Selected quad for the lemma: world_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
world_n	circle_n	great_a	pole_n	2,915	5	11.6366	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A64224	Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.	Taylor, John, mathematician.	1687 (1687)	Wing T534; ESTC R23734	190,995	514

a_o 49_o 47_o 30_o 39_o to_o tc._n ½_o v._o at_o e_o 15°_n 47_o 47_o which_o double_v be_v the_o angle_n at_o e_o 31°_n 34_o 34_o as_o be_v require_v prop._n viii_o case_n 8._o two_o angle_n and_o a_o side_n opposite_a to_o one_o of_o they_o be_v know_v to_o find_v the_o interjacent_a side_n in_o the_o triangle_n ade_n there_o be_v give_v the_o angle_n e_o 31°_n 34_o 34_o the_o angle_n d_o 130°_n 03_o 03_o and_o his_o opposite_a side_n ae_n 70°_n 00_o 00_o now_o the_o side_n ed_z be_v require_v first_o by_o prop._n 2._o case_n 2._o i_o find_v ad_fw-la oppose_v to_o e_o to_o be_v 40°_n 00_o 00_o and_o then_o work_v thus_o take_v the_o sum_n and_o difference_n of_o the_o angle_n than_o also_o find_v the_o difference_n of_o the_o two_o side_n 39_o 39_o 39_o now_o say_v as_o s._n ½_o x._o w_n d_o and_o e_o 49°_n 14_o 14_o 30_o to_o s._n ½_o z_o w_n d_o and_o e_o 80_o 48_o 30._o so_o be_v t_o ½_o x_o cr_n s._n ad_fw-la and_o ae_n 15_o 00_o 00_o to_o t._n ½_o cr_n s._n ed_z 19°_n 15_o 15_o 00_o which_o be_v double_v be_v the_o side_n ed_z 38°_n 30_o 30_o as_o require_v prop._n ix_o case_n 9_o two_o side_n and_o their_o include_v angle_n be_v know_v to_o find_v the_o three_o side_n in_o the_o triangle_n apz_n there_o be_v give_v the_o side_n 40._o zp_v 38°_n 30_o 30_o the_o side_n pa_n 70°_n and_o the_o angle_n p_o let_v be_v 31°_n 34_o and_o the_o side_n az_n be_v require_v the_o resolution_n of_o this_o case_n depend_v on_o the_o catholic_a proposition_n of_o the_o lord_n of_o marchiston_n by_o suppose_v the_o oblique-triangle_n to_o be_v divide_v by_o a_o suppose_a perpendicular_a fall_v either_o within_o or_o without_o the_o triangle_n into_o two_o rectangular_n now_o in_o the_o triangle_n azp_n let_v fall_v the_o perperpendicular_a zr_n so_o be_v the_o triangle_n azp_v divide_v into_o two_o rectangular_n arz_n and_o zrp_n now_o the_o side_n az_n may_v be_v find_v at_o two_o operation_n thus_o say_v as_o the_o radius_fw-la or_o s._n of_o 90°_n 00_o 00_o to_o sc._n of_o the_o include_v v._o p._n 31_o 34._o so_o be_v t._n of_o the_o lesser_a side_n pz_n 38_o 30_o to_o t._n of_o a_o four_o arch._n 34_o 08._o if_o the_o contain_v angle_n be_v less_o than_o 90°_n take_v this_o four_o arch_n from_o the_o great_a side_n but_o if_o it_o be_v great_a than_o 90°_n from_o its_o compliment_n unto_o 180°_n the_o remainder_n be_v the_o residual_a arch_n now_o again_o say_v as_o sc._n of_o the_o four_o arch._n 34°_n 08_o 08_o to_o sc._n residual_a arch._n 35_o 52_o so_o sc._n of_o the_o lesser_a side_n pz_n 38_o 30_o to_o sc._n az_n the_o side_n require_v 40_o 00_o triangle_n but_o note_n that_o many_o time_n the_o perpendicular_a will_v fall_v without_o the_o triangle_n as_o it_o do_v 41._o now_o within_o in_o such_o case_n the_o side_n of_o the_o triangle_n must_v be_v continue_v so_o will_v there_o be_v two_o rectangular_n the_o one_o include_v within_o the_o other_o as_o in_o the_o triangle_n hik_fw-mi the_o perpendicular_a let_v fall_n be_v km_n fall_v on_o the_o side_n he_o and_o so_o the_o two_o rectangular_n find_v thereby_o will_v be_v in_o k_o and_o kmh_o and_o so_o by_o the_o direction_n in_o the_o former_a proposition_n find_v out_o the_o side_n ik_fw-mi if_o require_v to_o be_v find_v prop._n x._o case_n 10._o two_o angle_n and_o their_o interjacent_a side_n know_v to_o find_v the_o three_o angle_n in_o the_o triangle_n azp_n there_o be_v give_v the_o side_n zp_v 38°_n 30_o 30_o the_o angle_n p_o 31°_n 34_o 34_o and_o the_o angle_n z_o 130°_n 03_o 03_o and_o the_o angle_n at_o a_o be_v require_v first_o the_o oblique-triangle_n azp_n be_v reduce_v into_o two_o rectangular_n arz_n and_o zrp_a by_o 40._o case_n 9_o aforegoing_a i_o find_v the_o angle_n rzp_n to_o be_v 64°_n 19_o 19_o in_o the_o triangle_n zrp_n which_o take_v out_o of_o angle_n azp_a 130°_n 03_o 03_o leave_v the_o angle_n azr_fw-fr 65°_n 44_o 44_o now_o the_o angle_n a_o be_v find_v by_o this_o analogy_n or_o proportion_n as_o s._n v._o pzr_n 64°_n 19_o 