co_n which_o be_v also_o equal_a to_o pi._n 13._o theor._n to_o divide_v a_o right_a line_n in_o two_o part_n so_o that_o the_o right_n angle_a figure_n make_v of_o the_o whole_a line_n and_o one_o part_n shall_v be_v equal_a to_o the_o square_n of_o the_o other_o part_n the_o right_a line_n give_v be_v ab_fw-la upon_o the_o same_o line_n ab_fw-la make_v a_o square_a as_o abcd_v and_o divide_v the_o side_n ad_fw-la in_o two_o equal_a part_n the_o midst_n be_v m_n from_o m_n draw_v a_o line_n to_o b_o and_o produce_v ad_fw-la to_o h_n so_o that_o mh_o be_v equal_a to_o mb_v and_o upon_o ah_o make_v a_o square_a as_o ahgf._n then_o extend_v gf_n to_o e_z and_o then_o be_v the_o right_a angle_a figure_n fc_n be_v make_v of_o the_o whole_a line_n fe_o which_o be_v equal_a to_o ab_fw-la and_o the_o part_n bf_n equal_a to_o the_o square_n of_o the_o other_o part_n of_o that_o be_v to_o the_o square_a ahgf._n forasmuch_o as_o by_o the_o last_o aforego_n the_o right_a angle_a figure_n comprehend_v of_o hd_a and_o ha_o or_o the_o right_a angle_a figure_n of_o hd_a and_o hg_a as_o the_o figure_n gh_v with_o the_o square_n of_o be_o be_v together_o equal_a to_o the_o square_n of_o he_o be_v equal_a to_o bm_v it_o follow_v that_o if_o we_o take_v away_o the_o square_a â—f_n be_o common_a to_o both_o that_o the_o square_a â—f_n ab_fw-la that_o be_v the_o square_a abcd_n be_v equal_a â—o_o the_o right_a angle_a figure_n hge_v and_o the_o â—ommon_a right_a angle_a figure_n ae_n be_v take_v from_o they_o both_o there_o shall_v remain_v the_o right_a angle_a figure_n fc_n equal_a to_o the_o square_a â—hfg_n which_o be_v to_o be_v prove_v 14_o theor._n to_o divide_v a_o right_a line_n give_v by_o extreme_a and_o mean_a proportion_n a_o right_a line_n be_v say_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n when_o the_o whole_a be_v to_o the_o great_a part_n as_o the_o great_a be_v to_o the_o less_o and_o thus_o a_o right_a line_n be_v divide_v as_o the_o right_a line_n ab_fw-la be_v divide_v in_o the_o precede_a diagram_n in_o the_o point_n f_o it_o be_v divide_v in_o extreme_a and_o mean_a proportion_n that_o be_v as_o ab_fw-la be_v to_o of_o so_o be_v of_o to_o bf_n demonstration_n forasmuch_o as_o the_o right_n line_v figure_n include_v with_o ab_fw-la and_o fb_fw-la as_o the_o figure_n fbce_n be_v equal_a to_o the_o square_n of_o of_o that_o be_v to_o the_o square_a afgh_o it_o follow_v by_o the_o eleven_o theorem_n of_o this_o chapter_n that_o the_o line_n ab_fw-la be_v divide_v in_o extreme_a and_o mean_a proportion_n that_o be_v as_o ab_fw-la be_v to_o of_o so_o be_v of_o to_o fb_v 15_o theor._n in_o all_o plain_a triangle_n a_o line_n draw_v parallel_n to_o any_o of_o the_o side_n cut_v the_o other_o two_o side_n proportional_o as_o in_o the_o plain_a triangle_n abc_n kl_n be_v parallel_n to_o the_o base_a bc_n it_o cut_v off_o from_o the_o side_n ac_fw-la one_o four_o and_o also_o it_o cut_v off_o from_o the_o side_n ab_fw-la one_o three_o part_n the_o reason_n be_v because_o the_o right_a line_n eh_n cut_v off_o one_o three_o part_n from_o the_o whole_a space_n dgfb_n &_o therefore_o it_o cut_v off_o one_o three_o part_n from_o all_o the_o line_n that_o be_v draw_v quite_o through_o that_o space_n and_o hereupon_o parallel_v line_n bound_v with_o parallel_n be_v equal_a as_o the_o parallel_n ed_z and_o gh_n be_v bound_v with_o the_o parallel_n dg_n and_o he_o be_v equal_a for_o since_o the_o whole_a line_n db_v and_o gf_n be_v equal_a de_fw-fr and_o gh_n be_v one_o four_o part_n thereof_o must_v needs_o be_v equal_a also_o 16_o theor._n equiangle_v triangle_n have_v their_o side_n about_o the_o equal_a angle_n proportional_a and_o contrary_o let_v abc_n and_o ade_n be_v two_o plain_a equiangle_v triangle_n so_o as_o the_o angle_n at_o b_o and_o d_o at_z a_o and_o a_o and_o also_o at_o c_z and_o e_z be_v equal_a one_o to_o the_o other_o i_o say_v their_o side_n about_o the_o equal_a angle_n be_v proportional_a that_o be_v 1_o as_o ab_fw-la be_v to_o bc_n so_o be_v ad_fw-la to_o ed._n 2_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v ad_fw-la to_o ae_n 3_o as_o ac_fw-la be_v to_o cb_n so_o be_v ae_n to_o ed._n demonstration_n because_o the_o angle_n bac_n and_o dae_n be_v equal_a by_o the_o proposition_n therefore_o if_o a_o +_o b_o be_v apply_v to_o ad_fw-la ac_fw-la shall_v fall_v in_o ae_n and_o by_o such_o application_n be_v this_o figure_n make_v in_o which_o because_o that_o ab_fw-la and_o ad_fw-la do_v meet_v together_o and_o also_o that_o the_o angle_n at_o b_o and_o d_o be_v equal_a by_o the_o proposition_n therefore_o the_o other_o side_n bc_n and_o de_fw-fr be_v parallel_n and_o by_o the_o last_o aforego_n bc_n cut_v the_o side_n ad_fw-la and_o ae_n proportional_o and_o therefore_o as_o ab_fw-la to_o ad_fw-la so_o be_v ac_fw-la to_o ae_n moreover_o by_o the_o point_n b_o let_v there_o be_v draw_v the_o right_a line_n bf_a parallel_n to_o the_o base_a ae_n and_o it_o shall_v cut_v the_o other_o two_o side_n proportional_o in_o the_o points_z b_o and_o f_o and_o therefore_o 1._o as_o ab_fw-la to_o ad_fw-la so_o be_v of_o to_o ed_z or_o thus_o as_o ab_fw-la to_o ad_fw-la so_o be_v cb_n to_o ed_z because_o that_o fe_o and_o bc_n be_v equal_a by_o the_o last_o aforego_n 1._o theor._n in_o all_o right_n angle_v plain_a triangle_n the_o side_n include_v the_o right_a angle_n be_v equal_a to_o the_o the_o three_o side_n in_o the_o right_n angle_v plain_a triangle_n abc_n right_o angle_v at_o b_o the_o side_n ab_fw-la and_o bc_n be_v equal_a in_o power_n to_o the_o three_o side_n ac_fw-la that_o be_v the_o square_n of_o the_o side_n ab_fw-la and_o bc_n to_o wit_n the_o square_n almb_n and_o bedc_n add_v together_o be_v equal_a to_o the_o square_n of_o the_o side_n ac_fw-la that_o be_v to_o the_o square_a acki_n demonstration_n 18._o theor._n the_o three_o angle_n of_o a_o right_n line_v triangle_n be_v equal_a to_o two_o right_a angle_n as_o in_o the_o follow_a plain_n triangle_n abc_n the_o three_o angle_n abc_n acb_n and_o cab_n be_v equal_a to_o two_o right_a angle_n let_v the_o side_n ab_fw-la be_v extend_v to_o d_o and_o let_v there_o be_v a_o semicircle_n draw_v upon_o the_o point_n b_o and_o let_v there_o be_v also_o dawn_n a_o line_n parallel_n unto_o ac_fw-la from_o b_o unto_o g._n demonstration_n i_o say_v that_o the_o angle_n gbd_v be_v equal_a to_o the_o angle_n bac_n by_o the_o 9_o the_o hereof_o and_o the_o angle_n cbg_n be_v equal_a to_o the_o angle_n acb_n by_o the_o same_o reason_n and_o the_o angle_v cbg_n and_o gbd_n be_v together_o equal_a to_o the_o angle_n cbd_v which_o be_v also_o equal_a to_o the_o angle_n abc_n by_o the_o 18_o the_o of_o the_o first_o and_o therefore_o the_o three_o angle_n of_o a_o right_n line_v triangle_n be_v equal_a to_o two_o right_a angle_n which_o be_v to_o be_v prove_v 19_o theor._n if_o a_o plain_a triangle_n be_v inscribe_v in_o a_o circle_n the_o angle_n opposite_a to_o the_o circumference_n be_v half_a as_o much_o as_o that_o part_n of_o the_o circumference_n which_o be_v opposite_a to_o the_o angle_n as_o if_o in_o the_o circle_n abc_n the_o circumference_n bc_n be_v 120_o degree_n than_o the_o angle_n bac_n which_o be_v opposite_a to_o that_o circumference_n shall_v be_v 60_o degree_n the_o reason_n be_v because_o the_o whole_a circle_n abc_n be_v 360_o degree_n and_o the_o three_o angle_n of_o a_o plain_a triangle_n can_v exceed_v 180_o degree_n or_o two_o right_a angle_n by_o the_o last_o aforego_n therefore_o as_o every_o arch_n be_v the_o one_o three_o of_o 360_o so_o every_o angle_n opposite_a to_o that_o arch_n be_v the_o one_o three_o of_o 180_o that_o be_v 60_o degree_n or_o thus_o from_o the_o angle_n abc_n let_v there_o be_v draw_v the_o diameter_n bed_n and_o from_o the_o centre_n e_o to_o the_o circumference_n let_v there_o be_v draw_v the_o two_o radii_fw-la or_o semidiameter_n ae_n and_o ac_fw-la i_o say_v then_o that_o the_o divide_a angle_n abdella_n and_o dbc_n be_v the_o one_o half_a of_o the_o angle_v aed_n and_o dec_n for_o the_o angle_v abe_n and_o bae_n be_v equal_a because_o their_o radii_fw-la ae_n and_o ebb_n be_v equal_a and_o also_o the_o angle_n aed_n be_v equal_a to_o the_o angle_v abe_n and_o bae_n add_v together_o for_o if_o you_o draw_v the_o line_n of_o parallel_n to_o ab_fw-la the_o angle_n feed_v shall_v be_v equal_a to_o the_o angle_n abe_n by_o the_o 9_o the_o hereof_o and_o by_o the_o like_a reason_n the_o angle_n aef_n be_v also_o equal_a to_o the_o angle_n bae_n and_o therefore_o the_o angle_n aed_n be_v equal_a to_o the_o
line_n tp_n be_v the_o difference_n of_o the_o two_o give_v tangent_n cg_n and_o bp_o be_v double_a to_o the_o right_a line_n bk_n be_v the_o tangent_fw-la of_o the_o difference_n of_o the_o two_o give_v arch_n or_o which_o be_v all_o one_o i_o say_v that_o the_o right_a line_n tp_n be_v equal_a to_o the_o right_a line_n mk_n demonstration_n then_o that_o the_o right_a line_n mt_n be_v equal_a to_o the_o right_a line_n ka_o be_v thus_o prove_v the_o right_a line_n ma_n be_v equal_a to_o the_o right_a line_n ka_o by_o the_o work_n but_o the_o right_a line_n mt_n be_v equal_a to_o the_o right_a line_n ma_n and_o therefore_o it_o be_v also_o equal_a to_o the_o right_a line_n ka_o that_o the_o right_a line_n mt_n be_v equal_a to_o the_o right_a line_n ma_n do_v thus_o appear_v for_o that_o the_o angle_v mat_n and_o mta_n be_v equal_a and_o therefore_o the_o side_n opposite_a unto_o they_o be_v equal_a for_o equal_a side_n subtend_v equal_a angle_n and_o the_o angle_v mta_n and_o mat_n be_v equal_a because_o the_o angle_n mta_n be_v equal_a to_o the_o angle_n tac_n by_o the_o like_a reason_n that_o the_o angle_n kpa_n be_v equal_a to_o the_o angle_n dac_n and_o the_o angle_n mat_n be_v equal_a to_o the_o angle_n tac_n by_o the_o proposition_n for_o the_o arch_n c_n and_o so_o be_v put_v to_o be_v equal_a therefore_o it_o follow_v that_o they_o be_v also_o equal_a one_o to_o another_o general_o therefore_o the_o difference_n of_o the_o tangent_n of_o two_o arch_n make_v a_o quadrant_n be_v double_a to_o the_o tangent_fw-la of_o the_o difference_n of_o those_o arch_n which_o be_v to_o be_v demonstrate_v and_o by_o consequence_n the_o tangent_n of_o two_o arch_n be_v give_v make_v a_o quadrant_n the_o tangent_fw-la of_o the_o difference_n of_o those_o arch_n be_v also_o give_v and_o contrary_o the_o tangent_fw-la of_o the_o difference_n of_o