chap._n 3._o of_o spherical_a triangle_n pag._n 13._o chap._n 4._o of_o the_o projection_n of_o the_o sphere_n in_o plano_fw-la pag._n 20._o chap._n 5._o of_o planometry_n and_o the_o centre_n of_o gravity_n pag._n 23._o ch._n 6._o of_o solid_a geometry_n p._n 29._o chap._n 7._o of_o the_o scale_n of_o ponderosity_n alias_o the_o stillyard_n p._n 43_o chap._n 8._o of_o the_o 4_o compendium_n for_o quadratique_a equation_n pag._n 45._o chap._n 9_o of_o recreative_a problem_n pag._n 47._o dary_n miscellany_n chap._n i._o of_o the_o inscription_n and_o circumscription_n of_o a_o circle_n 1._o forasmuch_o as_o the_o ratio_fw-la of_o a_o arch_a line_n to_o a_o right_a line_n be_v yet_o unknown_a it_o be_v absolute_o necessary_a that_o right_a line_n be_v apply_v to_o a_o circle_n for_o the_o calculation_n of_o triangle_n wherein_o arch_a line_n come_v in_o competition_n 2._o right_o line_n apply_v to_o a_o circle_n be_v chord_n sin_n tangent_n secant_v and_o verse_v sin_n 3._o the_o chord_n of_o a_o arch_n be_v a_o right_a line_n extend_v from_o one_o end_n of_o that_o arch_n to_o the_o other_o end_n thereof_o the_o sine_fw-la be_v a_o right_a line_n draw_v from_o one_o end_n of_o that_o arch_n perpendicular_o upon_o the_o diameter_n draw_v from_o the_o other_o end_n of_o that_o arch_n the_o tangent_fw-la be_v a_o right_a line_n touch_v one_o end_n of_o that_o arch_n extend_v till_o it_o concur_v with_o the_o secant_fw-la the_o secant_fw-la be_v a_o right_a line_n extend_v from_o the_o centre_n of_o the_o circle_n till_o it_o concur_v with_o the_o tangent_fw-la the_o verse_v sine_fw-la be_v a_o right_a line_n be_v a_o segment_n of_o the_o diameter_n draw_v from_o one_o end_n of_o that_o arch_n till_o it_o be_v cut_v by_o a_o perpendicular_a i._n e._n the_o sine_fw-la from_o the_o other_o end_n of_o that_o arch._n 4._o it_o be_v to_o be_v note_v by_o this_o definition_n in_o prop._n 3._o that_o the_o chord_n of_o a_o arch_n be_v common_a to_o two_o arch_n one_o of_o they_o be_v the_o compliment_n of_o the_o other_o to_o a_o whole_a circle_n and_o likewise_o the_o verse_v sine_fw-la be_v common_a to_o two_o arch_n one_o of_o they_o be_v the_o compliment_n of_o the_o other_o to_o a_o whole_a circle_n but_o the_o sine_fw-la of_o a_o arch_n be_v common_a to_o two_o arch_n one_o of_o they_o be_v the_o compliment_n of_o the_o other_o to_o a_o semicircle_n 5._o as_o the_o sum_n of_o two_o sines_n be_v to_o their_o difference_n so_o be_v the_o tangent_fw-la of_o the_o ½_n sum_n of_o those_o arch_n to_o the_o tangent_fw-la of_o their_o ½_n difference_n 6._o as_o the_o sum_n of_o two_o tangent_n be_v to_o their_o difference_n so_o be_v the_o sine_fw-la of_o the_o sum_n of_o those_o arch_n to_o the_o sine_fw-la of_o their_o difference_n 7._o as_o the_o sine_fw-la of_o the_o sum_n of_o two_o arch_n be_v to_o the_o sum_n of_o their_o sin_n so_o be_v the_o difference_n of_o those_o sin_n to_o the_o sine_fw-la of_o their_o difference_n 8._o if_o you_o put_v r_o =_o the_o radiu_o s_o of_o a_o circle_n a_o =_o a_o arch_n propose_v c_o =_o the_o chord_n of_o that_o arch_n s_o =_o the_o sine_fw-la of_o that_o arch_n t_o =_o the_o tangent_fw-la of_o that_o arch_n and_o z_o =_o the_o secant_fw-la of_o that_o arch._n then_o 9_o if_o twice_o three_o arch_n equi-different_a be_v propose_v then_o as_o the_o sine_fw-la of_o one_o of_o the_o mean_n be_v to_o the_o sum_n of_o the_o sin_n of_o its_o extreme_n so_o be_v the_o sine_fw-la of_o the_o other_o mean_v to_o the_o sum_n of_o the_o sin_n of_o its_o extreme_n 10._o and_o hence_o if_o a_o rank_n of_o arch_n be_v equi-different_a as_o the_o sine_fw-la of_o any_o arch_n in_o that_o rank_n be_v to_o the_o sum_n of_o the_o sin_n of_o any_o two_o arch_n equal_o remote_a from_o it_o on_o each_o side_n so_o be_v the_o sine_fw-la of_o any_o other_o arch_n in_o the_o say_a rank_n to_o the_o sum_n of_o the_o sin_n of_o two_o arch_n next_o to_o it_o on_o each_o side_n have_v the_o same_o common_a distance_n 11._o three_o arch_n equi-different_a be_v propose_v if_o you_o put_v z_o =_o the_o sine_fw-la of_o the_o great_a extreme_a y_fw-fr =_o the_o sine_fw-la of_o the_o lesser_a extreme_a m_n =_o the_o sine_fw-la of_o the_o mean_a m_o =_o the_o cousin_a thereof_o d_o =_o the_o sine_fw-la of_o the_o common_a difference_n d_o =_o the_o cousin_a thereof_o and_o r_o =_o the_o radius_fw-la 12._o from_o the_o last_o before_o