19_o to_o s._n v._o azr_n 65_o 44_o so_o be_v sc._n v._o at_o p_o 31_o 34_o to_o sc._n v._o at_o a_o 30_o 28_o which_o be_v require_v to_o be_v find_v out_o and_o know_v prop._n xi_o case_n 11._o three_o side_n give_v to_o find_v a_o angle_n in_o the_o triangle_n apz_n the_o side_n az_n be_v 40°_n 00_o 00_o the_o side_n zp_n be_v 38°_n 30_o 30_o the_o side_n ap_n be_v 70°_n 00_o 00_o and_o the_o angle_n z_o be_v require_v to_o find_v which_o do_v thus_o add_v the_o three_o side_n together_o and_o from_o half_a 40._o their_o sum_n deduct_v the_o side_n opposite_a to_o the_o require_a angle_n and_o then_o proceed_v as_o you_o see_v in_o the_o operation_n follow_v 40._o 40._o ½sum_fw-la be_v 65°_n 07_o 07_o 30_o the_o sc._n ½_o v._o at_o z_o which_o double_v be_v 130°_n 03_o 03_o 12_o the_o angle_n at_o z_o require_v prop._n xii_o case_n 12._o three_o angle_n give_v to_o find_v a_o side_n in_o the_o triangle_n azp_n the_o angle_n a_o be_v 30°_n 28_o 28_o 40._o 11_o the_o angle_n z_o 130°_n 03_o 03_o 12_o the_o angle_n p_o be_v 31°_n 34_o 34_o 26_o and_o the_o side_n az_n opposite_a to_o p_o be_v require_v this_o case_n be_v likewise_o perform_v as_o the_o former_a case_n or_o proposition_n the_o angles_n be_v convert_v into_o sides_n and_o the_o side_n into_o angle_n by_o take_v the_o compliment_n of_o the_o great_a angle_n unto_o 180°_n see_v the_o work_n 40._o 40._o which_o be_v double_v give_v the_o side_n az_v 40°_n 00_o require_v to_o be_v find_v out_o and_o know_v ☞_o but_o if_o the_o great_a side_n ap_n be_v require_v the_o operation_n will_v produce_v the_o complem_n 〈…〉_z thereof_o unto_o a_o semicircle_n or_o 180°_n therfo_o 〈…〉_z subtract_v it_o from_o 180°_n it_o leave_v the_o remain_v require_v side_n seek_v thus_o i_o have_v lay_v down_o all_o the_o case_n of_o triangle_n both_o rightlined_n and_o spherical_a either_o right_n or_o obliqueangled_n i_o may_v hereunto_o have_v annex_v many_o variety_n unto_o each_o case_n and_o some_o fundamental_a axiom_n which_o somewhat_o more_o will_v have_v illustrate_v and_o demonstrate_v those_o case_n and_o proportion_n but_o because_o of_o the_o smallness_n of_o this_o treatise_n which_o be_v intend_v more_o for_o practice_n than_o theory_n i_o have_v for_o brevity_n sake_n omit_v they_o and_o refer_v you_o for_o those_o thing_n to_o large_a author_n who_o have_v large_o discourse_v thereon_o to_o good_a purpose_n chap._n vi_o of_o astronomy_n defin._n astronomy_n be_v a_o art_n mathematical_a which_o measure_v the_o distinct_a course_n of_o time_n day_n year_n etc._n etc._n it_o show_v the_o distance_n magnitude_n natural_a motion_n appearance_n and_o passion_n proper_a unto_o the_o planet_n and_o fix_v star_n for_o any_o time_n past_a present_a and_o to_o come_v by_o this_o we_o be_v certify_v of_o the_o distance_n of_o the_o starry_a sky_n and_o of_o each_o planet_n from_o the_o centre_n of_o the_o earth_n and_o the_o magnitude_n of_o any_o fix_a star_n or_o planet_n in_o respect_n of_o the_o earth_n magnitude_n sect_n i._o of_o astronomical_a definition_n defin._n 1._o asphere_n or_o globe_n be_v a_o solid_a body_n contain_v only_o one_o superficies_n in_o who_o middle_a there_o be_v a_o point_n call_v the_o centre_n from_o which_o all_o right_a or_o straight_a line_n draw_v unto_o the_o circumference_n or_o superficies_n be_v equal_a 2._o the_o pole_n of_o the_o world_n be_v two_o fix_a point_n 42._o in_o the_o heaven_n diametrical_o opposite_a the_o one_o to_o the_o other_o the_o one_o call_v the_o arctic_a or_o north-pole_n note_v in_o the_o scheme_n by_o p._n the_o other_o be_v call_v the_o antarctic_a or_o south-pole_n as_o s._n and_o be_v not_o to_o be_v see_v of_o we_o be_v in_o the_o low_a hemisphere_n 3._o the_o axis_n of_o the_o world_n be_v a_o imaginary_a 42._o line_n draw_v from_o the_o north-pole_n through_o the_o centre_n of_o the_o earth_n unto_o the_o south-pole_n about_o which_o the_o diurnal_a motion_n be_v perform_v from_o the_o east_n to_o the_o west_n as_o the_o line_n ps_n 4._o the_o meridian_n be_v great_a circle_n concur_v 42._o and_o intersect_v one_o another_o in_o the_o pole_n of_o the_o world_n as_o pes_fw-la and_o pc_n s._n 5._o the_o equinoctial_a or_o equator_fw-la be_v a_o great_a circle_n 90_o deg_n distant_a from_o the_o pole_n of_o the_o world_n cut_v the_o meridian_n at_o right-angle_n and_o divide_v the_o world_n into_o two_o equal_a part_n call_v 42._o the_o northern_a and_o southern_a hemisphere_n as_o e_o ♎_o q._n in_o scheme_n 42._o 6._o the_o ecliptic_a be_v a_o great_a circle_n cross_v the_o equinoctial_a in_o the_o two_o