those_o two_o arch_n be_v give_v together_o with_o the_o tangent_fw-la of_o one_o of_o the_o arch_n the_o tangent_fw-la of_o the_o other_o arch_n be_v also_o give_v example_n let_v there_o be_v give_v the_o tang._n of_o 72_o de_fw-la 94_o m._n and_o the_o tang._n of_o its_o compliment_n that_o be_v of_o 17_o 6_o half_a the_o difference_n of_o these_o two_o arch_n be_v 27_o 94_o tangent_fw-la of_o 72_o de_fw-la 94_o m._n be_v 32586438_o tangent_fw-la of_o 17_o 6_o 306â—761_n their_o difference_n be_v 29517677_o the_o half_a whereof_o be_v 14758838_o the_o tangent_fw-la of_o 55_o de_fw-la 88_o min._n or_o let_v the_o tangent_fw-la of_o the_o great_a arch_n 72_o d._n 94_o m._n be_v give_v with_o the_o tangent_fw-la of_o the_o difference_n 55_o de_fw-la 88_o m._n and_o let_v the_o lesser_a arch_n 17_o de_fw-la 6_o m._n be_v demand_v tangent_fw-la of_o 72_o de_fw-la 94_o m._n be_v 32586438_o tang._n of_o 55_o de_fw-la 88_o m._n double_a be_v 29517676_o  _fw-fr  _fw-fr their_o difference_n be_v 03068762_o the_o tangent_fw-la of_o 17_o de_fw-la 6_o m._n or_o last_o let_v the_o lesser_a arch_n be_v give_v with_o the_o tangent_fw-la of_o the_o difference_n and_o let_v the_o great_a arch_n be_v demand_v tang._n of_o 55_o de_fw-la 88_o m._n the_o diff_n be_v 14758838_o  _fw-fr  _fw-fr which_o double_v be_v 29517676_o to_o which_o the_o tang_n of_o 17_o d._n 6_o m._n ad_fw-la 3068761_o their_o aggregate_v be_v 32586437_o the_o tangent_fw-la of_o 72_o degree_n 94_o minute_n theor._n 2._o the_o tangent_fw-la of_o the_o difference_n of_o two_o arch_n make_v a_o quadrant_n with_o the_o tangent_fw-la of_o the_o lesser_a arch_n make_v the_o secant_fw-la of_o the_o difference_n the_o reason_n be_v because_o the_o tangent_fw-la of_o the_o difference_n bl_n or_o boy_n that_o be_v the_o right_a line_n bk_n or_o bm_n with_o the_o tangent_fw-la of_o the_o lesser_a arch_n b_n that_o be_v with_o the_o right_a line_n bt_n make_v the_o right_a line_n mt_n which_o be_v equal_a to_o the_o secant_fw-la ak_n by_o the_o demonstration_n of_o the_o first_o theorem_fw-la therefore_o the_o tangent_fw-la of_o the_o difference_n of_o two_o arch_n make_v a_o quadrant_n and_o the_o tangent_fw-la of_o the_o lesser_a arch_n be_v give_v the_o secant_fw-la of_o the_o difference_n be_v also_o give_v and_o contrary_o for_o example_n let_v the_o tangent_fw-la of_o the_o former_a difference_n 55_o degree_n 88_o minute_n and_o the_o tangent_fw-la of_o the_o lesser_a arch_n 17_o degree_n â—â—_o minute_n be_v give_v i_o say_v the_o secant_fw-la of_o this_o difference_n be_v also_o give_v tang._n of_o the_o diff_n 55_o de_fw-la 88_o m._n be_v 14758838_o the_o tangent_fw-la of_o 17_o 06_o be_v 3068762_o  _fw-fr  _fw-fr their_o sum_n be_v the_o secant_fw-la of_o 55_o 88_o 17827600_o theor._n 3_o the_o tangent_fw-la of_o the_o difference_n of_o two_o arch_n make_v a_o quadrant_n with_o the_o secant_fw-la of_o their_o difference_n be_v equal_a to_o the_o tangent_fw-la of_o the_o great_a arch_n because_o the_o tangent_fw-la of_o the_o arch_n bl_n be_v the_o difference_n of_o the_o two_o arch_n bc_n and_o dc_o make_v a_o quadrant_n with_o the_o secant_fw-la of_o the_o same_o arch_n bl_n that_o be_v the_o right_a line_n bk_n with_o the_o right_a line_n ak_v be_v equal_a to_o the_o right_a line_n bp_o by_o the_o demonstration_n of_o the_o first_o theorem_fw-la therefore_o the_o tangent_fw-la of_o the_o difference_n of_o two_o arch_n make_v a_o quadrant_n be_v give_v with_o the_o secant_fw-la of_o their_o difference_n the_o tangent_fw-la of_o the_o great_a arch_n be_v also_o give_v for_o example_n let_v the_o tangent_fw-la of_o the_o difference_n be_v the_o tang_n of_o the_o arch_n of_o 55_o de_fw-la 88_o m._n viz._n  _fw-fr 14758838_o the_o secant_fw-la of_o this_o difference_n be_v 17827600_o their_o sum_n be_v the_o tang_n of_o 72_o 94_o 32586438_o the_o great_a of_o the_o two_o former_a give_v arch_n and_o now_o by_o the_o like_a reason_n these_o rule_n may_v be_v add_v by_o way_n of_o appendix_n rule_n i._n the_o double_a tangent_fw-la of_o a_o arch_n with_o the_o tangent_fw-la of_o half_a the_o compliment_n be_v equal_a to_o the_o tangent_fw-la of_o the_o arch_n compose_v of_o the_o arch_n give_v and_o half_a the_o compliment_n thereof_o for_o if_o the_o arch_n bl_n be_v put_v for_o the_o arch_n give_v the_o double_a tangent_fw-la thereof_o shall_v be_v tp_n by_o the_o demonstration_n of_o the_o first_o theorem_fw-la and_o the_o compliment_n of_o the_o arch_n bl_n shall_v be_v the_o arch_n lc_n who_o half_a be_v the_o arch_n ld_n or_o dc_o who_o tangent_fw-la be_v the_o right_a line_n gc_n or_o bt_n but_o tp_n add_v to_o bt_n make_v bp_o be_v the_o tangent_fw-la of_o the_o arch_a bd_o compose_v of_o the_o give_v arch_n bl_n and_o half_a the_o compliment_n ld_n therefore_o the_o double_a tangent_fw-la etc._n etc._n rule_n ii_o the_o tangent_fw-la of_o a_o arch_n with_o the_o tangent_fw-la of_o half_a the_o compliment_n be_v equal_a to_o the_o secant_fw-la of_o that_o arch_n for_o if_o you_o have_v the_o arch_n bl_n or_o boy_n for_o the_o arch_n give_v the_o tangent_fw-la of_o the_o arch_n give_v shall_v be_v bm_n the_o tangent_fw-la of_o half_a the_o compliment_n shall_v be_v bt_n which_o two_o tangent_n add_v together_o make_v the_o right_a line_n mt_n but_o the_o right_a line_n mt_n be_v equal_a to_o the_o right_a line_n ak_v by_o the_o demonstration_n of_o the_o first_o theorem_fw-la which_o right_a line_n ak_v be_v the_o secant_fw-la of_o the_o arch_n give_v bl_n by_o the_o proposition_n therefore_o the_o tangent_fw-la of_o a_o arch_n etc._n etc._n rule_n iii_o the_o tangent_fw-la of_o a_o arch_n with_o the_o secant_fw-la thereof_o be_v equal_a to_o the_o tangent_fw-la of_o a_o arch_n compose_v of_o the_o arch_n give_v and_o half_a the_o compliment_n for_o if_o you_o have_v the_o arch_n bl_n for_o the_o arch_n give_v bk_n shall_v be_v the_o tangent_fw-la and_o ak_v the_o secant_fw-la of_o that_o arch_n but_o the_o right_a line_n ak_v and_o kp_n be_v equal_a by_o the_o demonstration_n of_o the_o first_o theorem_fw-la therefore_o the_o tangent_fw-la of_o the_o arch_n give_v bl_n that_o be_v the_o right_a line_n bk_n with_o the_o secant_fw-la of_o the_o same_o arch_n that_o be_v ak_n be_v equal_a to_o the_o right_a line_n bp_o which_o be_v the_o tangent_fw-la of_o the_o arch_n bd_o be_v compose_v of_o the_o give_v arch_n bl_a and_o ld_n be_v half_o the_o compliment_n these_o rule_n be_v sufficient_a for_o the_o make_n of_o the_o table_n of_o natural_a sin_n tangent_n &_o secant_v the_o use_n whereof_o in_o the_o resolution_n of_o plain_a &_o spherical_a triangle_n shall_v now_o follow_v but_o because_o the_o right_n honourable_a john_n lord_n nepoir_n baron_n of_o marchiston_n have_v teach_v we_o how_o by_o borrow_a number_n call_v logarithme_n to_o perform_v the_o same_o after_o a_o more_o easy_a and_o compendious_a way_n we_o will_v first_o speak_v
fraction_n be_v in_o the_o calculation_n very_o tedious_a beside_o here_o no_o fraction_n almost_o be_v exquisite_o true_a therefore_o the_o radius_fw-la for_o the_o make_n of_o rhese_fw-mi table_n be_v to_o be_v take_v so_o much_o the_o more_o that_o there_o may_v be_v no_o error_n in_o so_o many_o of_o the_o figure_n towards_o the_o left_a hand_n as_o you_o will_v have_v place_v in_o the_o table_n and_o as_o for_o the_o number_n superfluous_a they_o be_v to_o be_v cut_v off_o from_o the_o right_a hand_n towards_o the_o left_a after_o the_o end_n of_o the_o supputation_n thus_o to_o find_v the_o number_n answer_v to_o each_o degree_n and_o minute_n of_o the_o quadrant_n to_o the_o radius_fw-la of_o 10000000_o or_o ten_o million_o i_o add_v eight_o cipher_n more_o and_o then_o my_o radius_fw-la do_v consist_v of_o sixteen_o place_n this_o do_v you_o must_v next_o find_v out_o the_o right_a sin_n of_o all_o the_o arch_n less_o than_o a_o quadant_n in_o the_o same_o part_n as_o the_o radius_fw-la be_v take_v of_o whatsoever_o bigness_n it_o be_v and_o from_o those_o right_a sin_n the_o tangent_n and_o secant_v must_v be_v find_v out_o 21._o the_o right_a sin_n in_o make_v of_o the_o table_n be_v either_o primary_n or_o secondary_a the_o primarie_a sin_n be_v those_o by_o which_o the_o rest_n be_v find_v and_o thus_o the_o radius_fw-la or_o whole_a sine_fw-la be_v the_o first_o primary_n sine_fw-la the_o which_o how_o great_a or_o little_o soever_o be_v equal_a to_o the_o side_n of_o a_o sixangled_n figure_n inscribe_v in_o a_o circle_n that_o be_v to_o the_o subtense_n of_o 60_o degree_n the_o which_o be_v thus_o demonstrate_v out_o of_o the_o radius_fw-la or_o subtense_n of_o 60_o degree_n the_o sine_fw-la of_o 30_o degree_n be_v easy_o find_v the_o half_a of_o the_o subtense_n be_v the_o measure_n of_o a_o angle_n at_o the_o circumference_n opposite_a thereunto_o by_o the_o 19_o of_o the_o second_o if_o therefore_o your_o radius_fw-la consist_v of_o 16_o place_n be_v 1000.0000.0000.0000_o the_o sine_fw-la of_o 30_o degree_n will_v be_v the_o one_o half_a thereof_o to_o wit_n 500.0000.0000.0000_o 22._o the_o other_o primary_n sin_n be_v the_o sin_n of_o 60_o 45_o 36_o and_o of_o 18_o degree_n be_v the_o half_a of_o the_o subtense_n of_o 120_o 90_o 72_o and_o of_o 36_o degree_n 23._o the_o subtense_n of_o 120_o degree_n be_v the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v find_v the_o rule_n subtract_v the_o square_a of_o the_o subtense_n of_o 60_o degree_n from_o the_o square_n of_o the_o diameter_n the_o square_a root_n of_o what_o remain_v be_v the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circleâ—_n or_o the_o subtense_n of_o 120_o degree_n the_o reason_n of_o the_o rule_n the_o subtense_n of_o a_o arch_n with_o the_o subtense_n of_o the_o compliment_n thereof_o to_o 180_o with_o the_o diameter_n make_v in_o the_o meeting_n of_o the_o two_o subtense_n a_o right_a angle_a triangle_n as_o the_o subtense_n ab_fw-la 60_o degree_n with_o the_o subtense_n ac_fw-la 120_o degree_n and_o the_o diameter_n cb_n make_v the_o right_a angle_a triangle_n abc_n right_o angle_v at_o a_o by_o the_o 19_o of_o the_o second_o and_o therefore_o the_o side_n include_v the_o right_a angle_n be_v equal_a in_o power_n to_o the_o three_o side_n by_o the_o 〈◊〉_d of_o the_o second_o therefore_o the_o square_a of_o ab_fw-la be_v take_v from_o the_o square_n of_o cb_n there_o remain_v the_o square_a of_o ac_fw-la who_o squar_fw-la root_n be_v the_o subtense_n of_o 〈◊〉_d