go_v it_o be_v evident_a that_o if_o two_o three_o i._o e_o either_o the_o former_a or_o the_o latter_a 60_o deg_fw-la or_o the_o former_a 30_o deg_fw-la and_o the_o latter_a 30_o deg_fw-la of_o the_o quadrant_n be_v complete_v with_o sines_n the_o remain_a three_o part_n of_o the_o quadrant_n maybe_o complete_v by_o addition_n or_o subduction_n only_o 13._o if_o in_o a_o circle_n two_o right_a line_n be_v inscribe_v cut_v each_o other_o the_o rectangle_v of_o the_o segment_n of_o each_o line_n be_v equal_a and_o the_o angle_n at_o the_o point_n of_o intersection_n be_v measure_v by_o the_o half-sum_n of_o its_o intercept_a arch_n 14._o if_o to_o a_o circle_n two_o right_a line_n be_v adscribe_v from_o a_o point_n without_o the_o rectangle_v of_o each_o line_n from_o the_o point_n assign_v to_o the_o convex_a and_o concave_a be_v equal_a and_o the_o angle_n at_o the_o assign_a point_n be_v measure_v by_o the_o half_a difference_n of_o its_o intercept_a arch_n 15._o if_o in_o a_o circle_n or_o a_o elipsis_n three_o right_a line_n shall_v be_v inscribe_v one_o of_o they_o cut_v the_o other_o two_o then_o the_o rectangle_v of_o the_o segment_n of_o each_o line_n so_o cut_v be_v direct_v proportional_a to_o the_o rectangle_v of_o the_o respective_a segment_n of_o of_o the_o cutter_n 16._o if_o a_o plain_a triangle_n be_v inscribe_v in_o a_o circle_n the_o angle_n be_v one_o half_a of_o what_o their_o opposite_a side_n do_v subtend_v 17._o therefore_o the_o angle_n of_o a_o plain_a triangle_n be_v equal_a to_o a_o semicircle_n 18._o and_o hence_o if_o a_o rectangled_a triangle_n be_v inscribe_v in_o a_o circle_n the_o hypothenuse_n thereof_o be_v the_o diameter_n of_o the_o circle_n 19_o as_o the_o diameter_n of_o a_o circle_n be_v to_o the_o chord_n of_o a_o arch_n so_o be_v that_o chord_n to_o the_o verse_v sine_fw-la of_o that_o arch._n 20._o and_o hence_o if_o from_o the_o right_a angle_n of_o a_o rectangled_a triangle_n a_o perpendicular_a be_v let_v fall_n upon_o the_o hypothenuse_n the_o hypothenuse_n be_v thereby_o cut_v according_a to_o the_o ratio_fw-la of_o the_o square_n of_o the_o side_n 21._o if_o in_o a_o circle_n any_o plain_a triangle_n be_v inscribe_v and_o a_o perpendicular_a be_v let_v fall_n upon_o one_o of_o the_o side_n from_o the_o opposite_a angular_a point_n then_o as_o that_o perpendicular_a be_v to_o one_o of_o the_o adjacent_a side_n so_o be_v the_o other_o adjacent_a side_n to_o the_o diameter_n of_o the_o circumscribr_a circle_n 22._o if_o a_o circle_n be_v inscribe_v within_o a_o plain_a triangle_n then_o as_o the_o perimeter_n be_v to_o the_o perpendicular_a so_o be_v the_o base_a on_o which_o it_o fall_v to_o the_o radius_fw-la of_o the_o inscribe_v circle_n 23._o if_o a_o quadrilateral_a figure_n be_v inscribe_v in_o a_o circle_n and_o interfect_a with_o diagonals_n the_o rectangle_n of_o the_o diagonals_n be_v equal_a to_o the_o two_o rectangle_v of_o the_o opposite_a side_n 24._o if_o a_o circle_n be_v both_o inscribe_v and_o circumscribe_v by_o two_o like_o ordinate_a polligon_n then_o as_o the_o co-versed_n sine_fw-la of_o the_o side_n of_o the_o inscribe_v be_v to_o the_o diameter_n so_o be_v the_o area_n of_o the_o inscribe_v to_o the_o area_n of_o the_o circumscribe_v 25._o if_o a_o ordinate_a polligon_n be_v both_o inscribe_v and_o circumscribe_v by_o two_o circle_n then_o as_o the_o diameter_n of_o the_o circumscribe_v be_v to_o the_o co-versed_n sine_fw-la of_o the_o side_n of_o the_o polligon_n so_o be_v the_o area_n of_o the_o circumscribe_v to_o the_o area_n of_o the_o inscribe_v 26._o in_o any_o right_n line_v figure_n if_o a_o circle_n be_v inscribe_v then_o as_o the_o peripheria_fw-la of_o the_o circle_n be_v to_o the_o area_n thereof_o so_o be_v the_o perimeter_n of_o the_o right_n line_v figure_n to_o the_o area_n thereof_o et_fw-la contempt_n 27._o but_o in_o all_o circle_n as_o the_o peripheria_fw-la be_v to_o the_o area_n so_o be_v 2_o to_o the_o radius_fw-la 28._o therefore_o in_o any_o right_n line_v figure_n if_o a_o circle_n be_v inscribe_v as_o 2._o be_v to_o the_o radius_fw-la so_o be_v the_o perimeter_n of_o the_o right_n line_v figure_n to_o the_o area_n thereof_o chap._n ii_o of_o plain_a triangle_n 1._o a_o triangle_n be_v a_o figure_n comprehend_v of_o three_o side_n and_o