opposite_a point_n aries_n and_o libra_n and_o make_v a_o angle_n therewith_o call_v its_o obliquity_n of_o 23°_n 30_o 30_o represent_v by_o ♋_o ♎_o ♑_o this_o circle_n be_v divide_v into_o 12_o sin_n each_o contain_v 30°_n 00_o 00_o as_o aries_n 42._o ♈_o taurus_n ♉_o gemini_fw-la ♊_o cancer_n ♋_o leo_n ♌_o virgo_fw-la ♍_o which_o be_v call_v northern_a sines_n libra_n ♎_o scorpio_n ♏_o sagitarius_n ♐_o capricornus_fw-la ♑_o aquarius_n ♒_o and_o pisces_fw-la ♓_o these_o be_v call_v southern_a sines_n 7._o the_o zodiac_n be_v a_o zone_n or_o girdle_n have_v 8_o deg_n of_o latitude_n on_o either_o side_n the_o ecliptic_a in_o which_o space_n the_o planet_n make_v their_o revolution_n this_o circle_n be_v a_o circle_n which_o regulate_v the_o year_n month_n and_o season_n etc._n and_o be_v distinguish_v 42._o in_o the_o scheme_n by_o the_o 12_o sin_n 8._o the_o colour_n be_v two_o meridian_n divide_v the_o ecliptic_a and_o the_o equinoctial_a into_o four_o equal_a part_n one_o of_o which_o pass_v by_o the_o equinoctial_a 42._o point_n aries_n and_o libra_n and_o be_v call_v the_o equinoctial_a colour_n as_o p_o ♎_o s._n the_o other_o by_o the_o begin_n of_o cancer_n and_o capricorn_n and_o be_v call_v the_o solstitial_a colour_n as_o p_o ♋_o s_o ♑_o 9_o the_o pole_n of_o the_o ecliptic_a be_v two_o point_n 23°_n 30_o 30_o distant_a from_o the_o pole_n of_o the_o world_n as_o i_o and_o k._n 42._o 10._o the_o tropic_n be_v two_o small_a circle_n parallel_n unto_o the_o equinoctial_a and_o distant_a therefrom_o 23°_n 30_o 30_o limit_v the_o sun_n great_a declination_n the_o northern_a tropic_n pass_v by_o the_o begin_n of_o cancer_n and_o be_v therefore_o call_v the_o tropic_a of_o cancer_n as_o ♋_o a_o d._n the_o southern_a tropic_n 42_o pass_v by_o the_o begin_n of_o capricorn_n and_o be_v therefore_o call_v the_o tropic_a of_o capricorn_n as_o b_o b_o ♑_o 11._o the_o polar_a circle_n be_v two_o small_a circle_n parallel_v to_o the_o equinoctial_a and_o distant_a therefrom_o 66°_n 30_o 30_o and_o from_o the_o pole_n of_o the_o world_n 23°_n 30_o 30_o that_o which_o be_v adjacent_a unto_o the_o north_n pole_n be_v call_v the_o arctic_a circle_n as_o g_z d_o i._n and_o the_o other_o the_o antarctic_a circle_n as_o kd_v m._n 42._o 12._o the_o zenith_n and_o the_o nadir_n be_v two_o point_n diametrical_o opposite_a the_o one_o to_o the_o other_o the_o zenith_n be_v the_o vertical_a point_n or_o the_o point_n over_o our_o head_n as_o z_o the_o nadir_n be_v opposite_a thereto_o as_o the_o point_n n._n 13._o the_o azimuth_n or_o vertical_a circle_n be_v great_a circle_n of_o the_o sphere_n concur_v and_o intersect_v each_o other_o in_o the_o zenith_n and_o nadir_n as_o z_o f_o n._n 14_o the_o horizon_n be_v a_o great_a circle_n 90_o deg_n 42._o distant_a from_o the_o zenith_n and_o nadir_n cut_v all_o the_o azimuth_n at_o rightangle_v and_o divide_v the_o world_n into_o two_o equal_a part_n the_o upper_z and_o visible_a hemisphere_n and_o the_o low_a and_o invisible_a hemisphere_n represent_v by_o h_n ♎_o r._n 15._o the_o meridian_n of_o a_o place_n be_v that_o meridian_n which_o pass_v by_o the_o zenith_n and_o nadir_n of_o 42._o the_o place_n as_o p_o z_o s_o n._n 16._o the_o alinicanthar_n or_o parallel_n of_o altitude_n be_v small_a circle_n parallel_v unto_o the_o horizon_n imagine_v to_o ●pass_n through_o every_o degree_n and_o minute_n of_o the_o meridian_n between_o the_o zenith_n and_o horizon_n b_o a_o f._n 17._o parallel_n of_o latitude_n or_o declination_n be_v small_a circle_n parallel_v unto_o the_o equinoctial_a they_o 42._o be_v call_v parallel_n of_o latitude_n in_o respect_n to_o any_o place_n on_o the_o earth_n and_o parallel_n of_o declination_n in_o respect_n of_o the_o sun_n or_o star_n in_o the_o heaven_n 18._o the_o latitude_n of_o a_o place_n be_v the_o height_n of_o the_o pole_n above_o the_o horizon_n or_o the_o distance_n between_o the_o zenith_n and_o the_o equinoctial_a 19_o the_o latitude_n of_o a_o star_n be_v the_o arch_n of_o a_o circle_n