the_o one_o above_o the_o other_o beneath_o the_o horizon_n imagine_v the_o globe_n to_o be_v so_o fix_v that_o it_o can_v move_v upon_o its_o pole_n then_o elevate_v or_o depress_v the_o globe_n so_o in_o the_o meridian_n that_o the_o point_n of_o intersection_n above_o the_o horizon_n may_v come_v under_o the_o zenith_n then_o will_v the_o pole_n of_o the_o world_n be_v elevate_v above_o the_o horizon_n to_o the_o new_a latitude_n seek_v and_o where_o the_o meridian_n of_o longitude_n that_o represent_v the_o plain_n intersect_v the_o horizon_n it_o show_v the_o new_a declination_n or_o it_o may_v be_v thus_o apprehend_v the_o distance_n between_o the_o pole_n of_o the_o world_n and_o that_o point_n of_o intersection_n that_o represent_v the_o zenith_n of_o the_o new_a latitude_n be_v the_o compliment_n of_o the_o say_a new_a latitude_n and_o the_o distance_n between_o that_o point_n and_o the_o equinoctial_a be_v the_o new_a latitude_n itself_o the_o new_a declination_n be_v the_o compliment_n of_o the_o angle_n between_o the_o plain_a and_o the_o meridian_n of_o the_o place_n a_o ark_n usual_o find_v in_o calculation_n under_o this_o denomination_n to_o find_v these_o ark_n by_o calculation_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 reinclination_n from_o the_o zenith_n to_o the_o tangent_fw-la of_o the_o meridional_a ark_n namely_o the_o ark_n of_o the_o meridian_n between_o the_o plain_a and_o the_o horizon_n and_o this_o be_v the_o first_o thing_n master_n gunter_n and_o other_o find_v for_o south_n recliner_n north_n incliner_n the_o one_o be_v the_o upper_a the_o other_o the_o under_o face_n get_v the_o difference_n between_o this_o ark_n and_o the_o latitude_n of_o the_o place_n the_o compliment_n of_o the_o say_a residue_n remainder_n or_o difference_n be_v the_o new_a latitude_n seek_v but_o for_o north_n recliner_n or_o south_n incliner_n the_o difference_n between_o this_o four_o arch_n and_o the_o compliment_n of_o the_o old_a latitude_n be_v the_o new_a latitude_n to_o find_v the_o new_a declination_n as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o reinclination_n so_o be_v the_o sine_fw-la of_o the_o old_a declination_n to_o the_o sine_fw-la of_o the_o new_a this_o method_n be_v hint_v to_o we_o in_o mr._n foster_n posthuma_fw-la also_o in_o his_o book_n of_o dial_v in_o anno_fw-la 1638_o where_o he_o refer_v lean_v plain_n to_o such_o a_o latitude_n wherein_o they_o may_v become_v east_n or_o west_n recliner_n but_o that_o method_n be_v to_o be_v desert_v as_o multiply_v more_o proportion_n than_o this_o and_o do_v not_o afford_v that_o instrumental_a ease_n for_o prick_v down_o the_o hour_n that_o this_o do_v affection_n determine_v such_o south_n recliner_n who_o meridional_a arch_n be_v less_o than_o the_o latitude_n pass_v beneath_o the_o pole_n and_o have_v the_o north_n pole_n elevate_v above_o they_o but_o if_o the_o meridional_a ark_n be_v great_a than_o the_o latitude_n they_o pass_v above_o the_o pole_n the_o north_n pole_n be_v elevate_v on_o the_o under_o face_n all_o other_o affection_n be_v before_o determine_v if_o the_o meridional_a arch_n be_v equal_a to_o the_o latitude_n the_o plain_a be_v a_o polar_a plain_n for_o plain_n lean_v southwards_o if_o the_o meridional_a arch_n be_v equal_a to_o the_o compliment_n of_o the_o latitude_n the_o plain_a be_v a_o equinoctial_a plain_n if_o it_o be_v more_o the_o plain_a have_v