compass_n thereof_o do_v cover_v all_o thing_n like_o a_o thing_n without_o measure_n and_o end_n but_o the_o elemental_a part_n be_v cover_v within_o the_o compass_n of_o the_o heaven_n the_o diameter_n thereof_o contain_v the_o diameter_n of_o the_o earth_n 23._o time_n their_o distinction_n according_a to_o their_o motion_n be_v in_o that_o the_o celestial_a part_n have_v a_o circular_a and_o a_o spherical_a motion_n the_o elemental_a a_o right_a motion_n more_o imperfect_a than_o the_o circular_a final_o they_o be_v distinguish_v according_a to_o their_o office_n for_o of_o those_o thing_n that_o be_v engender_v in_o the_o elemental_a part_n the_o heaven_n work_v by_o a_o continual_a motion_n be_v as_o it_o be_v the_o formal_a and_o efficient_a cause_n from_o whence_o life_n be_v derive_v and_o the_o elemental_a part_n which_o be_v subject_a unto_o passion_n and_o alteration_n be_v as_o it_o be_v the_o material_a cause_n from_o whence_o nourishment_n do_v proceed_v the_o reason_n of_o the_o sublunare_n or_o elemental_a region_n chap._n 9_o the_o elemental_a region_n which_o the_o heaven_n encompass_v comprehend_v within_o it_o the_o element_n wherein_o we_o be_v to_o consider_v the_o definition_n the_o number_n and_o the_o situation_n or_o order_n the_o element_n be_v simple_a body_n aswell_o in_o respect_n of_o the_o mix_a body_n which_o be_v understand_v to_o be_v compound_v of_o they_o as_o of_o the_o simple_a and_o least_o part_n as_o also_o in_o respect_n of_o the_o division_n for_o that_o they_o can_v be_v divide_v into_o body_n of_o diverse_a kind_n if_o they_o be_v give_v pure_a and_o without_o mixture_n for_o the_o use_n of_o live_a creature_n and_o thing_n grow_v do_v make_v they_o impure_a the_o element_n be_v 4._o in_o number_n find_v so_o to_o be_v by_o sense_n and_o by_o reason_n the_o element_n be_v find_v to_o be_v sour_a by_o sense_n which_o the_o physician_n do_v sollow_v first_o for_o that_o more_o simple_a body_n can_v be_v show_v 2_o nature_n have_v allot_v unto_o they_o certain_a place_n to_o the_o end_n that_o other_o thing_n may_v by_o they_o be_v breed_v and_o nourish_v 3_o nothing_o else_o can_v evident_o be_v show_v whereof_o other_o thing_n may_v be_v make_v 4_o in_o live_a creture_n there_o be_v certain_a part_n agreeable_a unto_o the_o nature_n of_o the_o several_a element_n the_o element_n be_v find_v to_o be_v four_o by_o two_o reason_n the_o former_a whereof_o be_v draw_v from_o the_o number_n of_o the_o four_o prime_a quality_n and_o the_o four_o fold_n possible_a knit_n together_o of_o they_o for_o heat_n may_v be_v join_v either_o with_o dryness_n which_o two_o make_v fire_n or_o else_o with_o moisture_n which_o two_o do_v make_v up_o aer_fw-la and_o cold_a may_v be_v join_v with_o moisture_n as_o it_o come_v to_o pass_v in_o the_o water_n or_o with_o dryness_n as_o in_o earth_n the_o late_a reason_n be_v take_v from_o the_o four_o fold_v difference_n of_o the_o right_a motion_n for_o the_o element_n be_v direct_o move_v either_o upwarde_a or_o downward_o such_o thing_n as_o move_v upward_o as_o light_v thing_n do_v be_v say_v so_o to_o do_v either_o simple_o as_o the_o fire_n which_o be_v the_o light_a of_o the_o rest_n or_o respectivelie_o as_o the_o aer_fw-la which_o be_v light_a than_o the_o water_n or_o the_o earth_n such_o thing_n as_o move_v downward_o as_o heanye_a thing_n do_v be_v say_v so_o to_o do_v either_o respectivelie_a as_o the_o water_n compare_v unto_o the_o fire_n and_o aer_fw-la or_o simple_o as_o the_o earth_n which_o be_v the_o heavy_a of_o all_o the_o rest_n the_o situation_n and_o order_n of_o the_o element_n be_v find_v either_o by_o their_o motion_n or_o else_o by_o the_o communication_n of_o their_o quality_n and_o first_o by_o the_o motion_n for_o inasmuch_o as_o the_o fire_n and_o the_o aer_fw-la do_v natural_o move_v upward_o the_o fire_n shall_v occupy_v the_o high_a place_n the_o aer_fw-la a_o upper_a place_n and_o for_o that_o the_o water_n and_o the_o earth_n do_v natural_o move_v downward_o the_o water_n shall_v possess_v a_o low_a place_n and_o the_o earth_n the_o low_a again_o the_o order_n of_o the_o element_n be_v find_v out_o by_o the_o communication_n of_o their_o quality_n for_o it_o be_v unfit_a that_o such_o thing_n as_o be_v mere_o contrary_a but_o such_o as_o in_o some_o sort_n can_v agree_v together_o shall_v be_v nigh_o one_o another_o the_o fire_n therefore_o shall_v be_v join_v unto_o the_o aer_fw-la by_o reason_n of_o