contain_v betwixt_o the_o centre_n of_o a_o star_n and_o the_o ecliptic_a line_n this_o circle_n make_v right-angle_n with_o the_o ecliptic_a be_v account_v either_o northward_o or_o southward_o according_a to_o the_o situation_n of_o the_o star._n 20._o longitude_n on_o earth_n be_v measure_v by_o a_o arch_n of_o the_o equinoctial_a contain_v between_o the_o primary_n meridian_n or_o meridian_n of_o that_o place_n where_o longitude_n be_v assign_v to_o begin_v and_o the_o meridian_n of_o any_o other_o place_n count_v always_o easterly_a 21._o the_o longitude_n of_o a_o star_n be_v that_o part_n of_o the_o ecliptic_a which_o be_v contain_v between_o the_o star_n place_n in_o the_o ecliptic_a and_o the_o beginning_n of_o aries_n count_v they_o according_a unto_o the_o succession_n of_o sines_n 22._o the_o altitude_n of_o the_o sun_n or_o star_n be_v the_o arch_n of_o a_o azimuth_n contain_v betwixt_o the_o centre_n of_o the_o sun_n or_o star_n and_o the_o horizon_n 23._o ascension_n be_v the_o rise_n of_o any_o star_n or_o part_n of_o the_o equinoctial_a to_o any_o degree_n above_o the_o horizon_n and_o descension_n be_v the_o set_n of_o it_o 24._o right_o ascension_n be_v the_o number_n of_o degree_n and_o minute_n of_o the_o equinoctial_a i._n e._n from_o the_o begin_n of_o aries_n which_o come_v unto_o the_o meridian_n with_o the_o sun_n or_o star_n or_o with_o any_o portion_n of_o the_o ecliptic_a 25._o oblique-ascension_n be_v a_o arch_n of_o the_o equinoctial_a between_o the_o begin_n of_o aries_n and_o that_o part_n of_o the_o equinoctial_a which_o rise_v with_o the_o centre_n of_o a_o star_n or_o with_o any_o portion_n of_o the_o ecliptic_a in_o a_o oblique_a sphere_n and_o oblique_a descension_n be_v that_o part_n of_o the_o equinoctial_a tha●_n set_v therewith_o 26._o the_o ascentional_a difference_n be_v a_o arch_n of_o the_o equinoctial_a be_v the_o difference_n betwixt_o the_o right_n and_o oblique-ascension_n 27._o the_o amplitude_n of_o the_o sun_n or_o star_n be_v 43._o the_o distance_n of_o the_o rise_n or_o set_v thereof_o from_o the_o east_n or_o west_n point_n of_o the_o horizon_n 28._o the_o parallax_n be_v the_o difference_n between_o the_o true_a and_o apparent_a place_n of_o the_o sun_n or_o star_n so_o the_o true_a place_n in_o respect_n of_o altitude_n be_v in_o the_o line_n ace_n or_o adg_n the_o sun_n or_o star_n be_v at_o c_o or_o d_o and_o the_o apparent_a place_n in_o the_o line_n bcf_n and_o bdh_n so_o likewise_o the_o angle_n of_o the_o parallax_n be_v acb_n or_o ecf_n and_o adb_n or_o gdb_v also_o in_o the_o say_a scheme_n abk_n represent_v a_o 43._o quadrent_fw-la of_o the_o globe_n or_o earth_n on_o the_o earth_n superficies_n a_o the_o centre_n of_o the_o earth_n and_o b_o any_o point_n of_o the_o earth_n surface_n 29._o the_o refraction_n of_o a_o star_n be_v cause_v by_o the_o atmosphere_n or_o vapourous_a thickness_n of_o the_o air_n near_o the_o earth_n superficies_n whereby_o the_o sun_n and_o star_n seem_v always_o to_o rise_v soon_o and_o and_o set_v late_a than_o real_o they_o do_v sect_n ii_o of_o astronomical_a proposition_n prop._n i._n the_o distance_n of_o the_o sun_n from_o the_o next_o equinoctial_a point_n either_o aries_n or_o libra_n be_v know_v to_o find_v his_o declination_n the_o analogy_n or_o proportion_n as_o radius_fw-la or_o s._n 90°_n to_o s._n of_o the_o sun_n distance_n from_o the_o next_o equinoctial_a point_n so_o it_o s._n of_o the_o sun_n be_v great_a declination_n to_o the_o s._n of_o the_o sun_n be_v present_a declination_n seek_v prop._n ii_o the_o sun_n place_n give_v to_o find_v his_o right-ascension_n this_o be_v the_o analogy_n or_o proportion_n as_o radius_fw-la or_o s._n 90°_n to_o t._n of_o the_o sun_n longitude_n from_o the_o next_o equinoctial_a point_n so_o be_v the_o sc._n of_o his_o great_a declination_n to_o t._n of_o his_o right-ascension_n from_o the_o next_o equinoctial_a point_n prop._n iii_o to_o find_v the_o sun_n place_n or_o longitude_n from_o aries_n his_o declination_n be_v give_v this_o be_v the_o analogy_n or_o proportion_n as_o s._n of_o the_o sun_n great_a declination_n to_o radius_fw-la or_o s._n 90°_n 00_o 00_o so_o be_v s._n of_o his_o present_a declination_n to_o s._n of_o the_o sun_n place_n or_o longitude_n from_o aries_n sin_n prop._n iu._n by_o know_v the_o sun_n declination_n to_o find_v his_o right_a ascension_n this_o be_v the_o analogy_n or_o proportion_n as_o radius_fw-la or_o s._n 90°_n to_o tc._n of_o the_o sun_n great_a declination_n so_o be_v t._n of_o the_o declination_n give_v to_o s._n of_o the_o sun_n right_a ascension_n require_v ♎_o prop._n v._o by_o know_v the_o latitude_n of_o a_o place_n and_o the_o sun_n declination_n to_o find_v the_o ascensional_a