degree_n or_o the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circle_n example_n let_v the_o diameter_n cb_n be_v 2000.0000_o 0000.0000_o the_o square_a thereof_o be_v 400000._o 00000.00000.00000.00000.00000_o the_o subtense_n of_o ab_fw-la be_v 100000.00000.00000_o the_o square_a thereof_o be_v 100000.00000.00000_o 00000.00000.00000_o which_o be_v substract_v from_o the_o square_n of_o cb_n the_o remainder_n be_v 300000.00000.00000.00000.00000.00000_o who_o square_a root_n 173205.08075.68877_o the_o subtense_n of_o 120_o degree_n consectary_n hence_o it_o follow_v that_o the_o subtense_n of_o a_o arch_n less_o than_o a_o semicircle_n be_v give_v the_o subtense_n of_o the_o compliment_n of_o that_o arch_n to_o a_o semicircle_n be_v also_o give_v 24._o the_o subtense_n of_o 90_o degree_n be_v the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v find_v the_o rule_n multiply_v the_o diameter_n in_o itself_o and_o the_o square_a root_n of_o half_a the_o product_n be_v the_o subtense_n of_o 90_o degree_n or_o the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n the_o reason_n of_o this_o rule_n the_o diagonal_a line_n of_o a_o square_n inscribe_v in_o a_o circle_n be_v two_o diameter_n and_o the_o right_a angle_a figure_n make_v of_o the_o diagonal_o be_v equal_a to_o the_o right_a angle_a figure_n make_v of_o the_o opposite_a side_n by_o the_o 20_o the_o of_o the_o second_o now_o because_o the_o diagonal_a line_n ab_fw-la and_o cd_o be_v equal_a it_o be_v all_o one_o whether_o i_o multiply_v ac_fw-la by_o itself_o or_o by_o the_o other_o diagonal_a cd_o the_o product_n will_v be_v still_o the_o same_o then_o because_o the_o side_n ab_fw-la ac_fw-la and_o bc_n do_v make_v a_o right_a angle_a triangle_n right_o angle_v at_o c_o by_o the_o 〈◊〉_d of_o the_o second_o &_o that_o the_o 〈◊〉_d ac_fw-la and_o â—b_n be_v equal_a by_o the_o work_n the_o half_a of_o the_o square_n of_o ab_fw-la must_v needs_o be_v the_o square_n of_o ac_fw-la or_o cb_n by_o the_o 17_o the_o of_o the_o second_o who_o square_a root_n the_o subtense_n of_o cb_n the_o side_n of_o a_o square_a or_o 90_o degree_n example_n let_v the_o diameter_n ab_fw-la be_v 200000.00000_o 00000_o the_o square_a thereof_o be_v 400000.00000_o 00000.00000.00000.00000_o the_o half_a whereof_o be_v 200000.00000.00000.00000.00000_o 00000._o who_o square_a root_n 14142â—_n 356â—3_n 73095._o be_v the_o subtense_n of_o 90_o degree_n or_o the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n 25._o the_o subtense_n of_o 36_o degree_n be_v the_o side_n of_o a_o decangle_n and_o may_v thus_o be_v find_v the_o rule_n divide_v the_o radius_fw-la by_o two_o then_o multiply_v the_o radius_fw-la by_o itself_o and_o the_o half_a thereof_o by_o itself_o and_o from_o the_o square_a root_n of_o the_o sum_n of_o these_o two_o product_n subtract_v the_o half_a of_o radius_fw-la what_o remain_v be_v the_o side_n of_o a_o decangle_n or_o the_o subtense_n of_o 36_o degree_n the_o reason_n of_o the_o rule_n for_o example_n let_v the_o radius_fw-la ebb_n be_v 100000.00000.00000_o then_o be_v bh_n or_o the_o half_a thereof_o 500000._o 00000.00000_o the_o square_a of_o ebb_n be_v 100000_o 00000.00000.00000.00000.00000_o and_o the_o square_a of_o bh_n 250000.00000.00000.00000_o 00000.00000.00000_o the_o sum_n of_o these_o two_o square_n viz_o 125000.00000.00000_o 00000_o 00000._o 00000_o be_v the_o square_a of_o he_o or_o hk_n who_o square_a root_n be_v 1118033â—_n 887â—9895_n from_o which_o deduct_v the_o half_a radius_fw-la bh_n 500000000000000_o and_o there_o remain_v 618033988749895_o the_o right_a line_n kb_n which_o be_v the_o side_n of_o a_o decangle_n or_o the_o subtense_n of_o 36_o degree_n 26_o the_o subtense_n of_o 72_o degree_n be_v the_o side_n of_o a_o pentagon_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v sound_a the_o rule_n subtract_v the_o side_n of_o a_o decangle_n from_o the_o diameter_n the_o remainder_n multiply_v by_o the_o radius_fw-la shall_v be_v the_o square_n of_o one_o side_n of_o a_o pentagon_n who_o square_a root_n shall_v be_v the_o side_n itself_o or_o subtense_n of_o 72_o degree_n the_o reason_n of_o the_o rule_n in_o the_o follow_a diagram_n let_v ac_fw-la be_v the_o side_n of_o a_o decangle_n equal_a to_o cx_o in_o the_o diameter_n and_o let_v the_o rest_n of_o the_o semicircle_n be_v bisect_v in_o the_o point_n e_o then_o shall_v either_o of_o the_o right_a line_n ae_n or_o ebb_n represent_v the_o side_n of_o a_o equilateral_a pentagon_n for_o ac_fw-la the_o side_n of_o a_o decangle_n subtend_v a_o arch_n of_o 36_o degree_n the_o ten_o part_n of_o a_o circle_n and_o therefore_o aeb_fw-mi the_o remain_a arch_n of_o a_o semicircle_n be_v 144_o degree_n the_o half_a whereof_o ae_n or_o ebb_n be_v 72_o degree_n the_o five_o part_n of_o a_o circle_n or_o side_n of_o a_o equilateral_a pentagon_n the_o square_a whereof_o be_v equal_a to_o the_o oblong_v make_v of_o db_n and_o bx_n demonstration_n draw_v the_o right_a line_n exit_fw-la ed_z and_o aec_fw-la then_o will_v the_o side_n of_o the_o angle_n ace_n and_o ecx_n be_v equal_a because_o cx_o be_v make_v equal_a to_o ac_fw-la and_o aec_fw-la common_a to_o both_o and_o the_o angle_n themselves_o be_v equal_a because_o they_o be_v in_o equal_a segment_n
of_o the_o same_o circle_n by_o the_o 19_o of_o the_o second_o and_o their_o base_n ae_n and_o exit_fw-la be_v equal_a by_o the_o 23_o of_o the_o second_o and_o because_o exit_fw-la be_v equal_a to_o ae_n it_o be_v also_o equal_a to_o ebb_v and_o so_o the_o triangle_n exb_n be_v equicrural_a and_o so_o be_v the_o triangle_n edb_n because_o the_o side_n ed_z and_o db_n be_v radii_fw-la and_o the_o angle_n at_o their_o base_n x_o and_o b_o e_o and_o b_o by_o the_o 24_o of_o the_o second_o and_o because_o the_o angle_n at_o b_o be_v common_a to_o both_o therefore_o the_o two_o triangle_n exb_n and_o edb_n be_v equiangle_v and_o their_o side_n proportional_a by_o the_o 18_o the_o and_o 16_o the_o theorem_n of_o the_o second_o chapter_n that_o be_v as_o db_v to_o ebb_v so_o be_v ebb_v to_o bx_v and_o the_o rectangle_n of_o db_n in_o bx_n be_v equal_a to_o the_o square_n of_o ebb_n who_o square_a root_n be_v the_o side_n ebb_n or_o subtense_n of_o 72_o degree_n example_n let_v ac_fw-la the_o side_n of_o a_o decangle_n or_o the_o subtense_n of_o 36_o degree_n be_v as_o before_o 618033988749895_o which_o be_v substract_v from_o the_o diameter_n bc_n 200000.00000_o 00000._o the_o remainder_n be_v xb_n 1381966011151105_o which_o be_v multiply_v by_o the_o radius_fw-la db_fw-la the_o product_n 1381966011251105_o 00000.00000.0000_o shall_v be_v the_o square_n of_o ebb_n who_o square_a root_n 1175570504584946_o be_v the_o right_a line_n ebb_n the_o side_n of_o a_o pentagon_n or_o subtense_n of_o 72_o degree_n consectary_n hence_o it_o follow_v that_o the_o subtense_n of_o a_o arch_n less_o than_o a_o semicircle_n be_v give_v the_o subtense_n of_o half_a the_o compliment_n to_o a_o semicircle_n be_v give_v also_o thus_o much_o of_o the_o primarie_n sines_n the_o secondary_a sin_n or_o all_o the_o sin_n remain_v may_v be_v find_v by_o these_o and_o the_o proposition_n follow_v 27._o the_o subtense_n of_o any_o two_o arch_n together_o less_o than_o a_o semicircle_n be_v give_v to_o find_v the_o subtense_n of_o both_o those_o arch_n the_o rule_n find_v the_o subtense_n of_o their_o compliment_n to_o a_o semicircle_n by_o the_o 23_o hereof_o then_o multiply_v each_o subtense_n give_v by_o the_o subtense_n of_o the_o compliment_n of_o the_o other_o subtense_n give_v the_o sum_n of_o both_o the_o product_n be_v divide_v by_o the_o diameter_n shall_v be_v the_o subtense_n of_o both_o the_o arch_n give_v the_o reason_n of_o the_o rule_n example_n let_v ai_fw-fr the_o side_n of_o a_o square_a or_o subtense_n of_o 90_o degree_n be_v 141421.35623.73059_o and_o eo_fw-la the_o side_n of_o a_o triangle_n or_o subtense_n of_o 120_o degree_n 173205.08075.68877_o the_o product_n of_o these_o two_o will_v be_v 2449489742783_o 77659465844164315._o let_v ae_n the_o side_n of_o a_o sixangled_a figure_n or_o the_o subtense_n of_o 60_o degree_n be_v 100000_o 00000.00000_o and_o io_o the_o side_n of_o a_o square_a or_o subtense_n of_o 90_o degree_n 141421.35623.73059_o the_o product_n of_o these_o two_o will_v be_v 141421.35623.73059_o 00000.00000.00000_o the_o sum_n of_o these_o two_o product_n 3863703305156272659465844164315_o and_o this_o sum_n divide_v by_o the_o diameter_n ao_o 200000.00000.00000_o leave_v in_o the_o quotient_a for_o the_o side_n ei_o or_o subtense_n of_o 150_o degree_n 1931851652578136._o the_o half_a whereof_o 965925826289068_o be_v the_o sine_fw-la of_o 75_o degree_n 28_o the_o subtense_n of_o any_o two_o arch_n less_o than_o a_o semicircle_n be_v give_v to_o find_v the_o subtense_n of_o the_o difference_n of_o those_o arch_n the_o rule_n find_v the_o subtense_n of_o their_o compliment_n to_o a_o semicircle_n by_o the_o 23_o hereof_o as_o before_o then_o multiply_v each_o subtense_n give_v by_o the_o subtense_n of_o the_o compliment_n of_o the_o other_o subtense_n give_v the_o lesser_a product_n be_v substract_v from_o the_o great_a and_o their_o difference_n divide_v by_o the_o diameter_n shall_v be_v the_o subtense_n of_o the_o difference_n of_o the_o arch_n give_v the_o reason_n of_o the_o rule_n let_v the_o subtense_n of_o the_o give_v arch_n be_v ae_n and_o ei_o and_o let_v the_o subtense_n seek_v be_v the_o right_a line_n ei_o then_o because_o the_o right_a angle_a figure_n make_v of_o the_o diagonal_o a_z and_o eo_fw-la be_v equal_a to_o the_o right_a angle_a figure_n make_v of_o their_o opposite_a side_n by_o the_o 20_o of_o the_o second_o therefore_o if_o i_o subtract_v the_o right_a angle_a figure_n make_v of_o ae_n and_o io_o from_o the_o right_a angle_a figure_n make_v of_o a_z and_o eo_fw-la the_o remainder_n will_v be_v the_o right_a angle_a figure_n of_o ao_o and_o ei_o which_o be_v divide_v by_o the_o diameter_n ao_o leave_v in_o the_o quotient_a ei._n example_n 29._o the_o sine_fw-la of_o a_o arch_n less_o than_o a_o quadrant_n be_v give_v together_o with_o the_o sine_fw-la of_o half_a his_o compliment_n to_o find_v the_o sine_fw-la of_o a_o arch_a equal_a to_o the_o commplement_n of_o the_o arch_n give_v and_o the_o half_a compliment_n add_v together_o the_o rule_n multiply_v the_o double_a of_o the_o sine_fw-la give_v by_o the_o sine_fw-la of_o half_a his_o compliment_n