the_o heat_n common_a unto_o they_o both_o the_o aer_fw-la unto_o the_o water_n by_o reason_n of_o the_o common_a moisture_n and_o the_o earth_n unto_o the_o water_n by_o reason_n of_o coldness_n common_a to_o they_o both_o the_o two_o fold_n difference_n of_o the_o celestial_a motion_n chap._n 10._o the_o whole_a frame_n of_o the_o world_n be_v carry_v round_o about_o with_o 2._o motion_n each_o of_o they_o be_v distinguish_v from_o the_o other_o in_o name_n and_o in_o reason_n the_o one_o of_o they_o be_v call_v the_o first_o and_o universal_a motion_n likewise_o the_o diurnal_a or_o worldly_a motion_n because_o it_o bring_v the_o day_n unto_o the_o world_n for_o in_o this_o motion_n the_o ☉_o and_z all_z the_o celestial_a body_n do_v every_o day_n arise_v and_o set_v they_o call_v it_o also_o the_o violent_a and_o rapt_a motion_n because_o by_o the_o violent_a swiftness_n thereof_o it_o carry_v with_o it_o the_o rest_n of_o the_o sphere_n the_o other_o be_v call_v the_o second_o and_o particular_a motion_n altogether_o contrary_a unto_o the_o former_a as_o by_o which_o all_o the_o particular_a orb_n do_v resist_v the_o universal_a motion_n they_o call_v it_o also_o sinister_a motus_fw-la the_o motion_n to_o the_o left_a hand_n as_o the_o former_a be_v in_o like_a sort_n call_v dexter_a that_o be_v the_o motion_n to_o the_o right_a hand_n the_o 2._o motion_n be_v also_o distinguish_v according_a to_o the_o reason_n or_o the_o substance_n in_o the_o which_o they_o be_v inherent_a for_o they_o differ_v the_o one_o from_o the_o other_o three_o way_n the_o first_o difference_n be_v in_o respect_n either_o of_o the_o whole_a for_o the_o diurnal_a motion_n be_v common_a unto_o all_o the_o celestial_a body_n or_o else_o of_o the_o part_n or_o star_n either_o fix_v or_o wander_a which_o have_v a_o motion_n peculiar_a and_o proper_a unto_o themselves_o the_o second_o difference_n be_v either_o in_o regard_n of_o the_o situation_n of_o the_o axe_n for_o the_o diurnal_a motion_n be_v make_v upon_o the_o axe_n and_o pole_n of_o the_o world_n and_o therefore_o the_o equator_fw-la divide_v it_o in_o the_o middle_n but_o the_o proper_a motion_n be_v make_v upon_o the_o axe_n and_o pole_n of_o the_o zodiac_n and_o therefore_o the_o zodiac_n do_v cut_v it_o in_o the_o middle_n or_o else_o it_o be_v in_o regard_n of_o the_o position_n of_o the_o term_n inasmuch_o as_o the_o diurnal_a revolution_n be_v make_v from_o the_o east_n unto_o the_o west_n or_o as_o plinte_n term_v it_o from_o the_o right_n toward_o the_o left_a hand_n but_o the_o proper_a revolution_n be_v from_o the_o west_n unto_o the_o east_n or_o from_o the_o left_a towards_o the_o right_a hand_n the_o three_o difference_n be_v in_o consideration_n of_o the_o swiftness_n for_o the_o diurnal_a motion_n fulfil_v his_o course_n within_o the_o space_n of_o 24._o common_a hour_n but_o the_o proper_a motion_n in_o diverse_a distance_n of_o time_n according_a to_o the_o largeness_n of_o the_o orb_n namely_o the_o orb_n of_o the_o fix_a star_n perform_v his_o circle_n in_o 36000._o year_n of_o ♄_o in_o 30_o of_o ♃_o in_o 12_o of_o ♂_o in_o 2._o year_n of_o the_o ☉_o in_o 365._o day_n and_o about_o 6._o hour_n of_o ♀_o in_o 384._o day_n after_o pliny_n the_o orb_n of_o ☿_o in_o as_o many_o day_n as_o the_o ☉_o and_o the_o orb_n of_o the_o ☽_o in_o 27._o day_n &_o 8._o hour_n the_o circular_a form_n and_o circular_a motion_n of_o the_o heaven_n chap._n 11._o the_o heaven_n be_v circular_a in_o motion_n and_o in_o figure_n the_o circular_a motion_n of_o the_o heaven_n be_v prove_v as_o well_o by_o 2._o experiment_n as_o by_o 2._o argument_n the_o one_o experiment_n be_v take_v from_o the_o star_n of_o the_o 8._o orb_n which_o both_o in_o their_o rise_n &_o set_v do_v always_o keep_v one_o &_o the_o same_o habitude_n both_o in_o regard_n of_o the_o earth_n and_o one_o to_o another_o which_o thing_n can_v agree_v with_o none_o other_o than_o a_o circular_a motion_n about_o the_o centre_n the_o other_o experiment_n be_v also_o take_v from_o the_o star_n of_o the_o 8._o orb_n always_o appear_v and_o retain_v in_o diverse_a place_n the_o same_o distance_n from_o the_o pole_n and_o onestom_n another_o which_o also_o agree_v with_o the_o circular_a motion_n only_o the_o first_o