less_o reclination_n than_o a_o equinoctial_a plain_n if_o it_o be_v less_o it_o have_v more_o and_o all_o affection_n necessary_a for_o place_v and_o calculate_v the_o meridian_n line_n be_v before_o determine_v this_o method_n of_o calculation_n find_v the_o substile_n distance_n from_o the_o meridian_n not_o from_o the_o plain_n perpetdicular_a wherefore_o it_o must_v be_v show_v how_o to_o place_v it_o in_o plain_n lean_v southwards_o for_o plain_n lean_v northwards_o use_v the_o former_a direction_n to_o place_v the_o substile_n in_o north_n recliner_n in_o these_o plain_n the_o meridian_n and_o substilar_n be_v to_o meet_v at_o the_o centre_n and_o not_o be_v draw_v through_o will_v make_v sometime_o a_o acute_a sometime_o a_o obtuse_a angle_n when_o the_o plain_n meridional_a ark_n be_v great_a than_o the_o colatitude_n they_o make_v a_o obtuse_a angle_n in_o this_o case_n have_v first_o place_v the_o meridian_n line_n above_o it_o prick_v off_o the_o compliment_n of_o the_o distance_n of_o the_o substile_n from_o meridian_n to_o a_o semicircle_n but_o when_o the_o meridional_a ark_n be_v less_o than_o the_o colatitude_n prick_v off_o the_o say_a distance_n itself_o above_o the_o meridian_n line_n in_o south_n incliner_n when_o the_o plain_n meridional_a ark_n be_v great_a than_o the_o colatitude_n the_o substile_n and_o meridian_n make_v a_o acute_a angle_n when_o it_o be_v equal_a to_o the_o colatitude_n they_o make_v a_o right_a angle_n when_o it_o be_v less_o than_o the_o colatitude_n they_o make_v a_o obtuse_a angle_n and_o must_v be_v prick_v off_o by_o the_o compliment_n of_o their_o distance_n to_o a_o semicircle_n the_o substile_n always_o lie_v on_o that_o side_n of_o the_o meridian_n opposite_a to_o the_o coast_n of_o declination_n a_o four_o method_n of_o calculation_n for_o lean_v plain_n a_o advertisement_n in_o this_o method_n of_o calculation_n for_o all_o plain_n lean_v northward_o both_o upper_a and_o under_o side_n their_o declination_n be_v the_o arch_n of_o the_o horizon_n between_o the_o north_n and_o the_o azimuth_n of_o the_o plain_n south_n pole_n so_o that_o their_o declination_n be_v always_o great_a than_o a_o quadrant_n but_o for_o all_o plain_n lean_v southwards_o both_o upper_a and_o under_o face_n their_o declination_n be_v the_o arch_n of_o the_o horizon_n between_o the_o north_n and_o the_o plain_n north_n pole_n wherefore_o it_o be_v always_o less_o than_o a_o quadrant_n in_o this_o sense_n declination_n be_v use_v in_o the_o follow_a proportion_n as_o the_o sine_fw-la of_o half_a the_o sum_n of_o the_o compliment_n both_o of_o the_o latitude_n and_o of_o the_o reclination_n be_v to_o the_o sine_fw-la of_o half_a their_o difference_n so_o be_v the_o contagent_n of_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 which_o may_v be_v of_o use_n to_o try_v the_o truth_n of_o these_o kind_n of_o plain_n 1._o the_o distance_n between_o the_o hour_n of_o 3_o and_o 4_o be_v equal_a to_o the_o distance_n between_o the_o hour_n of_o 1_o and_o 3._o 2._o the_o distance_n between_o the_o hour_n of_o 2_o and_o 4_o be_v double_a to_o the_o distance_n between_o the_o hour_n of_o 12_o and_o 2._o 3._o the_o distance_n between_o the_o hour_n of_o 11_o and_o 4_o be_v doubt_n to_o the_o distance_n between_o 12_o and_o 3_o or_o to_o that_o between_o 9_o and_o 12_o and_o so_o equal_a to_o the_o distance_n between_o 9_o and_o 3_o also_o equal_a to_o