the_o product_v divide_v by_o the_o radius_fw-la will_v leave_v in_o the_o quotient_a a_o number_n which_o be_v add_v to_o the_o sine_fw-la of_o the_o half_a compliment_n shall_v be_v the_o sine_fw-la of_o the_o arch_n seek_v the_o reason_n of_o the_o rule_n that_o be_v as_o a_z be_v to_o be_o so_o be_v cp_n to_o pm_n and_o so_o be_v ps_n to_o pn_n and_o then_o by_o composition_n as_o a_z be_o so_o be_v c_n to_o mn_v now_o then_o let_v es_fw-ge be_v the_o arch_n give_v and_o si_fw-it they_fw-mi compliment_n thereof_o to_o a_o quadrant_n then_o be_v cg_n or_o ib_n be_v equal_a to_o ay_o the_o half_a of_o the_o say_a compliment_n si_fw-it and_o be_o be_v the_o sine_fw-la thereof_o and_o the_o sine_fw-la of_o es_fw-ge be_v the_o right_a line_n his_o and_o the_o double_a c_n mn_n be_v the_o difference_n between_n be_o the_o sine_fw-la of_o cg_n or_o ib_n and_o a_o the_o sine_fw-la of_o sb_n and_o ai_fw-fr be_v the_o radius_fw-la and_o it_o be_v already_o prove_v that_o ai_fw-fr be_v in_o proportion_n too_o be_o as_o c_n be_v to_o mn_v therefore_o if_o you_o multiply_v be_o by_o sc_n and_o divide_v the_o product_n by_o ai_fw-fr the_o quotient_n will_v be_v nm_n which_o be_v add_v to_o be_o do_v make_v a_o the_o sine_fw-la or_o the_o arch_n seek_v example_n letoy_n es_fw-ge the_o arch_n give_v be_v 84_o degree_n and_o the_o sine_fw-la thereof_o 9945219_o which_o double_v be_v 19890438_o the_o sine_fw-la of_o 3_o degree_n the_o half_a compliment_n be_v 523360_o by_o which_o the_o double_a sine_fw-la of_o 84_o degree_n be_v multiply_v the_o product_n will_v be_v 104098â—9_n 631680_o which_o divide_v by_o the_o radius_fw-la the_o quotient_n will_v be_v 10409859_o from_o which_o also_o cut_v off_o the_o last_o figure_n because_o the_o sine_fw-la of_o 3_o degree_n be_v at_o first_o take_v 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the_o obtuse_a angle_n b_o be_v the_o great_a angle_n in_o the_o give_v triangle_n and_o let_v he_o be_v the_o measure_n of_o the_o angle_n at_o c_o kl_n be_v equal_a to_o the_o arch_n de_fw-fr because_o kd_v and_o le_z be_v quadrant_n and_o their_o common_a compliment_n be_v ld_n lm_o be_v equal_a to_o the_o arch_a fg_n because_o lg_n and_o fm_n be_v quadrant_n and_o their_o common_a compliment_n be_v lf_a km_n be_v equal_a to_o the_o arch_n he_o because_o ki_v and_o mh_o be_v quadrant_n and_o their_o common_a compliment_n be_v kh_n therefore_o the_o side_n of_o the_o triangle_n klm_n be_v equal_a to_o the_o angle_n of_o the_o triangle_n abc_n take_v for_o the_o great_a angle_n abc_n the_o compliment_n thereof_o fbg_n and_o by_o the_o like_a reason_n it_o may_v be_v demonstrate_v that_o the_o side_n of_o the_o triangle_n abc_n be_v equal_a to_o the_o angle_n of_o the_o triangle_n klm_n for_o the_o side_n ac_fw-la be_v equal_a to_o the_o arch_n di_fw-it be_v the_o measure_n of_o the_o 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distance_n of_o each_o hour_n add_v the_o natural_a tangent_n of_o those_o distance_n as_o here_o you_o see_v so_o be_v the_o table_n prepare_v for_o use_n by_o which_o you_o may_v easy_o frameâ—_n the_o dial_n to_o what_o greatness_n you_o will_v after_o this_o manner_n hour_n equ_n do_v tang._n 4_o 8_o 35_o 0_o 700_o 5_o 7_o 20_o 0_o 364_o 6_o 6_o 5_o 0_o 087_o  _fw-fr  _fw-fr meridian_n substile_fw-fr 7_o 5_o 10_o 0_o 176_o 8_o 4_o 25_o 0_o 166_o 9_o 3_o 40_o 0_o 839_o 10_o 2_o 55_o 0_o 1.428_o 11_o 1_o 70_o 0_o 2.747_o 12_o 12_o 85_o 0_o 11.430_o the_o geometrical_a projection_n proportion_n the_o plane_n bcde_v whereon_o you_o will_v draw_v the_o dial_n to_o what_o scantling_n you_o think_v fit_a let_v up_o represent_v the_o horizontal_a line_n upon_o any_o part_n thereof_o as_o at_o p_o make_v choice_n of_o a_o fit_a place_n for_o the_o perpendicular_a stile_n though_o afterward_o you_o may_v use_v another_o form_n near_o about_o the_o upper_a part_n of_o the_o plane_n because_o the_o great_a angle_n between_o the_o two_o meridian_n make_v the_o substile_a which_o must_v pass_v through_o the_o point_n p_o to_o fall_n so_o near_o the_o 6_o of_o clock_n hour_n as_o that_o there_o may_v be_v but_o one_o hour_n place_v above_o it_o if_o you_o desire_v to_o have_v the_o hour_n of_o 11_o upon_o the_o plane_n which_o be_v more_o useful_a than_o 4_o let_v p_o be_v the_o centre_n and_o with_o any_o chord_n the_o great_a the_o better_o make_v two_o obscure_a arch_n one_o above_o the_o horizontal_a line_n the_o other_o under_o it_o and_o with_o the_o same_o chord_n set_v off_o the_o arch_n of_o 51.70_o which_o be_v the_o angle_n between_o the_o substile_a and_o horizon_n and_o be_v the_o compliment_n of_o the_o angle_n between_o the_o substile_a and_o meridian_n and_o set_v it_o from_o five_o to_o t_o both_o way_n then_o draw_v the_o straight_a line_n tpt_v which_o shall_v be_v the_o substile_a of_o this_o dial_n which_o be_v 5_o inch_n and_o 66_o hundred_o part_n for_o the_o distance_n of_o 11_o a_o clock_n from_o the_o point_n h_n and_o will_v be_v the_o same_o with_o those_o point_n set_v off_o by_o the_o natural_a tangent_n in_o the_o table_n have_v do_v with_o this_o equinoctial_a you_o must_v do_v the_o like_a with_o another_o to_o find_v the_o place_n whereof_o it_o will_v be_v necessary_a first_o to_o know_v the_o length_n of_o the_o whole_a line_n from_o h_n the_o equinoctial_a to_o the_o centre_n of_o the_o dial_n in_o part_n of_o the_o perpendicular_a stile_n po_n if_o you_o will_v work_v by_o the_o scale_n of_o inch_n or_o else_o the_o length_n in_o natural_a tangent_n if_o you_o will_v use_v a_o diagonall_a scale_n first_o therefore_o to_o find_v the_o length_n thereof_o in_o inch-measure_n we_o have_v give_v in_o the_o right_n angle_v plain_a triangle_n hop_v the_o base_a open_a and_o the_o angle_n at_o o_o to_o find_v hp_n and_o in_o the_o triangle_n open_v centre_n we_o have_v give_v the_o perpendicular_a open_a and_o the_o angle_n po_n centre_n the_o compliment_n of_o the_o former_a to_o find_v h_n centre_n wherefore_o by_o the_o first_o case_n of_o right_n angle_v plain_a triangle_n as_o the_o radius_fw-la 90_o 10.000000_o be_v to_o the_o base_a open_v 206_o 2.313867_o so_o be_v the_o tang_n of_o hop_v 3.97_o 8.841364_o  _fw-fr  _fw-fr to_o the_o perpendicular_a ph14_n 1.155231_o again_o as_o the_o radius_fw-la 90_o 10.000000_o be_v to_o the_o perpend_v open_v 206_o 2.313867_o so_o be_v the_o tang_n po_n centre_n 86.3_o 11.158636_o  _fw-fr  _fw-fr to_o the_o base_a p_o centre_n 2972_o 3.472403_o add_v the_o two_o line_n of_o 014_o and_o 2972_o together_o and_o you_o have_v the_o whole_a line_n h_n centre_n 2986_o in_o part_n of_o the_o radius_fw-la po_n viz._n 29_o inch_n and_o 86_o part_n out_o of_o this_o line_n abate_v what_o part_n you_o will_v suppose_v 343_o that_o be_v 3_o inch_n and_o 43_o part_n and_o then_o the_o remainder_n will_v be_v 2643._o now_o if_o you_o set_v 343_o from_o h_n to_o i_o the_o triangle_n io_o centre_n will_v be_v equiangle_v with_o the_o former_a and_o i_o centre_n be_v give_v to_o find_v lo_o the_o proportion_n be_v as_o h_n centre_n the_o first_o base_n 2986_o co_fw-la be_v 6.524911_o be_v to_o ho_o the_o first_o perpend_v 206._o 2.313867_o so_o be_v i_o centre_n the_o 2d_o base_a 2643_o 3.422097_o  _fw-fr  _fw-fr to_o io_o the_o 2d_o perpend_v 182_o 2.260875_o have_v thus_o find_v the_o length_n of_o io_o to_o be_v one_o inch_n and_o 82_o part_n make_v that_o the_o radius_fw-la and_o then_o nt4_n shall_v be_v a_o tangent_fw-la line_n thereunto_o upon_o which_o according_a to_o this_o new_a radius_fw-la set_v off_o the_o hour-distance_n before_o find_v and_o so_o have_v you_o 2_o prick_n by_o which_o you_o may_v draw_v the_o height_n of_o the_o stile_n oh_o and_o the_o hour-line_n for_o the_o dial._n the_o length_n of_o h_n centre_n in_o natural_a tangent_n be_v thus_o find_v hp_n 069_o be_v the_o tangent_fw-la line_n of_o the_o angle_n hop_v 3_o deg_n 97_o min._n and_o by_o the_o same_o reason_n p_o centre_n 14421_o be_v the_o tangent_fw-la line_n of_o po_n centre_n 86.3_o the_o compliment_n of_o the_o other_o and_o therefore_o these_o two_o tangent_n add_v together_o do_v make_v 14490_o the_o length_n of_o the_o substile_a h_n centre_n that_o be_v 14_o time_n the_o radius_fw-la and_o 49_o part_n out_o of_o which_o subtract_v what_o number_n of_o part_n you_o will_v the_o rest_n be_v the_o distance_n from_o the_o second_o equinoctial_a to_o the_o centre_n in_o natural_a tangent_n suppose_v 158_o to_o be_v substract_v that_o be_v one_o radius_fw-la and_o 58_o part_n which_o set_v from_o h_n to_o t_n in_o proportion_n to_o the_o radius_fw-la ho_o and_o from_o the_o point_n
with_o eeee_o the_o quadrato_fw-la cube_fw-la with_o eeeee_o my_o first_o divisor_n i_o note_v with_o ffff_n because_o this_o equation_n be_v quadrato_fw-la quadratick_a and_o 5_o my_o second_o divisor_n i_o note_v with_o cc_o because_o the_o divisor_n itself_o be_v cubick_a these_o thing_n premise_v i_o proceed_v thus_o first_o i_o multiply_v 405_o which_o be_v 5_o aaaa_n or_o 5_o time_n the_o quadrato_fw-la quadrate_n of_o 3_o by_o e_o that_o be_v by_z 4_o and_o the_o product_n thereof_o 1620_o i_o set_v under_o my_o divisor_n correct_v so_o as_o the_o last_o figure_n thereof_o may_v stand_v under_o the_o first_o figure_n of_o the_o three_o quadrato_fw-la cubick_a number_n and_o against_o this_o number_n i_o put_v in_o the_o margin_n 5_o aaaae_fw-la that_o be_v five_o time_n the_o quadrato_fw-la quadrate_n of_o 3_o multiply_v by_o 4_o next_o 270_o ten_o time_n the_o cube_fw-la of_o 3_o by_o 16_o the_o square_a of_o 4_o and_o this_o product_v 4320_o i_o set_v under_o the_o former_a a_o place_n forward_a and_o 90_o which_o be_v 10_o time_n the_o square_a of_o 3_o i_o multiply_v by_o 64_o the_o cube_fw-la of_o 4_o &_o this_o product_n 5760_o i_o set_v under_o the_o last_o a_o place_n forward_a than_o that_o and_o 15_o which_o be_v 5_o time_n 3_o i_o multiply_v by_o 256_o the_o quadrato_fw-la quadrate_n of_o 4_o &_o the_o product_n thereof_o 3840_o i_o set_v under_o the_o three_o product_v a_o place_n forward_a and_o 1024_o the_o quadrato_fw-la cube_fw-la of_o four_o under_o that_o last_o i_o multiply_v four_o the_o last_o figure_n place_v in_o the_o quotient_a by_o 5_o my_o divisor_n and_o the_o last_o figure_n of_o this_o product_n i_o set_v under_o 5_o my_o divisor_n and_o suppose_v cipher_n to_o be_v thereunto_o annex_v i_o collect_v these_o several_a product_n into_o one_o sum_n