equator_fw-la and_o true_a place_n of_o the_o starie_n it_o be_v divide_v into_o the_o septentrional_a and_o meridional_a declination_n the_o relation_n that_o the_o star_n have_v unto_o the_o zodiac_n be_v also_o two_o fold_n either_o according_a to_o the_o longitude_n of_o the_o zodrake_n or_o else_o according_a to_o the_o transuerse_a distance_n toward_o either_o of_o the_o pole_n in_o the_o longitude_n of_o the_o zodiac_n we_o be_v to_o consider_v the_o name_n and_o the_o definition_n it_o be_v call_v longitude_n for_o that_o it_o be_v recove_v longwaies_o on_o the_o circumference_n of_o the_o ecliptic_a it_o be_v also_o call_v the_o true_a motion_n of_o the_o star_n it_o be_v define_v to_o be_v the_o ark_n of_o the_o zodiac_n intercept_v between_o the_o head_n of_o ♈_o and_o the_o section_n of_o a_o great_a circle_n pass_v through_o the_o pole_n of_o the_o zodiac_n and_o the_o true_a place_n of_o the_o star_n in_o the_o transuerse_a distance_n we_o be_v to_o note_v the_o name_n the_o definition_n and_o the_o division_n it_o be_v call_v latitude_n because_o it_o be_v reckon_v according_a to_o the_o position_n that_o it_o have_v from_o some_o one_o side_n of_o the_o ecliptic_a it_o be_v define_v to_o be_v the_o ark_n of_o a_o great_a circle_n draw_v through_o the_o pole_n of_o the_o zodiac_n and_o the_o true_a place_n of_o the_o star_n the_o say_a ark_n be_v intercept_v between_o the_o zodiac_n and_o the_o centre_n of_o the_o star_n it_o be_v divide_v into_o the_o septentrional_a latitude_n when_o the_o star_n be_v under_o the_o northerly_a sign_n and_o into_o the_o meridional_a latitude_n when_o they_o be_v in_o the_o southerly_a sign_n of_o the_o propertion_n and_o supputation_n of_o the_o declination_n of_o every_o point_n of_o the_o ecliptic_a or_o the_o regard_n of_o the_o part_n of_o the_o zodtake_n unto_o the_o equator_fw-la chap._n 20._o in_o the_o declination_n of_o any_o point_n of_o the_o ecliptic_a 2._o thing_n be_v to_o be_v observe_v the_o proportion_n and_o the_o supputation_n in_o the_o proportion_n we_o may_v note_v also_o 2._o thing_n for_o either_o they_o have_v none_o obliquation_n or_o else_o their_o obliquation_n be_v equal_a those_o that_o have_v none_o obliquation_n be_v the_o head_n of_o ♈_o and_o ♎_o as_o be_v the_o common_a intersection_n of_o the_o equator_fw-la and_o the_o zodiac_n those_o that_o have_v equal_a obliquation_n be_v such_o as_o be_v equal_o distant_a from_o the_o equator_fw-la and_o they_o be_v either_o great_a obliquation_n or_o else_o the_o great_a the_o great_a obliquation_n be_v those_o that_o have_v any_o distance_n less_o than_o the_o great_a from_o either_o of_o the_o section_n and_o of_o that_o sort_n there_o be_v always_o four_o the_o great_a obliquation_n be_v those_o that_o have_v the_o great_a distance_n from_o the_o equator_fw-la as_o the_o head_n of_o ♋_o that_o s_o the_o summer_n solstice_n and_o the_o head_n of_o ♑_o that_o be_v the_o winter_n solstice_n the_o supputation_n be_v make_v either_o by_o the_o table_n of_o declination_n or_o of_o sines_n the_o table_n of_o declination_n be_v calculate_v in_o sundry_a place_n by_o astronomer_n and_o they_o consist_v of_o the_o 2._o side_n the_o area_n and_o of_o the_o two_o extremity_n or_o end_n the_o side_n be_v either_o at_o the_o right_a hand_n or_o at_o the_o left_a that_o at_o the_o left_a hand_n to_o be_v enter_v into_o when_o you_o have_v the_o sign_n in_o the_o top_n of_o the_o table_n and_o that_o on_o the_o right_a hand_n when_o the_o sign_n be_v in_o the_o foot_n thereof_o the_o area_n be_v that_o wherein_o at_o the_o common_a angle_n the_o declination_n be_v find_v the_o 2._o extremity_n be_v those_o that_o contain_v the_o sign_n of_o which_o extremity_n the_o one_o be_v call_v the_o top_n or_o upper_a part_n the_o other_o the_o foot_n or_o the_o nether_a patte_n of_o the_o table_n the_o supputation_n that_o be_v make_v by_o the_o table_n of_o sines_n be_v perform_v by_o the_o help_n of_o the_o rule_n of_o 4._o proportion_n all_o number_n wherein_o 3._o number_n be_v give_v and_o four_o be_v to_o be_v seek_v out_o the_o 3._o number_n give_v must_v contain_v the_o right_n sine_fw-la of_o the_o whole_a quadrant_a or_o of_o the_o semidiameter_n the_o right_a sine_fw-la of_o the_o great_a declination_n of_o the_o ☉_o and_o the_o right_a sine_fw-la of_o the_o distance_n of_o the_o point_n of_o the_o ecliptic_a give_v from_o the_o first_o section_n of_o the_o zodiac_n and_o the_o equator_fw-la the_o four_o number_n produce_v by_o