edge_n shall_v show_v the_o hour_n of_o the_o day_n now_o to_o draw_v the_o hour-line_n with_o the_o radius_fw-la of_o your_o line_n of_o chord_n on_o m_n strike_v the_o arch_a qn_n which_o divide_v into_o 5_o equal_a part_n in_o the_o point_n ●_o ●_o ●_o etc._n etc._n then_o lay_v a_o ruler_n from_o m_n unto_o each_o of_o those_o point_n and_o it_o will_v cut_v the_o line_n jk_n in_o the_o point_n point_n point_n point_n etc._n etc._n through_o which_o point_n by_o prop._n 4._o 4._o 1._o chap._n 4._o draw_v parallel_n to_o 6_o i_o 6_o as_o the_o line_n 77_o 88_o etc._n etc._n which_o shall_v be_v the_o true_a hour-line_n of_o a_o east_n plane_n from_o 6_o 66._o in_o the_o morning_n till_o 11_o before_o noon_n then_o for_o the_o hour-line_n of_o 4_o and_o 5_o you_o must_v prick_v off_o 5_o as_o far_o from_o 6_o as_o 6_o be_v from_o 7_o and_o 4_o as_o far_z as_o 6_o be_v from_o 8_o and_o draw_v the_o hour-line_n 55_o and_o 44_o as_o before_o thus_o be_v your_o dial_n complete_v and_o in_o the_o form_n of_o which_o you_o have_v make_v both_o a_o east_n and_o a_o west_n dial_n which_o be_v the_o same_o in_o all_o respect_n only_o whereas_o the_o arch_n h_n g_o through_o which_o the_o equinoctial_a pass_v in_o the_o east_n dial_n be_v describe_v on_o the_o right_a hand_n of_o the_o plane_n in_o the_o west_n it_o must_v be_v draw_v on_o the_o left_a hand_n and_o the_o hour-line_n 4_o 5_o 6_o 7_o 8_o 9_o 10_o and_o 11_o in_o the_o forenoon_n in_o the_o east_n dial_n must_v be_v 8_o 7_o 6_o 5_o 4_o 3_o 2_o and_o 1_o in_o the_o west_n in_o 67._o the_o afternoon_n as_o in_o the_o figure_n plain_o appear_v now_o you_o may_v find_v the_o distance_n of_o the_o hour-line_n from_o the_o substile_a by_o this_o analogy_n or_o proportion_n as_o the_o radius_fw-la to_o the_o height_n of_o the_o style_n so_o be_v the_o tangent_fw-la of_o any_o hour_n distance_n from_o 6_o to_o the_o distance_n thereof_o from_o the_o substile_a prop._n iu._n how_o to_o draw_v the_o hour-line_n on_o a_o direct_a south_n and_o north_n plane_n this_o plane_n or_o dial_n must_v stand_v upright_o have_v his_o face_n or_o plane_n if_o it_o be_v a_o south_n dial_n direct_o opposite_a unto_o the_o south_n but_o if_o a_o north_n plane_n direct_o opposite_a unto_o the_o north_n now_o admit_v it_o be_v require_v to_o make_v a_o direct_a south_n dial_n 68_o for_o the_o latitude_n of_o 51°_n 32_o 32_o to_o make_v which_o first_o describe_v the_o circle_n abcd_n to_o represent_v a_o e●ect_n direct_a south_n plane_n cross_v it_o with_o the_o diameter_n cb_n and_o ad_fw-la then_o out_o of_o your_o line_n of_o chord_n take_v 38°_n 28_o 28_o the_o compliment_n of_o the_o latitude_n and_o set_v it_o from_o a_o unto_o a_o and_o from_o b_o unto_z b_o then_o lay_v a_o ruler_n from_o c_o unto_o a_o and_o it_o will_v cut_v the_o meridian_n ard_n in_o p_o the_o pole_n of_o plate_n five_o page_n 261_o as_o radius_fw-la or_o s._n 90°_n to_o sc._n of_o the_o latitude_n so_o be_v t._n of_o the_o hour_n from_o noon_n to_o t._n of_o the_o hour-line_n from_o the_o meridian_n prop._n v._o how_o to_o draw_v the_o hour-line_n on_o a_o horizontal_n plane_n this_o horizontal_n plane_n or_o dial_n be_v one_o of_o the_o best_a and_o most_o useful_a dial_n in_o our_o oblique_a hemisphere_n admit_v it_o be_v require_v to_o make_v a_o horizontal_n dial_n for_o the_o latitude_n of_o 51°_n 32_o 32_o to_o make_v which_o first_o describe_v the_o circle_n ab_fw-la 70._o cd_o which_o represent_v your_o horizontal_n plane_n then_o cross_v it_o with_o the_o two_o diameter_n arc_n and_o brd_n then_o take_v 51°_n 32_o 32_o out_z of_o your_o line_n of_o chord_n and_o set_v it_o from_o b_o to_z a_o and_o from_o c_o to_z b_o then_o lay_v a_o ruler_n from_o a_o unto_o a_o and_o it_o will_v cut_v the_o meridian_n bd_o in_o p_o the_o pole_n of_o the_o world_n then_o lay_v a_o ruler_n from_o