and_o their_o aggreagate_n 20000021135424_o be_v five_o root_n more_o one_o quadrato_fw-la quadrate_n under_o which_o i_o draw_v a_o line_n and_o seek_v the_o five_o cube_n to_o be_v substract_v thus_o first_o i_o multiply_v 135_o which_o be_v thrice_o the_o square_a of_o three_o multiply_v by_o five_o my_o cubick_a divisor_n by_o four_o the_o figure_n last_o place_v in_o the_o quotient_a and_o the_o product_n thereof_o 540_o i_o set_v under_o the_o last_o sum_n so_o as_o the_o last_o figure_n thereof_o may_v be_v under_o the_o first_o figure_n of_o the_o three_o cube_fw-la next_o i_o multiply_v 45_o that_o be_v five_o time_n the_o triple_a of_o three_o by_o 16_o the_o square_a of_o four_o and_o this_o product_v 720_o i_o set_v under_o the_o former_a a_o place_n forward_a and_o under_o that_o 320_o which_o be_v five_o time_n the_o cube_fw-la of_o 4_o a_o place_n forward_a too_o these_o product_n draw_v into_o one_o sum_n do_v make_v 61520_o the_o five_o cube_n to_o the_o substracted_a from_o the_o five_o root_n more_o one_o quadrato_fw-la quadrate_n before_o find_v which_o be_v do_v the_o remainder_n will_v be_v 19938501135424_o and_o this_o remainder_n be_v substract_v from_o the_o figure_n of_o the_o subtense_n give_v over_o the_o head_n thereof_o the_o remainder_n will_v be_v 450.79600_o 59892_o and_o because_o such_o a_o substraction_n may_v be_v convenient_o make_v i_o conclude_v that_o i_o have_v find_v the_o true_a quotient_a and_o so_o have_v i_o wrought_v twice_o 34_o the_o sin_n of_o two_o arch_n equal_o distant_a on_o both_o side_n from_o 60_o degree_n be_v give_v to_o find_v the_o sine_fw-la of_o the_o distance_n the_o rule_n take_v the_o difference_n of_o the_o sin_n give_v and_o that_o difference_n shall_v be_v the_o sine_fw-la of_o the_o arch_n seek_v the_o reason_n of_o the_o rule_n let_v cn_fw-la and_o pn_n be_v the_o two_o arch_n give_v and_o equal_o distant_a from_o 60_o deg_n mn_v that_o be_v equal_o distant_a on_o both_o side_n from_o the_o point_n m._n and_o let_v the_o right_a line_n ck_a and_o pl_z be_v the_o sin_n of_o those_o arch_n be_v draw_v perpendicular_a to_o the_o right_a line_n a_fw-la and_o thereupon_o parallel_v to_o one_o another_o moreover_o let_v the_o right_a line_n pt_v be_v draw_v perpendicular_a upon_o the_o right_a line_n ck_n and_o so_o parallel_v to_o the_o right_a line_n kl_n than_o this_o right_a line_n tp_n cut_v from_o the_o right_a line_n ck_n another_o line_n tk_n equal_a unto_o pl_z by_z the_o 15_o of_o the_o second_o and_o leave_v the_o right_a line_n tc_n for_o the_o difference_n of_o the_o sin_n ck_a and_o pl._n last_o the_o sin_n of_o the_o distance_n of_o either_o of_o they_o from_o 60_o degree_n let_v be_v the_o right_a line_n cd_o or_o dp_n i_o say_v that_o the_o right_a line_n tc_n be_v equal_a to_o the_o right_a line_n cd_o or_o dp_n demonstration_n example_n let_v the_o arch_n cn_fw-la be_v 70_o degree_n pn_n 50_o cm_o or_o pm_n 10_o degree_n for_o so_o many_o degree_n be_v the_o arch_n of_o 70_o degree_n and_o 50_o degree_n distant_a from_o the_o arch_n of_o 60_o degree_n on_o both_o side_n and_o let_v first_o the_o sin_n of_o 70_o degree_n and_o 10_o degree_n be_v give_v and_o let_v the_o sine_fw-la of_o 50_o degree_n be_v demand_v from_o the_o sine_fw-la of_o 70d._o ck_n 9396926_o subtract_v the_o sine_fw-la of_o 10d._o cd_o or_o et_fw-la 1736482_o  _fw-fr  _fw-fr the_o remainder_n will_v be_v the_o sine_fw-la of_o 50d._o tk_n or_o pl_z 7660444_o then_o let_v the_o sine_fw-la of_o 70_o degree_n and_o 50_o degree_n be_v give_v and_o let_v the_o sine_fw-la of_o ten_o degree_n be_v demand_v from_o the_o sine_fw-la of_o 70_o degree_n ck_o 9396926_o subtract_v the_o sine_fw-la of_o 50d._o tk_n or_o pl_z 7660444_o  _fw-fr  _fw-fr remainder_n be_v the_o sine_fw-la of_o 10d_o cd_o 1736482_o last_o let_v the_o sin_n of_o 50_o degree_n and_o 10_o degree_n be_v give_v and_o let_v the_o sine_fw-la of_o 70_o degree_n be_v demand_v to_o the_o sine_fw-la of_o 50d_o pl_z or_o tk_n 7660444_o add_v the_o sine_fw-la of_o 10d_o dp_n or_o tc_n 1736482_o  _fw-fr  _fw-fr their_o sum_n will_v be_v the_o sine_fw-la of_o 70d_o 9396926_o and_o thus_o far_o of_o the_o make_n of_o the_o table_n of_o right_a sin_n the_o table_n of_o verse_v sin_n be_v not_o necessary_a as_o have_v be_v say_v chap._n iu._n by_o the_o table_n of_o sines_n to_o make_v the_o table_n of_o tangent_n and_o secant_v 1._o as_o the_o sine_fw-la of_o the_o compliment_n be_v to_o the_o sine_fw-la of_o a_o arch_n so_o be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o that_o arch_n 2._o as_o the_o sine_fw-la of_o the_o compliment_n be_v to_o the_o radius_fw-la so_o be_v the_o radius_fw-la to_o the_o secant_fw-la of_o that_o arch_n for_o by_o the_o 16_o the_o of_o the_o second_o 1._o as_o the_o sine_fw-la of_o the_o compliment_n ab_fw-la be_v to_o the_o sine_fw-la ca_n so_o be_v the_o radius_fw-la bd_o or_o bc_n to_o df_n the_o tangent_fw-la 2._o as_o the_o sine_fw-la of_o the_o compliment_n ab_fw-la be_v to_o the_o radius_fw-la bd_o or_o bc_n so_o be_v the_o radius_fw-la bc_n to_o the_o secant_fw-la bf_n example_n let_v the_o tangent_fw-la and_o secant_fw-la of_o the_o arch_n cd_o 30_o degree_n be_v seek_v for_o the_o sine_fw-la ac_fw-la 30_o degree_n be_v 5000000_o the_o sine_fw-la of_o the_o compliment_n ab_fw-la 60_o degree_n be_v 8660254._o now_o than_o if_o you_o multiply_v the_o sine_fw-la ac_fw-la 5000000_o by_o the_o radius_fw-la cb_n 10000000_o the_o product_n will_v be_v 50000000000000_o which_o divide_v by_o the_o sine_fw-la of_o the_o compliment_n ab_fw-la 8660254_o the_o quotient_n will_v be_v 5773503_o the_o right_a line_n fd_n or_o the_o tangent_fw-la of_o the_o arch_n of_o 30_o degree_n 2._o as_o the_o sine_fw-la of_o the_o compliment_n ab_fw-la 8660254_o be_v to_o the_o radius_fw-la db_fw-la 10000000_o so_o be_v the_o radius_fw-la bc_n 10000000_o to_o fb_v the_o secant_fw-la of_o the_o arch_n of_o 30_o degree_n and_o so_o for_o any_o other_o but_o with_o more_o ease_n by_o the_o help_n of_o these_o theorem_n follow_v theorem_fw-la 1._o the_o difference_n of_o the_o tangent_n of_o any_o two_o arch_n make_v a_o quadrant_n be_v double_a to_o the_o tangent_fw-la of_o the_o difference_n of_o those_o arch_n the_o declaration_n let_v the_o two_o arch_n make_v a_o quadrant_n be_v cd_o and_o bd_o who_o tangent_n be_v cg_n and_o bp_o and_o let_v b_n be_v a_o arch_n make_v equal_a to_o cd_o and_o then_o sd_z will_v be_v the_o arch_n of_o the_o difference_n of_o the_o two_o give_v arch_n cd_o or_o b_n and_o bd._n and_o also_o let_v the_o tangent_fw-la bt_n be_v equal_a to_o the_o tangent_fw-la cg_n and_o then_o the_o right_a line_n tp_n will_v be_v the_o difference_n of_o the_o tangent_n give_v cg_n or_o bt_n and_o bp_o last_o let_v the_o arch_n bl_a and_o boy_n who_o tangent_n be_v bk_n and_o bm_n be_v make_v equal_a to_o the_o arch_n sd_z i_o say_v the_o right_a
like_o triangle_n by_o the_o 22_o of_o the_o second_o and_o their_o side_n proportional_a that_o be_v ab_fw-la  _fw-fr ab_fw-la bc_n  _fw-fr ao_o  _fw-fr  _fw-fr  _fw-fr ac_fw-la  _fw-fr de_fw-fr cf_n  _fw-fr cf_n and_o therefore_o the_o oblong_v of_o bc×ac_n ao_o ×_o de_fw-fr and_o ab_fw-la ×_o cf_n be_v equal_a and_o the_o side_n of_o equal_a rectangle_v figure_n reciprocal_o proportional_a that_o be_v as_o bc_n ao_o ∷_o de_fw-fr ac_fw-la or_o as_o ao_o bc_n ∷_o ac_fw-la de._n if_o therefore_o you_o multiply_v ao_o the_o half_a radius_fw-la by_o de_fw-fr the_o sine_fw-la of_o the_o arch_n give_v and_o divide_v the_o product_n by_o bc_n the_o sine_fw-la of_o half_a the_o arch_n give_v the_o quotient_n shall_v be_v ac_fw-la the_o sine_fw-la compliment_n of_o half_a the_o give_v arch_n or_o if_o you_o multiply_v bc_n the_o sine_fw-la of_o a_o arch_n by_o ac_fw-la the_o sine_fw-la compliment_n of_o the_o same_o arch_n and_o divide_v the_o product_n by_o ao_o the_o half_a radius_fw-la the_o quotient_n shall_v be_v de_fw-fr the_o sine_fw-la of_o the_o double_a arch_n and_o therefore_o the_o sin_n of_o 45_o degree_n be_v give_v or_o the_o logarithme_n of_o those_o sin_n the_o rest_n may_v be_v find_v by_o the_o rule_n of_o proportion_n for_o illustration_n sake_n we_o will_v add_v a_o example_n in_o natural_a and_o artificial_a number_n natural_a as_o bc_n 28_o 46947_o be_v to_o ao_o 30_o 50000_o so_o be_v de_fw-fr 56_o 82903_o to_o ac_fw-la 62_o 88294_o logarith_n as_o bc_n 28_o 9.671609_o  _fw-fr  _fw-fr be_v to_o ao_o 30_o 9.698970_o so_o be_v de_fw-fr 56_o 9.918574_o  _fw-fr  _fw-fr to_o ac_fw-la 62._o 9.945935_o 18._o §_o the_o composition_n of_o the_o natural_a tangent_n and_o secant_v by_o the_o first_o and_o second_o of_o the_o four_o be_v thus_o to_o be_v make_v 1._o as_o the_o sine_fw-la of_o the_o compliment_n be_v to_o the_o sine_fw-la of_o a_o arch_n so_o be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o that_o arch_n 2._o as_o the_o sine_fw-la of_o the_o compliment_n be_v to_o the_o radius_fw-la so_o be_v the_o radius_fw-la to_o the_o secant_fw-la of_o that_o arch_n and_o by_o the_o same_o rule_n may_v be_v also_o make_v the_o artificial_a but_o with_o more_o ease_n as_o by_o example_n it_o will_v appear_v let_v the_o tangent_fw-la of_o 30_o degree_n be_v seek_v  _fw-fr logarith_n as_o the_o cousin_a of_o 60_o degree_n 9.937531_o  _fw-fr  _fw-fr be_v to_o the_o sine_fw-la of_o 30_o 9.698970_o so_o be_v the_o radius_fw-la 10.000000_o  _fw-fr  _fw-fr to_o the_o tangent_fw-la of_o 30_o 9.761439_o and_o thus_o have_v make_v the_o artificial_a tangent_n of_o 45_o degree_n the_o other_o 45_o be_v but_o the_o arithmetical_a compliment_n of_o the_o former_a take_v as_o have_v be_v show_v in_o the_o eight_o rule_n of_o the_o five_o chapter_n again_o let_v the_o secant_fw-la of_o 30_o degree_n be_v seek_v as_o the_o cousin_a of_o 60_o degree_n 9.937531_o  _fw-fr  _fw-fr be_v to_o the_o radius_fw-la 10.000000_o so_o be_v the_o radius_fw-la 10.000000_o  _fw-fr  _fw-fr  _fw-fr 20.000000_o  _fw-fr  _fw-fr to_o the_o secant_fw-la of_o 30_o 10.062469_o and_o thus_o the_o radius_fw-la be_v add_v to_o the_o arithmetical_a compliment_n of_o the_o sine_fw-la of_o a_o arch_n their_o aggregate_v be_v the_o secant_fw-la of_o the_o compliment_n of_o that_o arch_n and_o this_o be_v sufficient_a for_o the_o construction_n of_o the_o natural_a and_o artificial_a canon_n how_o to_o find_v the_o sine_fw-la tangent_fw-la or_o secant_fw-la of_o any_o arch_n give_v in_o the_o canon_n