multiplitation_n and_o division_n be_v the_o right_a sine_fw-la of_o the_o declination_n seek_v who_o subtended_a ark_n declare_v the_o number_n of_o degree_n of_o the_o 2._o circle_n call_v the_o colour_n distinguish_v the_o equinoctial_a and_o solstitial_a point_n chap._n 21._o forasmuch_o as_o there_o be_v certain_a point_n of_o the_o zodiac_n and_o the_o equator_fw-la more_o notable_a than_o the_o rest_n therefore_o the_o astronomer_n have_v think_v good_a to_o fit_v unto_o those_o point_n 2._o circle_n whereof_o we_o may_v consider_v the_o reason_n of_o their_o name_n their_o definition_n their_o number_n their_o figuration_n or_o description_n and_o their_o use_n they_o be_v terméd_v colour_n that_o be_v imperfect_a in_o 3._o respect_n 1._o because_o they_o appear_v always_o incomplete_a or_o maim_v the_o which_o thing_n notwithstanding_o seem_v to_o be_v common_a with_o diverse_a other_o circle_n 2._o because_o they_o have_v some_o part_n that_o do_v never_o arise_v 3._o because_o they_o be_v carry_v about_o after_o a_o imperfect_a manner_n &_o not_o according_a to_o the_o position_n of_o longitude_n as_o the_o motion_n of_o the_o heaven_n be_v the_o definition_n contain_v their_o magnitude_n their_o intersection_n and_o their_o motion_n as_o touch_v their_o magnitude_n they_o be_v of_o the_o number_n of_o the_o great_a circle_n as_o touch_v their_o intersection_n they_o cut_v one_o another_o in_o both_o the_o pole_n of_o the_o world_n at_o spherical_a right_a angle_n in_o their_o motion_n they_o be_v move_v together_o with_o the_o sphere_n their_o number_n be_v two_o whereof_o the_o one_o pass_v through_o the_o equinoctial_a point_n and_o the_o pole_n of_o the_o world_n and_o be_v call_v either_o the_o equinoctial_a colour_n or_o the_o distinguisher_n of_o the_o equinoctial_o the_o other_o pass_v through_o the_o solstitial_a point_n and_o the_o pole_n both_o of_o ecliptic_a and_o of_o the_o world_n and_o be_v call_v both_o the_o solstitial_a colour_n the_o distinguisher_n of_o the_o solstice_n and_o also_o the_o circle_n of_o the_o great_a declination_n their_o figuration_n be_v describe_v by_o the_o semidiameter_n of_o the_o world_n who_o revolution_n be_v ful_o perform_v through_o the_o pole_n of_o the_o world_n and_o the_o equinoctial_a point_n make_v the_o equinoctial_a colour_n but_o pass_v through_o the_o pole_n of_o the_o world_n and_o the_o solstitial_a point_n it_o make_v the_o solstitial_a colour_n their_o use_n be_v manifold_a but_o principal_o in_o 3._o thing_n 1._o in_o distinguish_v the_o equinoctial_a and_o solstitial_a point_n 2._o in_o reckon_n aswell_o the_o quantity_n of_o the_o great_a declination_n of_o the_o ☉_o by_o the_o ark_n intercept_v between_o the_o equator_fw-la and_o the_o ecliptic_a as_o the_o space_n comprehend_v between_o the_o pole_n of_o the_o world_n and_o the_o pole_n of_o the_o ecliptic_a which_o be_v always_o equal_a unto_o the_o ark_n of_o the_o great_a declination_n 3._o for_o better_a understanding_n of_o the_o ascension_n and_o descension_n of_o the_o sign_n of_o the_o meridian_n chap._n 22._o the_o ☉_o carry_v about_o by_o the_o first_o motion_n when_o it_o be_v at_o the_o high_a design_v a_o point_n of_o a_o circle_n who_o definition_n variety_n and_o office_n we_o be_v to_o consider_v the_o definition_n take_v hold_n both_o of_o the_o name_n thereof_o and_o of_o the_o matter_n itself_o it_o be_v call_v the_o circle_n meridian_n meridional_a and_o merinoctiall_a the_o circle_n of_o the_o midday_n and_o midnight_n either_o because_o it_o divide_v both_o the_o day_n and_o the_o night_n into_o 2._o equal_a part_n the_o one_o ascend_v the_o other_o descend_v or_o else_o because_o so_o often_o as_o the_o ☉_o according_a to_o the_o first_o motion_n be_v under_o the_o meridian_n it_o be_v then_o either_o midday_n or_o else_o midnight_n the_o matter_n itself_o be_v that_o according_a whereto_o it_o be_v define_v to_o be_v one_o of_o the_o great_a circle_n draw_v through_o the_o pole_n of_o the_o world_n and_o the_o vertical_a point_n of_o any_o place_n give_v and_o stand_v still_o when_o the_o sphere_n be_v move_v the_o variety_n of_o the_o meridian_n by_o reason_n of_o the_o round_a figure_n of_o the_o earth_n be_v either_o none_o at_o all_o or_o manifold_a it_o be_v none_o at_o all_o either_o in_o regard_n of_o reason_n or_o of_o sense_n it_o be_v