a_o unto_z b_o and_o it_o will_v cut_v abdella_n the_o meridian_n in_o the_o point_n ae_n where_o the_o equinoctial_a cut_v the_o meridian_n then_o through_o the_o three_o point_n a_o ae_n and_o c_o draw_v 70._o the_o equinoctial_a circle_n who_o centre_n be_v at_o h_n and_o find_v as_o in_o the_o former_a proposition_n then_o divide_v the_o semicircle_n adc_n into_o 12_o equal_a part_n in_o the_o point_n ●_o ●_o ●_o etc._n etc._n then_o lay_v a_o ruler_n to_o r_o the_o centre_n of_o the_o plane_n and_o on_o those_o point_n so_o shall_v the_o equinoctial_a circle_n aaec_fw-la be_v by_o it_o divide_v into_o 12_o unequal_a part_n in_o the_o point_n point_n point_n point_n point_n etc._n etc._n then_o a_o ruler_n lay_v unto_o p_o the_o pole_n of_o the_o world_n and_o those_o point_n shall_v cut_v the_o semicircle_n cda_n in_o those_o point_n i_o i_o i_o etc._n etc._n last_o from_o the_o centre_n r_o and_o through_o those_o point_n let_v there_o be_v draw_v right_a line_n which_o shall_v be_v the_o true_a hour-line_n of_o such_o a_o horizontal_n plane_n from_o 6_o in_o the_o morning_n until_o 6_o at_o night_n but_o for_o the_o hour_n of_o 4_o and_o 5_o in_o 70._o the_o morning_n and_o 7_o and_o 8_o in_o the_o evening_n they_o be_v delineate_v by_o produce_v 4_o and_o 5_o in_o the_o evening_n through_o the_o centre_n r_o and_o 7_o and_o 8_o in_o the_o morning_n extend_v they_o out_o unto_o the_o other_o side_n of_o the_o plane_n so_o shall_v you_o have_v those_o hour-line_n also_o on_o your_o plane_n delineate_v as_o you_o see_v in_o the_o figure_n the_o style_n of_o this_o plane_n may_v be_v a_o thin_a plate_n of_o brass_n cut_v exact_o unto_o the_o quantity_n of_o a_o angle_n of_o 51°_n 32_o 32_o and_o set_v perpendicular_a on_o the_o meridian_n line_n for_o the_o form_n of_o this_o style_n take_v out_o of_o your_o line_n of_o chord_n 51°_n 32_o 32_o and_o set_v it_o from_o d_o unto_z e_z and_o draw_z re_z which_o shall_v be_v the_o axis_n of_o the_o style_n you_o may_v also_o prefix_v the_o half_n and_o quarter_n of_o hour_n in_o the_o very_a same_o manner_n as_o the_o hour_n themselves_o be_v draw_v now_o to_o find_v out_o the_o distance_n of_o the_o hour-line_n from_o the_o meridian_n say_v as_o the_o radius_fw-la or_o s._n 90°_n to_o the_o s._n of_o the_o latitude_n so_o be_v the_o t._n of_o the_o hour_n from_o noon_n to_o the_o t._n of_o the_o hour-line_n from_o the_o meridian_n line_n these_o kind_n of_o dial_n be_v so_o frequent_o use_v 70._o with_o we_o in_o this_o oblique_a sphere_n for_o the_o help_n of_o the_o speedy_a delineat_n of_o they_o i_o have_v annex_v hereunto_o the_o table_n of_o longomontanus_n wherein_o the_o hour-line_n for_o many_o latitude_n be_v calculate_v a_o table_n show_v the_o distance_n of_o the_o hour-line_n from_o the_o meridian_n in_o these_o degree_n of_o latitude_n a_o horizontal_n dial_n latitude_n the_o hour_n from_o the_o meridian_n a_o south_n erect_v dial_n latitude_n xi_o i._o x._o two_o ix_o three_o viij_o four_o seven_o v._n vi_fw-la d_o m_o d_o m_o d_o m_o d_o m_o d_o m_o d_o m_o 30_o 7_o 38_o 16_o 6_o 26_o 34_o 40_o 54_o 61_o 49_o 90_o 00_o 60_o 31_o 7_o 51_o 16_o 34_o 27_o 14_o 41_o 42_o 62_o 28_o 90_o 00_o 59_o 32_o 8_o 4_o 17_o 1_o 27_o 53_o 42_o 30_o 63_o 6_o 90_o 00_o 58_o 33_o 8_o 17_o 17_o 27_o 28_o 34_o 43_o 17_o 63_o 45_o 90_o 00_o 57_o 34_o 8_o 30_o 17_o 54_o 29_o 13_o 44_o 5_o 64_o 42_o 90_o 00_o 56_o 35_o 8_o 43_o 18_o 20_o 29_o 49_o 44_o 46_o 64_o 56_o 90_o 00_o 55_o 36_o 8_o 56_o 18_o 45_o 30_o 25_o 45_o 28_o 65_o 27_o 90_o 00_o 54_o 37_o 9_o 9_o 19_o 9_o 31_o 1_o 46_o 9_o 65_o 58_o 90_o 00_o 53_o 38_o 9_o 21_o 19_o 34_o 31_o 37_o 46_o 50_o 66_o 29_o 90_o 00_o 52_o 39_o 9_o 33_o 19_o 57_o 32_o 9_o 47_o 26_o 66_o 55_o 90_o 00_o 51_o 40_o 9_o 46_o 20_o 20_o 32_o 40_o 48_o 1_o 67_o 20_o 90_o 00_o 50_o 41_o 9_o 58_o 20_o 43_o 33_o 14_o 48_o 37_o 67_o 45_o 90_o 00_o 49_o 42_o 10_o 10_o 21_o 7_o 33_o 47_o 49_o 13_o 68_o 11_o 90_o 00_o 48_o 43_o 10_o 22_o 21_o 29_o 34_o 17_o 49_o 44_o 68_o 32_o 90_o 00_o 47_o 44_o 10_o 24_o 21_o 50_o 34_o 46_o 50_o 14_o 68_o 52_o 90_o 00_o 46_o 45_o 10_o 43_o 22_o 12_o 35_o 15_o 50_o 45_o 69_o 14_o 90_o 00_o 45_o 46_o 10_o 54_o 22_o 33_o 35_o 44_o 51_o 16_o 69_o 37_o 90_o 00_o 44_o 47_o 11_o 5_o 22_o 33_o 36_o 10_o 51_o 43_o 69_o 53_o 90_o 00_o 43_o 48_o 11_o 16_o 23_o 12_o 36_o 35_o 52_o 9_o 70_o 10_o 90_o 00_o 42_o 49_o 11_o 26_o 23_o 32_o 37_o 1_o 52_o 35_o 70_o 28_o