herewith_o print_v shall_v be_v show_v in_o the_o preface_n thereunto_o here_o follow_v the_o use_n of_o the_o natural_a and_o artificial_a number_n both_o first_o in_o the_o resolve_v any_o triangle_n and_o then_o in_o astronomy_n dial_v and_o navigation_n chap._n vi_o the_o use_n of_o the_o table_n of_o natural_a and_o artificial_a sin_n and_o tangent_n and_o the_o table_n of_o logarithme_n in_o the_o dimension_n i._o of_o plain_a right_a angle_a triangle_n the_o measure_n or_o resolve_v of_o triangle_n be_v the_o find_v out_o of_o the_o unknown_a side_n or_o angle_n thereof_o by_o three_o thing_n know_v whether_o angle_n or_o side_n or_o both_o and_o this_o by_o the_o help_n of_o that_o precious_a gem_n in_o arithmetic_n for_o the_o excellency_n thereof_o call_v the_o golden_a rule_n which_o teach_v of_o four_o number_n proportional_a one_o to_o another_o any_o three_o of_o they_o be_v give_v to_o find_v out_o a_o four_o and_o also_o of_o these_o table_n aforesaid_a of_o triangle_n as_o have_v be_v say_v there_o be_v two_o sort_n plain_a and_o spherical_a a_o triangle_n upon_o a_o plain_a be_v right_o line_v upon_o the_o sphere_n circular_a right_o line_v triangle_n be_v right_o angle_v or_o oblique_a a_o right_a angle_a right-lined_n triangle_n we_o speak_v of_o first_o who_o side_n then_o relate_v to_o a_o circle_n be_v inscribe_v total_o or_o partial_o total_o if_o the_o side_n subtend_v the_o right_a angle_n be_v make_v the_o radius_fw-la of_o a_o circle_n and_o then_o all_o the_o side_n be_v call_v sines_n as_o in_o the_o triangle_n abc_n partial_o if_o either_o of_o the_o side_n adjacent_a to_o the_o right_a angle_n be_v make_v the_o radius_fw-la of_o a_o circle_n and_o then_o one_o side_n of_o the_o triangle_n be_v the_o radius_fw-la or_o whole_a sine_fw-la the_o short_a of_o the_o other_o two_o side_n be_v a_o tangent_fw-la and_o the_o long_a a_o secant_fw-la now_o according_a as_o the_o right_a angle_a triangle_n be_v suppose_v whether_o to_o be_v total_o or_o but_o partial_o inscribe_v in_o a_o circle_n so_o be_v the_o trouble_n of_o find_v the_o part_n unknown_a more_o or_o less_o whether_o side_n or_o angle_n for_o if_o the_o triangle_n be_v suppose_v to_o be_v total_o inscribe_v in_o a_o circle_n we_o be_v in_o the_o solution_n thereof_o confine_v to_o the_o table_n of_o sines_n only_o because_o all_o the_o side_n of_o such_o a_o triangle_n be_v sin_n but_o if_o the_o triangle_n be_v suppose_v to_o be_v but_o partial_o inscribe_v in_o a_o circle_n we_o be_v leave_v at_o liberty_n to_o use_v the_o table_n of_o sines_n tangent_n or_o secant_v as_o we_o shall_v find_v to_o be_v most_o convenient_a for_o the_o work_n in_o a_o right_n angle_v plain_a triangle_n either_o all_o the_o angle_n with_o one_o side_n be_v give_v and_o the_o other_o two_o side_n be_v demand_v i_o say_v all_o the_o angle_n because_o one_o of_o the_o acute_a angle_n be_v give_v the_o other_o be_v give_v also_o by_o consequence_n or_o else_o two_o side_n with_o one_o angle_n that_o be_v the_o right_a angle_n be_v give_v and_o the_o other_o two_o angle_n with_o the_o three_o side_n be_v demand_v in_o both_o which_o case_n this_o axiom_n follow_v be_v well_o nigh_o sufficient_a the_o first_o axiom_n in_o all_o plain_a triangle_n the_o side_n be_v in_o portion_n one_o to_o another_o as_o be_v the_o sin_n oâ—_n the_o angle_n opposite_a to_o those_o side_n as_o in_o the_o triangle_n abc_n the_o side_n ab_fw-la be_v in_o proportion_n to_o the_o side_n ac_fw-la as_o the_o sine_fw-la of_o the_o angle_n at_o b_o be_v in_o proportion_n to_o the_o sine_fw-la of_o the_o angle_n at_o c_o and_o so_o of_o the_o rest_n demonstration_n the_o circle_n adf_n be_v circumscribe_v about_o the_o triangle_n abc_n the_o side_n ab_fw-la be_v make_v the_o chord_n or_o subtense_n of_o the_o angle_n acb_n that_o be_v of_o the_o arch_n ab_fw-la which_o be_v opposite_a to_o the_o angle_n acb_n the_o side_n ac_fw-la be_v make_v the_o subtense_n of_o the_o angle_n abc_n and_o the_o side_n bc_n be_v make_v the_o subtense_n of_o the_o angle_n bac_n and_o be_v the_o double_a measure_n thereof_o by_o the_o 19_o theorem_fw-la of_o the_o second_o chapter_n therefore_o the_o side_n ab_fw-la be_v in_o proportion_n to_o the_o side_n ac_fw-la as_o the_o subtense_n of_o the_o angle_n acb_n be_v in_o proportion_n to_o the_o subtense_n of_o the_o angle_n abc_n but_o half_o the_o subtense_n of_o the_o angle_n acb_n be_v the_o sine_fw-la of_o the_o angle_n acb_n and_o half_a the_o subtense_n of_o the_o angle_n abc_n be_v the_o sine_fw-la of_o the_o angle_n abc_n now_o as_o the_o whole_a be_v to_o the_o whole_a so_o be_v the_o half_a to_o the_o half_a therefore_o in_o all_o plain_a triangle_n etc._n etc._n the_o first_o consectary_n the_o angle_n of_o a_o plain_a triangle_n and_o one_o side_n be_v give_v the_o reason_n of_o the_o other_o side_n be_v also_o give_v the_o second_o consectary_n two_o side_n of_o a_o plain_a triangle_n with_o a_o angle_n opposite_a to_o one_o of_o they_o be_v give_v the_o reason_n of_o the_o other_o angle_n be_v also_o give_v by_o this_o proportion_n if_o the_o side_n of_o a_o triangle_n be_v require_v put_v the_o angle_n opposite_a to_o the_o give_v side_n in_o the_o first_o place_n if_o a_o
proportion_n to_o his_o opposite_a side_n the_o base_a so_o be_v radius_fw-la to_o his_o opposite_a side_n the_o hypothenusal_a and_o thus_o you_o see_v that_o the_o hypothenusal_a may_v be_v find_v without_o the_o trouble_n of_o square_n the_o side_n and_o thence_o extract_v the_o square_a root_n and_o hence_o also_o all_o the_o case_n of_o a_o right_n angle_v plain_a triangle_n may_v be_v resolve_v several_a way_n that_o be_v to_o say_v 1._o in_o a_o plain_a right_a angle_a triangle_n the_o angle_n and_o one_o side_n be_v give_v every_o of_o the_o other_o side_n be_v give_v by_o a_o threefold_a proportion_n that_o be_v as_o you_o shall_v put_v for_o the_o radius_fw-la either_o the_o side_n subtend_v the_o right_a angle_n or_o the_o great_a or_o lesser_a side_n include_v the_o right_a angle_n 2._o any_o of_o the_o two_o side_n be_v give_v either_o of_o the_o acute_a angle_n be_v give_v by_o a_o double_a proportion_n that_o be_v as_o you_o shall_v put_v either_o this_o or_o that_o side_n for_o the_o radius_fw-la to_o make_v this_o clear_a we_o will_v first_o set_v down_o the_o ground_n or_o reason_n for_o vary_v of_o the_o term_n of_o proportion_n and_o then_o the_o proportion_n themselves_o in_o every_o case_n according_a to_o all_o the_o variation_n the_o reason_n for_o vary_v of_o the_o term_n of_o proportion_n be_v chief_o three_o the_o first_o reason_n be_v because_o the_o radius_fw-la of_o a_o circle_n do_v bear_v a_o threefold_a proportion_n to_o a_o sine_fw-la tangent_fw-la or_o secant_fw-la and_o contrariwise_o a_o sine_fw-la tangent_fw-la or_o secant_fw-la have_v a_o threefold_a proportion_n to_o radius_fw-la by_o the_o second_o axiom_n of_o this_o chapter_n for_o as_o sine_fw-la bc_n to_o rad._n ac_fw-la in_o the_o 1._o triangle_n so_o rad._n bc_n to_o secant_fw-la ac_fw-la in_o the_o 3d._n tri_fw-la so_o tang_n bc_n to_o secant_fw-la ac_fw-la in_o the_o 2d_o tri_fw-la &_o contra_fw-la again_o as_o tang_n bc_n to_o rad._n ab_fw-la in_o the_o 2d_o triang_v so_o rad._n bc_n to_o tang_n ab_fw-la in_o the_o 3d._n trian_a so_o sine_fw-la bc_n to_o sine_z ba_z in_o the_o first_o triang_n &_o contra_fw-la last_o as_o secant_fw-la ac_fw-la to_o rad._n bc_n in_o the_o 3d._n tri_fw-la so_o rad._n ac_fw-la to_o sine_fw-la bc_n in_o the_o first_o trian_a so_o secant_fw-la ac_fw-la to_o tang_n bc_n in_o the_o 2d_o tri_fw-la &_o contra_fw-la hence_o then_o as_o the_o sine_fw-la of_o a_o arch_n or_o ang_fw-mi be_v to_o rad._n so_o rad._n to_o the_o secant_fw-la comp_n of_o that_o arch_n &_o so_o be_v the_o tang_n of_o that_o arch_n to_o his_o sec_fw-la &_o contr_n also_o as_o the_o tang_n of_o a_o arch_n or_o ang_fw-mi be_v to_o rad._n so_o be_v rad._n to_o the_o tangent_fw-la compl_a thereof_o and_o so_o be_v the_o sine_fw-la thereof_o to_o the_o sine_fw-la of_o its_o compliment_n &_o contra_fw-la last_o as_o the_o secant_fw-la of_o a_o arch_n or_o ang_fw-mi to_o rad._n so_o be_v radius_fw-la to_o the_o sine_fw-la compl_a thereof_o and_o so_o be_v secant_fw-la compliment_n to_o tangent_fw-la compliment_n thereof_o &_o contra_fw-la example_n let_v there_o be_v give_v the_o angle_n at_o the_o perpendicular_a 41_o degree_n 60_o minute_n and_o the_o base_a 768_o pace_n to_o find_v the_o perpendicular_a first_o by_o the_o natural_a number_n as_o the_o secant_fw-la of_o bac_n 41_o d._n 60m_n 13372593_o  _fw-fr  _fw-fr be_v to_o radius_fw-la 10000000_o so_o be_v the_o base_a ab_fw-la 768_o  _fw-fr  _fw-fr to_o the_o perpendicular_a bc_n â—74_n 574_o by_o the_o artificial_a as_o the_o secant_fw-la of_o bac_n 41.60_o 10.1262157_o  _fw-fr  _fw-fr be_v to_o radius_fw-la 10.0000000_o so_o be_v the_o base_a 768_o 2.8853612_o  _fw-fr  _fw-fr  _fw-fr 12.8853612_o to_o the_o perpendicular_a 574_o 2.7591455_o second_o by_o the_o natural_a number_n as_o the_o radius_fw-la 10000000_o to_o the_o cousin_a of_o bac_n 41.60_o 7477981_o so_o be_v the_o base_a ab_fw-la 768_o to_o the_o perpendicular_a bc_n 574_o by_o the_o artificial_a as_o the_o radius_fw-la 10.0000000_o to_o the_o cousin_a of_o bac_n 41.60_o 9.8737843_o so_o be_v the_o base_a ab_fw-la 768_o 2.8853612_o to_o the_o perpendicular_a bc_n 574_o 2.7591455_o three_o by_o the_o natural_a number_n as_o the_o co-secant_a of_o bac_n 41.60_o 15061915_o be_v to_o the_o co-tang_n of_o bac_n 41.60_o 11263271_o so_o be_v the_o base_a ab_fw-la 768_o to_o the_o perpendicular_a bc_n 574_o by_o the_o artificial_a as_o the_o co-secant_a of_o bac_n 41.60_o 10._o 1778802_o be_v to_o the_o co-tang_n of_o bac_n 41.60_o 10.0516645_o so_o be_v the_o base_a ab_fw-la 768_o 2.8853612_o to_o the_o perpendicular_a bc_n 574_o 2.7591455_o corollary_n hence_o it_o be_v evident_a that_o radius_fw-la be_v a_o mean_a proportional_a between_o the_o sine_fw-la of_o a_o arch_n and_o the_o secant_fw-la compliment_n of_o the_o same_o arch_n also_o between_o the_o tangent_fw-la of_o a_o arch_n and_o the_o tangent_fw-la of_o the_o compliment_n of_o the_o same_o arch_n the_o second_o reason_n the_o sin_n of_o several_a arch_n and_o the_o secant_v of_o their_o compliment_n be_v reciprocal_o proportional_a that_o be_v as_o the_o sine_fw-la of_o a_o arch_n or_o angle_n be_v to_o the_o sine_fw-la of_o another_o