do_v ascend_v therewith_o and_o then_o it_o be_v say_v to_o have_v a_o oblique_a ascension_n by_o reason_n of_o the_o more_o oblique_a angle_n that_o it_o make_v with_o the_o horizon_n the_o culmination_n be_v define_v either_o the_o pass_v of_o some_o point_n of_o the_o zodiac_n or_o of_o the_o world_n by_o the_o meridian_n circle_n or_o else_o the_o degree_n of_o the_o equator_fw-la which_o with_o the_o portion_n of_o the_o zodiac_n give_v pass_v through_o the_o meridian_n the_o set_n of_o a_o sign_n or_o of_o any_o point_n of_o the_o heaven_n offer_v 2._o thing_n unto_o our_o consideration_n the_o definition_n and_o the_o diversity_n thereof_o the_o definition_n be_v either_o according_a to_o the_o name_n or_o according_a to_o the_o matter_n according_a to_o to_o the_o name_n it_o be_v call_v the_o descension_n or_o set_n which_o we_o measure_v by_o the_o ark_n of_o the_o equator_fw-la descend_v therewith_o according_a to_o the_o matter_n it_o be_v define_v to_o be_v the_o ark_n of_o the_o equator_fw-la comprehend_v between_o the_o sign_n or_o point_n set_v and_o the_o head_n of_o ♈_o the_o consideration_n of_o the_o diversity_n of_o set_v be_v either_o that_o a_o great_a part_n of_o the_o equator_fw-la descend_v with_o the_o sign_n or_o point_n of_o the_o heaven_n and_o then_o it_o be_v say_v to_o have_v a_o right_n or_o along_o and_o slow_a descension_n or_o else_o that_o a_o less_o portion_n of_o the_o equator_fw-la setterh_fw-mi therewith_o and_o then_o it_o be_v say_v to_o have_v a_o oblique_a or_o a_o short_a and_o swift_a descension_n of_o the_o diversity_n of_o ascension_n descension_n and_o culmination_n in_o a_o right_a sphere_n chap._n 34._o the_o zodiac_n in_o a_o right_a sphere_n be_v fit_v unto_o the_o equal_a conversion_n of_o the_o equator_fw-la and_o together_o with_o the_o part_n thereof_o pass_v by_o the_o east_n or_o the_o west_n or_o the_o midst_n of_o heaven_n both_o in_o the_o quadrant_n or_o quarter_n and_o in_o the_o sign_n the_o quadrant_n in_o equal_a space_n of_o time_n do_v ascend_v and_o descend_v or_o do_v pass_v through_o the_o midst_n of_o heaven_n begin_v either_o at_o the_o solstitial_a point_n namely_o at_o the_o head_n of_o ♋_o &_o ♑_o and_o count_v to_o the_o end_n of_o â™_o &_o ♓_o or_o else_o beginning_n at_o the_o equinoctial_a point_n which_o be_v the_o head_n of_o ♈_o &_o ♎_o and_o count_v to_o the_o end_n of_o ♊_o &_o â™_o the_o sign_n apply_v unto_o the_o motion_n of_o the_o equinoctial_a be_v consider_v either_o whole_a or_o in_o part_n the_o sign_n consider_v whole_o have_v relation_n either_o unto_o the_o equator_fw-la or_o unto_o the_o zodiac_n the_o sign_n in_o their_o relation_n unto_o the_o equator_fw-la do_v ascend_v inequal_o for_o some_o of_o they_o do_v rise_v right_o and_o some_o oblique_o those_o that_o have_v right_a ascension_n be_v ♊_o ♋_o â™_o ♑_o with_o the_o which_o there_o do_v coascende_v 32._o gr_n 11._o mi._n of_o the_o equator_fw-la those_o that_o have_v oblique_a ascension_n be_v ♈_o ♎_o ♓_o â™_o wherewith_o there_o do_v coascende_v 27._o gr_n 54._o mi._n of_o the_o equator_fw-la and_o ♉_o â™_o ♌_o â™’_o wherewith_o there_o arise_v 29._o gr_n 54._o mi._n thereof_o the_o sign_n in_o their_o relation_n unto_o the_o zodiac_n or_o consider_v several_o apart_o have_v ascension_n either_o equal_a or_o inequal_a one_o unto_o another_o they_o have_v equal_a ascension_n that_o come_v forth_o in_o equal_a time_n and_o they_o be_v either_o opposite_a in_o the_o diameter_n or_o equal_o distant_a from_o the_o equinoctial_a point_n as_o be_v ♓_o ♈_o â™’_o ♉_o ♑_o ♊_o â™_o ♋_o â™_o ♌_o ♎_o â™_o they_o have_v inequal_a ascension_n that_o neither_o be_v opposite_a nor_o equal_o distant_a from_o the_o aforesaid_a point_n the_o sign_n consider_v in_o their_o part_n have_v also_o relation_n either_o unto_o the_o equator_fw-la or_o unto_o the_o zodiac_n the_o part_n have_v relation_n unto_o the_o equator_fw-la do_v as_o before_o ascend_v inequal_o and_o that_o either_o right_o or_o oblique_o the_o part_n have_v right_a ascension_n be_v comprehend_v within_o the_o four_o sign_n next_o unto_o the_o 2._o solstitialle_n the_o part_n have_v oblique_a ascension_n be_v contain_v within_o the_o sign_n next_o unto_o the_o equinoctial_a point_n on_o each_o side_n those_o part_n of_o the_o sign_n that_o have_v relation_n unto_o the_o zodiac_n have_v their_o ascension_n partly_o equal_a and_o partly_o inequal_a