the_o distance_n between_o 4_o and_o 5._o 4._o the_o distance_n between_o the_o hour_n of_o 11_o and_o 5_o be_v double_a to_o the_o distance_n between_o the_o hour_n of_o 11_o and_o 4_o as_o also_o to_o the_o distance_n between_o the_o hour_n of_o 4_o and_o 5_o and_o between_o 9_o and_o 3_o and_o quadruple_a to_o the_o distance_n between_o 12_o and_o 9_o or_o between_o 12_o and_o 3._o these_o be_v absolute_o true_a as_o may_v be_v find_v by_o compare_v the_o difference_n of_o the_o respective_a tangent_n from_o the_o natural_a table_n to_o draw_v a_o polar_a direct_a south_n dial_n have_v draw_v the_o plain_n perpendicular_a in_o the_o middle_n of_o the_o plain_a let_v that_o be_v the_o hour_n line_n of_o 12_o then_o assume_v the_o stile_n to_o be_v of_o any_o convenient_a parallel_n height_n that_o will_v suit_v the_o plain_a make_v that_o radius_fw-la divide_v a_o tangent_fw-la line_n into_o hour_n and_o quarter_n by_o the_o former_a direction_n and_o prick_v they_o down_o on_o the_o plain_a upon_o a_o line_n draw_v perpendicular_a to_o the_o meridian_n or_o hour_n line_n of_o 12_o on_o each_o side_n thereof_o and_o through_o the_o point_n so_o prick_v off_o draw_v line_n parallel_v to_o the_o meridian_n line_n and_o they_o shall_v be_v the_o hour_n line_n require_v as_o in_o the_o example_n to_o describe_v a_o equinoctial_a dial_n such_o direct_a 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 such_o plain_n draw_v a_o horizontal_a line_n and_o cross_v the_o same_o with_o a_o perpendicular_a or_o vertical_a at_o the_o intersection_n set_v v_o at_o the_o upper_a end_n of_o the_o vertical_a line_n set_v s_o at_o the_o low_a end_n n_o at_o the_o east_n end_n of_o the_o horizontal_a line_n set_v e_o at_o the_o west_n end_n w_o prick_v off_o the_o declination_n of_o the_o plain_n in_o its_o proper_a coast_n from_o s_o or_o n_o to_o d_o and_o draw_v dv_o through_o the_o centre_n the_o same_o way_n count_v off_o the_o latitude_n to_o l_o and_o from_o it_o draw_v a_o line_n into_o the_o centre_n in_o the_o same_o quarter_n make_v a_o geometrical_a square_n of_o any_o proportion_n at_o pleasure_n so_o that_o two_o side_n thereof_o may_v be_v parallel_n to_o the_o horizontal_a and_o vertical_a line_n at_o the_o intersection_n of_o one_o of_o the_o side_n thereof_o with_o the_o vertical_a set_v a_o and_o of_o the_o other_o side_n with_o the_o horizontal_a line_n set_v b_o 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 to_o prick_v off_o the_o substilar_n for_o latitude_n above_o 45d_o take_v bf_o and_o prick_v it_o on_o the_o line_n of_o declination_n beyond_o the_o centre_n from_o five_o to_o o_o and_o from_o the_o point_n o_o draw_v the_o line_n oc_o parallel_n to_o the_o vertical_a line_n and_o produce_v beyond_o o_o if_o need_v require_v and_o thereon_o from_o c_o to_o i_o prick_v the_o side_n of_o the_o square_n and_o a_o line_n draw_v into_o the_o centre_n shall_v be_v the_o substilar_a line_n but_o for_o latitude_n under_o 45d_o prick_n the_o side_n of_o the_o square_a from_o v_o to_o o_o and_o draw_v oc_o as_o before_o and_o make_v ci_o equal_a to_o af_o and_o a_o line_n from_o i_o draw_v into_o the_o centre_n shall_v be_v the_o substilar_a line_n style_v height_n from_o the_o point_n i_o erect_v the_o extent_n oc_o perpendicular_o to_o the_o substilar_a oi_o at_o the_o extremity_n thereof_o set_v k_o from_o 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