arch_n or_o angle_v so_o be_v the_o secant_fw-la of_o the_o compliment_n of_o that_o other_o to_o the_o co-secant_a of_o the_o former_a for_o by_o the_o forego_n corollary_n radius_fw-la be_v the_o mean_a proportional_a between_o the_o sine_fw-la of_o any_o arch_n and_o the_o co-secant_a of_o the_o same_o arch_n therefore_o whatsoever_o sine_fw-la be_v multiply_v by_o the_o secant_fw-la of_o the_o compliment_n be_v equal_a to_o the_o square_n of_o radius_fw-la so_o that_o all_o rectangle_v make_v of_o the_o sin_n of_o arch_n and_o of_o the_o secant_v of_o their_o compliment_n be_v equal_a one_o to_o another_o but_o equal_a rectangle_v have_v their_o side_n reciprocal_o pro_fw-la portional_a by_o the_o ten_o theorem_n of_o the_o second_o chapter_n therefore_o the_o sin_n of_o several_a arch_n etc._n etc._n the_o three_o reason_n the_o tangent_n of_o several_a arch_n and_o the_o tangent_n of_o their_o compliment_n be_v reciprocal_o proportional_a that_o be_v as_o the_o tangent_fw-la of_o a_o arch_n or_o angle_n be_v to_o the_o tangent_fw-la of_o another_o arch_n or_o angle_n so_o be_v the_o co-tangent_a of_o that_o other_o to_o the_o co-tangent_a of_o the_o former_a for_o by_o the_o forego_n corollary_n radius_fw-la be_v the_o mean_a proportional_a between_o the_o tangent_fw-la of_o every_o arch_n and_o the_o tangent_fw-la of_o his_o compliment_n therefore_o the_o rectangle_n make_v of_o any_o tangent_fw-la and_o of_o the_o tangent_fw-la of_o his_o compliment_n be_v equal_a to_o the_o square_n of_o radius_fw-la so_o that_o all_o rectangle_v make_v of_o the_o tangent_n of_o arch_n and_o of_o the_o tangent_n of_o their_o compliment_n be_v equal_a one_o to_o another_o but_o equal_a rectangle_v etc._n etc._n as_o before_o to_o these_o three_o reason_n a_o four_o may_v be_v add_v for_o in_o the_o rule_n of_o proportion_n wherein_o there_o be_v always_o four_o term_n three_o give_v the_o four_o demand_v it_o be_v all_o one_o whether_o of_o the_o two_o middle_a term_n be_v put_v in_o the_o second_o or_o three_o place_n for_o it_o be_v all_o one_o whether_o i_o shall_v say_v as_o 2_o to_z 4_o so_o 5_o to_z 10_o or_o say_v as_o 2_o to_z 5_o so_o 4_o to_z 10_o and_o from_o hence_o every_o example_n in_o any_o triangle_n may_v be_v vary_v and_o thus_o you_o see_v the_o reason_n of_o vary_v the_o term_n of_o proportion_n we_o come_v now_o to_o show_v you_o the_o various_a proportion_n themselves_o of_o the_o several_a case_n in_o right_a angle_a plain_a triangle_n right_o angle_v plain_a triangle_n may_v be_v distinguish_v into_o seven_o case_n whereof_o those_o in_o which_o a_o side_n be_v require_v viz._n three_o may_v be_v find_v by_o a_o triple_a proportion_n and_o those_o in_o which_o a_o angle_n be_v require_v viz._n three_o may_v be_v find_v by_o a_o double_a proportion_n case_n 1._o the_o angle_n and_o base_a give_v to_o find_v the_o perpendicular_a first_o as_o sine_z the_o angle_n at_o the_o perpendicular_a be_v to_o the_o base_a so_o be_v sine_fw-la the_o angle_n at_o the_o base_a to_o the_o perpendicular_a or_o second_o thus_o as_o radius_fw-la to_o the_o base_a so_o tangent_fw-la the_o angle_n at_o the_o base_a to_o the_o perpendicular_a or_o three_o thus_o as_o the_o tangent_fw-la of_o the_o angle_n at_o the_o perpendicular_a be_v to_o the_o base_a so_o be_v radius_fw-la to_o the_o perpendicular_a case_n 2._o the_o angle_n and_o base_a give_v to_o find_v the_o hypothenusal_a first_o as_o the_o sine_fw-la of_o the_o angle_n at_o the_o perpendicular_a be_v to_o the_o base_a so_o be_v radius_fw-la to_o the_o hypothenusal_a or_o second_o thus_o as_o radius_fw-la be_v to_o the_o base_a so_o the_o secant_fw-la of_o the_o angle_n at_o the_o base_a to_o
and_o lm_o the_o tangent_fw-la thereof_o also_o df_n be_v the_o sine_fw-la of_o the_o perpendicular_a db_n and_o kb_n be_v the_o tangent_fw-la thereof_o i_o say_v then_o as_o ad_fw-la be_v to_o fd_n so_o be_v a_z to_z ih_o by_o the_o 16_o the_o theorem_a of_o the_o second_o chapter_n and_o because_o it_o be_v all_o one_o whether_o of_o the_o mean_a proportional_n be_v put_v in_o the_o second_o place_n therefore_o i_o may_v say_v as_o ad_fw-la the_o sine_fw-la of_o the_o hypothenusal_a be_v in_o proportion_n to_o ai_fw-fr radius_fw-la so_o be_v fd_v the_o sine_fw-la of_o the_o perpendicular_a to_o ih_v the_o sine_fw-la of_o the_o angle_n at_o the_o base_a 2._o in_o all_o rectangled_a spherical_a triangle_n have_v the_o same_o acute_a angle_n at_o the_o base_a the_o sin_n of_o the_o base_n and_o the_o tangent_n of_o the_o perpendicular_o be_v proportional_a for_o as_o ab_fw-la to_o kb_v so_o be_v be_o to_o ml_o by_o the_o 16_o the_o theorem_fw-la of_o the_o second_o chapter_n or_o which_o be_v all_o one_o as_o ab_fw-la the_o sine_fw-la of_o the_o base_a be_v in_o proportion_n too_o be_o radius_fw-la so_o be_v bk_n the_o tangent_fw-la of_o the_o perpendicular_a to_o ml_o the_o tangent_fw-la of_o the_o angle_n at_o the_o base_a 3._o if_o â—_o circle_n of_o the_o sphere_n be_v so_o order_v that_o the_o first_o intersect_v the_o second_o the_o second_o the_o three_o the_o three_o the_o four_o the_o four_o the_o five_o and_o the_o five_o the_o five_o at_o right_a angle_n the_o right_a angle_a triangle_n make_v by_o their_o intersection_n do_v all_o consist_v of_o the_o same_o circular_a part_n as_o in_o this_o scheme_n let_v igab_n be_v the_o first_o circle_n blf_o the_o second_o fec_n the_o three_o gad_n the_o four_o hlei_fw-mi the_o five_o then_o do_v these_o five_o circle_n retain_v the_o condition_n require_v the_o first_o intersect_v the_o second_o in_o b_o the_o second_o the_o three_o in_o f_o the_o three_o the_o four_o in_o c_o the_o four_o the_o five_o in_o h_n the_o five_o the_o angle_n we_o mark_v or_o note_v intersection_n at_o b_o f_o â—_o to_o a_o quadrant_a as_o angle_n therefore_o i_o sayâ—nt_v as_o the_o completriangle_n make_v by_o the_o interior_a ad_fw-la we_o write_v circle_n namely_o abdella_n dâ—_n write_v compl_a egi_fw-la and_o gca_n do_v all_o coâ—â—d_n ab_fw-la besame_a circular_a part_n for_o the_o circuâ—â—â—_n 〈◊〉_d in_o every_o of_o these_o triangle_n be_v as_o hâ—â—d_v by_o pear_v in_o abdella_n be_v ab_fw-la bd_o c_o bda_n c_o ad_fw-la c_o daâ—_n dhl_n c_o hld_n c_o ld_n c_o ldh_n dh_n hl_n lfe_n con_fw-mi elf_n lf_n fe_o con_fw-mi fel_n c_o el_n egi_fw-la ig_n con_fw-mi ige_v c_o ge_n con_fw-mi gei_n je_n gca_n c_o give_v c_o agc_fw-fr gc_n ca_n c_o cag_n where_o you_o may_v observe_v that_o the_o side_n ab_fw-la in_o the_o first_o triangle_n be_v equal_a to_o compl_a hld_v in_o the_o second_o or_o compl_a elf_n in_o the_o three_o or_o ig_v in_o the_o four_o or_o come_v give_v in_o the_o five_o and_o so_o of_o the_o rest_n to_o express_v this_o more_o plain_o ab_fw-la in_o the_o first_o triangle_n be_v the_o compliment_n of_o the_o angle_n hld_v in_o the_o second_o or_o the_o compliment_n of_o the_o angle_n elf_n in_o the_o three_o or_o the_o side_n ig_n in_o the_o four_o or_o the_o compliment_n of_o the_o hypothenusal_a give_v in_o the_o five_o and_o from_o these_o premise_n be_v deduce_v this_o universal_a proposition_n 4._o the_o sine_fw-la of_o the_o middle_a part_n and_o radius_fw-la be_v reciprocal_o proportional_a with_o the_o tangent_n of_o the_o extreme_n conjunct_a and_o with_o the_o co-sine_n of_o the_o extreme_n disjunct_n namely_o as_o the_o radius_fw-la to_o the_o tangent_fw-la of_o one_o of_o the_o extreme_n conjoin_v so_o be_v tangent_fw-la of_o the_o other_o extreme_a conjoin_v to_o the_o sine_fw-la of_o the_o middle_a part_n and_o also_o as_o the_o radius_fw-la to_o the_o cousin_a of_o one_o of_o the_o extreme_n disjoined_n so_o the_o cousin_a of_o the_o other_o extreme_a disjoined_n to_o the_o sine_fw-la of_o the_o middle_a part_n therefore_o if_o the_o middle_a part_n be_v seek_v the_o radius_fw-la must_v be_v in_o the_o first_o place_n if_o either_o of_o the_o extreme_n the_o other_o extreme_a must_v be_v in_o the_o first_o place_n for_o the_o better_a demonstration_n hereof_o it_o be_v first_o to_o be_v understand_v that_o a_o right_n angle_v spherical_a triangle_n have_v five_o part_n beside_o the_o right_a angle_n as_o the_o triangle_n abdella_n in_o the_o former_a diagram_n right_o angle_v at_o b_o have_v first_o the_o side_n ab_fw-la second_o the_o angle_n at_o a_o three_o the_o hypothenusal_a ad_fw-la four_a the_o angle_n adb_n five_o the_o side_n db._n three_o of_o these_o part_n which_o be_v far_a from_o the_o right_a angle_n we_o mark_v or_o noâ—e_v by_o their_o compliment_n to_o a_o quadrant_a as_o the_o angle_n bid_v we_o account_v as_o the_o compliment_n to_o the_o same_o angle_n for_o ad_fw-la we_o write_v comp_n ad_fw-la and_o for_o adb_n we_o write_v compl_a adb_n but_o the_o two_o side_n db_n and_o ab_fw-la be_v next_o to_o the_o right_a angle_n 〈…〉_z be_v not_o note_v by_o their_o compliment_n of_o these_o five_o part_n two_o be_v always_o give_v to_o find_v a_o three_o and_o of_o these_o three_o one_o be_v in_o the_o middle_n and_o the_o other_o two_o be_v extreme_n either_o adjacent_a to_o that_o middle_a one_o or_o opposite_a to_o it_o if_o the_o part_n give_v and_o require_v be_v all_o conjoin_v together_o the_o middle_n be_v the_o middle_a part_n conjunct_a and_o the_o extreme_n the_o extreme_a part_n conjunct_a if_o again_o any_o of_o the_o part_n give_v or_o require_v be_v disjoined_n that_o which_o stand_v by_o itself_o be_v the_o middle_a part_n disjoined_n and_o the_o extreme_n be_v extreme_a part_n disjoined_n thus_o if_o there_o be_v give_v in_o the_o triangle_n abdella_n the_o side_n ab_fw-la the_o angle_n at_o a_o to_o find_v the_o hypothenusal_a ad_fw-la there_o the_o angle_n at_o a_o be_v in_o the_o middle_n and_o the_o side_n ad_fw-la and_o ab_fw-la be_v adjacent_a to_o it_o and_o therefore_o the_o middle_a part_n be_v call_v the_o middle_n conjunct_a and_o the_o extreme_n be_v the_o extreme_n conjunct_a but_o if_o there_o be_v give_v the_o side_n ab_fw-la the_o hypothenusal_a ad_fw-la to_o find_v the_o angle_n at_o d_o here_o ab_fw-la be_v the_o middle_a part_n dis-junct_a because_o it_o be_v disjoined_n from_o the_o side_n ad_fw-la by_o the_o angle_n at_o a_o and_o from_o the_o angle_n at_o d_o by_o the_o side_n db_n for_o the_o right_a angle_n be_v not_o reckon_v among_o the_o circular_a part_n and_o here_o the_o extreme_n be_v extreme_n dis-junct_a these_o thing_n premise_v we_o come_v now_o to_o demonstrate_v the_o proposition_n