90_o 00_o 41_o 50_o 11_o 36_o 23_o 51_o 37_o 27_o 53_o 1_o 70_o 43_o 90_o 00_o 40_o 51_o 11_o 46_o 24_o 9_o 37_o 50_o 53_o 24_o 70_o 58_o 90_o 00_o 39_o 52_o 11_o 56_o 24_o 26_o 38_o 13_o 53_o 46_o 71_o 12_o 90_o 00_o 38_o 53_o 12_o 5_o 24_o 44_o 38_o 36_o 54_o 8_o 71_o 27_o 90_o 00_o 37_o 54_o 12_o 14_o 25_o 2_o 38_o 59_o 54_o 30_o 71_o 41_o 90_o 00_o 36_o 55_o 12_o 23_o 25_o 18_o 39_o 18_o 54_o 50_o 71_o 53_o 90_o 00_o 35_o 56_o 12_o 32_o 25_o 33_o 39_o 38_o 55_o 9_o 72_o 4_o 90_o 00_o 34_o 57_o 12_o 46_o 25_o 49_o 39_o 58_o 55_o 28_o 72_o 16_o 90_o 00_o 33_o 58_o 12_o 48_o 26_o 5_o 40_o 18_o 55_o 46_o 72_o 27_o 90_o 00_o 32_o 59_o 13_o 56_o 26_o 19_o 40_o 36_o 56_o 1_o 72_o 38_o 90_o 00_o 31_o 60_o 13_o 58_o 26_o 30_o 40_o 53_o 56_o 15_o 72_o 47_o 90_o 00_o 30_o prop._n vi_o how_o to_o draw_v the_o hour-line_n on_o a_o erect_a decline_a plane_n these_o plane_n be_v make_v to_o set_v on_o the_o side_n of_o house_n wherein_o the_o meridian_n be_v always_o a_o perpendicular_a 71._o draw_v on_o the_o plane_n in_o who_o top_n be_v the_o centre_n where_o the_o substile_a and_o the_o hour-line_n all_o meet_a now_o before_o we_o can_v delineate_v the_o hour-line_n on_o any_o such_o plane_n two_o thing_n must_v be_v give_v as_o the_o latitude_n of_o the_o place_n and_o the_o plane_n declination_n by_o have_v which_o we_o must_v find_v these_o three_o thing_n viz._n the_o pole_n height_n above_o the_o plane_n the_o distance_n of_o the_o substile_a from_o the_o meridian_n and_o the_o plane_n difference_n of_o longitude_n for_o the_o find_n of_o which_o requisite_n by_o geometrical_a projection_n we_o describe_v on_o the_o dial_n plane_n these_o circle_n of_o the_o sphere_n viz._n the_o horizon_n meridian_n and_o equinoctial_a which_o be_v describe_v in_o their_o true_a position_n on_o the_o plane_n we_o proceed_v thus_o 71._o admit_v it_o be_v require_v to_o make_v a_o direct_a south_n dial_n on_o a_o erect_a direct_v south_n plane_n decline_a westward_o 24°_n 20_o 20_o in_o the_o latitude_n of_o 51°_n 32_o 32_o now_o in_o order_n to_o find_v the_o requisite_n before_o mention_v describe_v the_o circle_n zhno_n and_o cross_v it_o with_o the_o two_o diameter_n zqn_o and_o h_n qo_o now_o z_o be_v the_o zenith_n n_o the_o nadir_n zqn_n the_o hour-line_n of_o 12_o hqo_o the_o horizon_n now_o see_v the_o plane_n decline_v s._n w._n 34°_n 20_o 20_o make_v na_fw-mi and_o ob_fw-la each_o equal_a to_o 34_o 20_o then_o a_o ruler_n lay_v from_o z_o to_o a_o will_v cut_v the_o horizon_n in_o saint_n the_o south_n point_n of_o the_o horizon_n through_o which_o draw_v the_o meridian_n zsn_a who_o centre_n be_v at_o y_o find_v as_o in_o the_o four_o proposition_n aforego_v then_o a_o ruler_n lay_v from_o z_o to_o b_o will_v cut_v the_o horizon_n in_o w_n the_o west_n point_v thereof_o now_o the_o horizon_n and_o the_o meridian_n be_v project_v on_o the_o plane_n take_v out_o of_o your_o line_n of_o chord_n 51°_n 32_o 32_o which_o place_n from_o h_n unto_z c_o and_o from_o n_o unto_z d_o then_o lay_v a_o ruler_n from_o w_n unto_z c_o and_o it_o cut_v the_o meridian_n in_o p_o the_o pole_n of_o the_o world._n then_o through_o p_o and_o q_o 71._o draw_v the_o line_n pqd_v which_o represent_v the_o axis_n of_o the_o world_n and_o the_o substilar_a line_n of_o the_o dial_n then_o lay_v a_o ruler_n from_o w_n to_z d_o it_o cut_v the_o meridian_n in_o ae_n so_o be_v w_n ae_n two_o point_n through_o which_o the_o equinoctial_a must_v pass_v who_o centre_n be_v find_v as_o afore_o to_o be_v at_o m_n be_v always_o in_o the_o axis_n of_o the_o world_n so_o have_v you_o on_o your_o plane_n the_o horizon_n hqo_n the_o meridian_n zpsae_fw-la n_o and_o the_o equinoctial_a laekwg_fw-mi describe_v on_o the_o plane_n as_o require_v now_o first_o to_o find_v the_o pole_n height_n above_o the_o plane_n which_o in_o this_o scheme_n be_v represent_v by_o bp_o lay_v a_o ruler_n from_o g_o unto_z p_o and_o it_o shall_v cut_v the_o plane_n in_o five_o then_o measure_v the_o distance_n bv_n on_o your_o line_n of_o chord_n and_o you_o 71._o will_v find_v it_o to_o contain_v 34°_n 33_o 33_o which_o be_v the_o pole_n height_n above_o the_o plane_n second_o to_o find_v the_o