part_n have_v equal_a ascension_n be_v these_o the_o first_o degree_n be_v equal_a unto_o the_o first_o degree_n of_o the_o opposite_a sign_n and_o the_o first_o degree_n unto_o the_o last_o of_o another_o sign_n equidistant_a from_o the_o equinoctial_a point_n part_n have_v inequal_a ascension_n be_v those_o in_o who_o neither_o opposition_n fall_v out_o nor_o equidistancie_n of_o the_o diversity_n of_o ascension_n and_o descension_n in_o a_o oblique_a sphere_n chap._n 35._o in_o the_o oblique_a situation_n of_o the_o sphere_n we_o consider_v either_o the_o proportion_n of_o the_o ascension_n or_o of_o the_o descension_n of_o the_o zodiac_n the_o ascension_n be_v compare_v and_o apply_v either_o unto_o the_o equator_fw-la or_o one_o with_o another_o or_o unto_o the_o ascension_n of_o a_o right_a sphere_n be_v compare_v unto_o the_o equator_fw-la they_o be_v either_o equal_a or_o inequal_a unto_o the_o ascension_n thereof_o in_o their_o equality_n they_o be_v number_v either_o in_o the_o northern_a semicircle_n from_o the_o head_n of_o ♈_o unto_o the_o end_n of_o â™_o or_o from_o the_o head_n of_o ♎_o unto_o the_o end_n of_o ♓_o in_o their_o inequality_n they_o be_v reckon_v either_o in_o the_o whole_a semicircle_n begin_v not_o in_o the_o equinoctial_a point_n but_o else_o where_o or_o else_o the_o reckon_n be_v make_v in_o some_o of_o their_o part_n in_o their_o comparison_n one_o with_o another_o they_o be_v either_o equal_a or_o inequal_a when_o they_o be_v equal_a they_o be_v reckon_v in_o some_o 2._o concordant_a arke_n of_o the_o ecliptic_a as_o in_o ♈_o ♓_o 14._o gr_n 50._o mi._n ♉_o â™’_o 18._o gr_n 51._o mi._n ♊_o ♑_o 27_o gr_n 16._o mi._n ♋_o â™_o 36._o gr_n 58._o mi._n ♌_o â™_o 40._o gr_n 57_o mi._n â™_o ♎_o 40._o gr_n 58._o mi._n in_o the_o latitude_n of_o 40._o gr_n when_o they_o be_v inequal_a they_o be_v reckon_v either_o in_o part_n not_o equidistant_a or_o in_o the_o semicircle_n either_o ascendent_n or_o descendent_a the_o semicircle_n ascendent_a be_v from_o the_o head_n of_o ♑_o unto_o the_o end_n of_o ♊_o and_o that_o ascend_v more_o oblique_a and_o swift_a the_o descendent_a semicircle_n be_v from_o the_o head_n of_o ♋_o unto_o the_o end_n of_o â™_o &_o it_o ascend_v more_o right_a and_o slow_a when_o the_o ascension_n be_v compare_v unto_o the_o ascension_n in_o a_o right_a sphere_n they_o be_v either_o less_o or_o more_o oblique_a or_o great_a or_o right_a than_o the_o say_a ascension_n in_o a_o right_a sphere_n the_o less_o or_o more_o oblique_a fall_n out_o in_o the_o north_n semicircle_n the_o great_a or_o more_o right_a happen_v in_o the_o south_n semicircle_n the_o distance_n between_o the_o ascension_n of_o each_o sphere_n be_v call_v the_o difference_n of_o ascension_n the_o descension_n of_o the_o zodiac_n be_v unto_o the_o ascension_n thereof_o either_o equal_a or_o inequal_a they_o be_v equal_a either_o in_o regard_n of_o the_o moity_n of_o the_o ecliptic_a comprehend_v between_o the_o equinoctial_a point_n or_o else_o according_a to_o the_o equidistant_n or_o opposite_a part_n of_o the_o zodiac_n the_o descension_n of_o the_o zodiac_n be_v inequal_a be_v compare_v either_o unto_o the_o right_a sphere_n or_o unto_o the_o same_o climate_n the_o descension_n of_o a_o oblique_a sphere_n be_v more_o oblique_a than_o the_o descension_n of_o a_o right_a sphere_n whereunto_o they_o be_v compare_v when_o as_o the_o ascension_n in_o a_o oblique_a sphere_n be_v more_o right_a than_o in_o a_o right_a sphere_n the_o descension_n of_o a_o oblique_a sphere_n be_v more_o right_a than_o the_o descension_n of_o a_o right_a sphere_n when_o as_o the_o ascension_n in_o a_o oblique_a sphere_n be_v more_o oblique_a then_o in_o a_o right_a sphere_n the_o inequal_a descension_n of_o the_o zodiac_n compare_v unto_o the_o same_o climate_n be_v to_o be_v note_v either_o in_o the_o part_n of_o the_o zodiac_n which_o descend_v oblique_a do_v rise_v right_a such_o as_o be_v the_o part_n of_o the_o descend_a semicircle_n or_o else_o in_o the_o part_n of_o the_o zodiac_n which_o descend_v right_o do_v rise_v oblique_a and_o such_o be_v the_o part_n of_o the_o ascend_a semicircle_n of_o the_o natural_a day_n and_o of_o the_o inequality_n and_o difference_n thereof_o chap._n 36._o out_o of_o the_o premise_n we_o may_v not_o unfit_o derive_v some_o matter_n concern_v the_o day_n whereof_o there_o be_v two_o sort_n the_o one_o