distance_n of_o the_o substile_a from_o the_o meridian_n represent_v in_o the_o scheme_n by_o the_o arch_a zb_n or_o nd_o which_o measure_v as_o afore_o will_v appear_v to_o be_v 18°_n 08_o 08_o the_o distance_n of_o the_o substile_a from_o the_o meridian_n three_o to_o find_v the_o plane_n difference_n of_o longitude_n which_o in_o the_o scheme_n be_v represent_v by_o the_o angle_n aepk_v lay_v a_o ruler_n from_o p_o unto_o ae_n and_o it_o cut_v the_o plane_n in_o x_o then_o measure_v the_o arch_n dx_o as_o afore_o and_o so_o will_v you_o find_v the_o plane_n difference_n of_o longitude_n to_o be_v 30°_n 00_o 00_o thus_o by_o geometrical_a projection_n have_v we_o find_v 71._o all_o the_o three_o requisite_n now_o to_o find_v they_o by_o arithmetical_a calculation_n observe_v these_o analogy_n or_o proportion_n 1._o for_o the_o pole_n height_n above_o the_o plane_n say_v as_o radius_fw-la or_o s._n 90°_n to_o sc._n of_o the_o latitude_n 38°_n 28_o 28_o so_o be_v sc._n of_o the_o declination_n 65°_n 40_o 40_o to_o s._n of_o the_o pole_n height_n above_o the_o plane_n 34°_n 33_o 33_o 2._o for_o the_o distance_n of_o the_o substile_a from_o the_o meridian_n say_v as_o the_o radius_fw-la or_o s._n 90°_n 00_o 00_o to_o the_o s._n of_o the_o plane_n declination_n 24°_n 20_o 20_o so_o be_v tc._n of_o the_o latitude_n 38°_n 28_o 28_o to_o the_o t._n of_o the_o substilar_a distance_n from_o the_o meridian_n 18°_n 10_o 10_o 3._o for_o the_o plane_n difference_n of_o longitude_n say_v as_o the_o sc._n of_o the_o latitude_n 38°_n 28_o 28_o 71._o to_o the_o radius_fw-la or_o s._n 90°_n 00_o 00_o so_o be_v s._n of_o the_o substilar_a distance_n 18°_n 10_o 10_o to_o the_o s._n of_o the_o difference_n of_o longitude_n 30_o deg._n or_o it_o may_v be_v find_v thus_o as_o the_o s._n of_o the_o latitude_n to_o the_o radius_fw-la so_o be_v the_o t._n of_o the_o declination_n to_o the_o t._n of_o the_o difference_n of_o longitude_n require_v these_o thing_n find_v we_o come_v now_o to_o show_v how_o the_o hour-line_n may_v be_v project_v to_o project_v which_o observe_v first_o to_o lay_v a_o ruler_n from_o p_o the_o pole_n of_o the_o world_n to_o ae_n the_o intersection_n of_o 71._o the_o equinoctial_a with_o the_o meridian_n and_o it_o will_v cut_v the_o plane_n in_o x_o where_o begin_v to_o divide_v the_o semicircle_n l_o x_o g_o into_z 12_o equal_a part_n in_o the_o point_n ●_o ●_o ●_o ●_o etc._n etc._n then_o lay_v a_o ruler_n from_o q_o to_o every_o of_o those_o part_n and_o it_o shall_v cut_v the_o equinoctial_a and_o divide_v it_o into_o 12_o unequal_a part_n in_o the_o point_n point_n point_n point_n point_n etc._n etc._n then_o a_o ruler_n lay_v from_o p_o the_o pole_n of_o the_o world_n unto_o each_o of_o these_o point_n it_o will_v divide_v the_o plane_n into_o 12_o unequal_a part_n in_o the_o point_n i_o i_o i_o i_o etc._n etc._n then_o by_o a_o ruler_n lay_v from_o the_o centre_n qui_fw-fr to_o those_o point_n draw_v right_a line_n which_o shall_v be_v the_o true_a hour-line_n proper_a unto_o such_o a_o decline_a plane_n as_o you_o see_v plain_o demonstrate_v by_o the_o scheme_n now_o the_o substilar_a line_n fall_v in_o this_o dial_n just_a on_o the_o hour-line_n of_o 2_o in_o the_o afternoon_n because_o the_o plane_n decline_v westerly_a the_o angle_n of_o the_o style_n be_v dqr_n 34°_n 33_o 33_o which_o may_v be_v either_o a_o plate_n or_o wyre_n bring_v into_o such_o a_o angle_n which_o must_v be_v place_v perpendicular_a to_o the_o plane_n and_o direct_o over_o the_o substilar_a line_n qd_n 2._o now_o the_o distance_n of_o the_o hour-line_n from_o the_o substilar_a line_n may_v also_o be_v find_v by_o this_o analogy_n or_o proportion_n as_o the_o radius_fw-la 71._o to_o the_o s._n height_n of_o the_o pole_n above_o the_o plane_n so_o be_v the_o t._n of_o the_o hour-line_n from_o the_o meridian_n of_o the_o plane_n to_o the_o t._n of_o the_o hour-line_n from_o the_o substile_a thus_o have_v you_o complete_v your_o dial_n as_o you_o see_v in_o the_o scheme_n and_o here_o you_o may_v take_v notice_n that_o have_v finish_v a_o west_n decliner_n you_o have_v also_o make_v a_o east_n decliner_n if_o you_o only_o convert_v the_o hour-line_n of_o the_o west_n decliner_n in_o such_o manner_n as_o you_o see_v in_o fig._n 72._o on_o the_o east_n 72._o