itself_o consist_v of_o two_o part_n first_o we_o will_v prove_v that_o the_o sine_fw-la of_o the_o middle_a part_n and_o radius_fw-la be_v proportional_a with_o the_o tangent_n of_o the_o extreme_n conjunct_a the_o middle_a part_n be_v either_o one_o of_o the_o side_n or_o one_o of_o the_o oblique_a angle_n or_o the_o hypothenusal_a case_n 1._o let_v the_o middle_a part_n be_v a_o side_n as_o in_o the_o right_n angle_v spherical_a triangle_n abdella_n of_o the_o last_o diagram_n let_v the_o perpendicular_a ab_fw-la be_v the_o middle_a part_n the_o base_a db_n and_o comp_n a_o the_o extreme_a conjunct_a than_o i_o say_v that_o the_o rectangle_n of_o the_o sine_fw-la of_o ab_fw-la and_o radius_fw-la be_v equal_a to_o the_o rectangle_n of_o the_o tangent_fw-la of_o db_n and_o the_o tangent_fw-la of_o the_o compliment_n of_o dab_n for_o by_o the_o second_o proposition_n of_o this_o chapter_n as_o the_o sine_fw-la of_o ab_fw-la be_v in_o proportion_n to_o radius_fw-la so_o be_v the_o tangent_fw-la of_o db_n to_o the_o tangent_fw-la of_o the_o angle_n at_o a._n therefore_o if_o you_o put_v the_o three_o term_n in_o the_o second_o place_n it_o will_v be_v as_o the_o sine_z of_o ab_fw-la to_o the_o tangent_fw-la of_o db_n so_o be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o angle_n at_o a._n but_o radius_fw-la be_v a_o mean_a proportional_a between_o the_o tangent_fw-la of_o a_o arch_n and_o the_o tangent_fw-la of_o the_o compliment_n of_o the_o same_o arch_n by_o the_o corollary_n of_o the_o first_o reason_n of_o the_o second_o axiom_n of_o plain_a triangle_n and_o therefore_o as_o radius_fw-la be_v to_o the_o tangent_fw-la of_o the_o angle_n at_o a_o so_o be_v the_o tangent_fw-la compliment_n of_o the_o same_o angle_n at_o a_o unto_o radius_fw-la therefore_o as_o the_o sine_fw-la of_o ab_fw-la be_v in_o proportion_n to_o the_o tangent_fw-la of_o db_fw-la so_o be_v the_o co-tangent_a of_o the_o angle_n at_o a_o to_o radius_fw-la and_o therefore_o the_o rectangle_n of_o ab_fw-la radius_fw-la be_v equal_a to_o the_o rectangle_n of_o the_o
right_a ascension_n seek_v probl._n 5._o the_o latitude_n of_o the_o place_n and_o declination_n of_o the_o sun_n give_v to_o find_v the_o ascensionall_a difference_n or_o time_n of_o the_o sun_n rise_v before_o or_o after_o the_o hour_n of_o six_o the_o ascensionall_a difference_n be_v nothing_o else_o but_o the_o difference_n between_o the_o ascension_n of_o any_o point_n in_o the_o ecliptic_a in_o a_o right_a sphere_n and_o the_o ascension_n of_o the_o same_o point_n in_o a_o oblique_a sphere_n as_o in_o the_o annex_v diagram_n agev_n represent_v the_o meridian_n emt_fw-mi the_o horizon_n gmcv_n the_o equator_fw-la a_o the_o north_n pole_n vt_fw-la the_o compliment_n of_o the_o pole_n elevation_n bc_n the_o sun_n declination_n db_v a_o arch_n of_o the_o ecliptic_a dc_o the_o right_a ascension_n mc_n the_o ascensionall_a difference_n then_o in_o the_o right_a angle_a triangle_n bmc_n we_o have_v limit_v 1_o the_o angle_n bmc_n the_o compliment_n of_o the_o pole_n elevation_n 38_o deg_n 47_o min._n 2_o the_o perpendicular_a bc_n the_o sun_n declination_n 20_o deg_n 22_o min._n hence_o to_o find_v mc_n the_o ascensional_a difference_n by_o the_o 6_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o tangent_fw-la of_o bc_n 20.22_o 9.566231_o so_o be_v co-tangent_a of_o bmc_n 38.47_o 10.099861_o to_o the_o sine_fw-la of_o mc_n 27.62_o 9.666092_o probl._n 6._o the_o latitude_n of_o the_o place_n and_o the_o sun_n declination_n give_v to_o find_v his_o amplitude_n the_o sun_n amplitude_n be_v a_o arch_n of_o the_o horizon_n intercept_v between_o the_o equator_fw-la and_o the_o point_n of_o rise_v that_o be_v in_o the_o precede_a diagram_n the_o arch_a mb_n therefore_o in_o the_o right_n angle_v spherical_a triangle_n mbc_n have_v the_o angle_n bmc_n the_o height_n of_o the_o equator_fw-la 38_o deg_n 47_o min._n and_o bc_n the_o sun_n declination_n 20_o de_fw-la 22_o m._n give_v the_o hypothenusal_a mb_n may_v be_v find_v by_o the_o 5_o case_n of_o right_n angle_v spherical_a triangle_n for_o as_o the_o sine_fw-la of_o bmc_fw-la 38.47_o 9.793863_o be_v to_o the_o radius_fw-la 90_o 10.000000_o so_o be_v the_o sine_fw-la of_o bc_n 20.22_o 9.538606_o to_o the_o sine_fw-la of_o mb_n 33.75_o 9.744743_o probl._n 7._o the_o latitude_n of_o the_o place_n and_o the_o sun_n declination_n give_v to_o find_v the_o time_n when_o he_o will_v be_v east_n or_o west_n let_v abcd_n in_o the_o annex_v diagram_n represent_v the_o meridian_n bd_o the_o horizon_n fg_v the_o equator_fw-la hnk_v a_o arch_n of_o a_o meridian_n ac_fw-la the_o azimuth_n of_o east_n and_o west_n or_o first_o vertical_a em_n a_o parallel_n of_o declination_n then_o in_o the_o right_n angle_v spherical_a triangle_n ahn_n we_o have_v know_v 1_o the_o perpendicular_a ah_o the_o compliment_n of_o the_o pole_n elevation_n 38_o deg_n 47_o mi._n 2_o the_o hypothenusal_a hn_n the_o compliment_n of_o the_o sun_n declination_n 69_o deg_n 78_o m._n hence_o the_o angle_n ahn_n may_v be_v find_v by_o the_o 13_o case_n of_o right_n angle_v spherical_a triangle_n as_o the_o radius_fw-la 90_o 10,000000_o to_o the_o tangent_fw-la of_o ah_o 38.47_o 9.900138_o so_o be_v the_o co-tangent_a hn_n 69.78_o 9.566231_o to_o the_o cousin_a of_o ahn_n 72.98_o 9.466369_o who_o be_v compliment_n nhz_n 17_o degr_n 2_o min._n be_v convert_v into_o time_n give_v one_o hour_n 13_o minute_n or_o centesme_n of_o a_o hour_n and_o so_o much_o be_v it_o after_o six_o in_o the_o morning_n when_o the_o sun_n will_v be_v due_a east_n and_o before_o six_o at_o night_n when_o he_o will_v be_v due_a west_n probl._n 8._o the_o latitude_n of_o the_o place_n and_o declination_n of_o the_o sun_n give_v to_o find_v his_o altitude_n when_o he_o come_v to_o be_v due_a east_n or_o west_n in_o the_o right_n angle_v spherical_a triangle_n nqz_n of_o the_o last_o diagram_n we_o have_v limit_v 1._o the_o perpendicular_a qn_n the_o sun_n declination_n 2._o the_o angle_n at_o the_o base_a nzq_n the_o pole_n elevation_n 51_o degr_n 53_o min._n hence_o to_o find_v the_o hypothenusal_a nz_n by_o the_o five_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o sine_fw-la of_o the_o ang_fw-mi nzq_fw-fr 51.53_o 9.893725_o be_v to_o the_o radius_fw-la 90_o 10.000000_o so_o be_v the_o sine_fw-la of_o nq_fw-fr 20.22_o 9.538606_o to_o the_o sine_fw-la of_o nz_n 26.20_o 9.644881_o probl._n 9_o the_o latitude_n of_o the_o place_n and_o declination_n of_o the_o sun_n give_v to_o find_v the_o sun_n azimuth_n at_o the_o hour_n of_o six_o in_o the_o right_n angle_v spherical_a triangle_n aih_n of_o the_o seven_o problem_n we_o have_v know_v 1._o the_o base_a ah_o the_o compliment_n of_o the_o pole_n elevation_n 38_o degr_n 47_o min._n and_o the_o perpendicular_a ih_v the_o compliment_n of_o the_o sun_n declination_n 69_o degr_n 78_o min._n hence_o to_o find_v the_o angle_n at_o the_o base_a hai_o the_o sun_n azimuth_n at_o the_o hour_n of_o six_o by_o the_o 11_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o sine_fw-la of_o ah_o 38.47_o 9.793863_o so_o the_o co-tangent_a of_o he_o 69.78_o 9.566231_o to_o the_o co-tangent_a of_o hai_o 77._o 9.360094_o probl._n 10._o the_o pole_n elevation_n with_o the_o sun_n altitude_n and_o declination_n give_v to_o find_v the_o sun_n azimuth_n in_o the_o oblique_a angle_a spherical_a triangle_n ahs_n in_o the_o diagram_n of_o the_o seven_o problem_n we_o have_v know_v the_o side_n ah_o the_o compliment_n of_o the_o pole_n elevation_n 38_o deg_n 47_o min._n his_o the_o compliment_n of_o the_o sun_n declination_n 74_o deg_n 83_o min._n and_o the_o side_n sa_o the_o compliment_n of_o the_o sun_n altitude_n 57_o deg_n 53_o min_fw-mi to_o find_v the_o angle_n sah_n now_o then_o by_o the_o 11_o case_n of_o oblique_a angle_a spherical_a triangle_n i_o work_v as_o be_v there_o direct_v  _fw-fr shall_z 74.83_o  _fw-fr  _fw-fr ha_o 38,47_o 9.793863_o  _fw-fr sa_n 57.53_o 9_o 9â—6174_n  _fw-fr  _fw-fr  _fw-fr sum_n of_o the_o side_n 170.83_o 19.720037_o half_a sum_n 85._o 41._o 50_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr dif_a of_o ha_o &_o half_a sum_n 46.91.50_o 9.863737_o dif_a of_o sa_o &_o half_a sum_n 27.88.50_o 9.669990_o the_o double_a radius_fw-la  _fw-fr 20.000000_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr their_o sum_n 39.533727_o from_o whence_o subtract_v  _fw-fr 19.720037_o there_o rest_v  _fw-fr 19.813690_o the_o half_a whereof_o  _fw-fr 9.906845_o be_v the_o sine_fw-la of_o 53_o deg_n 80_o min._n which_o double_v be_v 107_o deg_n 60_o min._n the_o sun_n azimuth_n from_o the_o north_n and_o 72_o deg_n 40_o min._n the_o compliment_n thereof_o to_o a_o semicircle_n be_v the_o sun_n azimuth_n from_o the_o south_n chap._n ii_o the_o art_n of_o shadow_n common_o call_v dial_v plain_o show_v out_o of_o the_o sphere_n the_o true_a ground_n and_o reason_n of_o make_v all_o kind_n of_o dial_n that_o any_o plain_n be_v capable_a of_o problem_n 1._o how_o to_o divide_v diverse_a line_n and_o make_v a_o chord_n to_o any_o proportion_n give_v forasmuch_o as_o there_o be_v continual_a use_n both_o of_o scale_n and_o chord_n in_o draw_v the_o scheme_n and_o dial_n follow_v it_o will_v be_v necessary_a first_o to_o show_v the_o make_n of_o they_o that_o such_o as_o can_v have_v the_o benefit_n of_o the_o skilful_a artificer_n labour_n may_v by_o their_o own_o pain_n supply_v that_o defect_n draw_v therefore_o upon_o a_o piece_n of_o paper_n or_o pasteboard_n a_o straight_a line_n of_o what_o length_n you_o please_v divide_v this_o line_n into_o 10_o equal_a part_n and_o each_o 10_o into_o 10_o more_o so_o be_v your_o line_n divide_v into_o 100_o equal_a part_n by_o help_n where_o of_o a_o line_n of_o chord_n to_o any_o proportion_n may_v be_v thus_o make_v first_o prepare_v a_o table_n therein_o set_v down_o the_o degree_n half_n and_o quarter_n if_o you_o please_v from_o one_o to_o 90._o unto_o each_o degree_n and_o part_n of_o a_o degree_n join_v the_o chord_n proper_a to_o it_o which_o be_v the_o natural_a sine_fw-la of_o half_a the_o arch_n double_v by_o the_o 19_o of_o the_o second_o of_o the_o first_o part_n if_o you_o double_v then_o the_o natural_a sin_n of_o 5._o 10._o 20._o 30._o degree_n you_o shall_v produce_v the_o chord_n of_o 10._o 20._o 40._o 60._o degree_n thus_o 17364_o the_o sine_fw-la of_o 10_o de_fw-la be_v double_v the_o sum_n will_v be_v 34â—28_n the_o chord_n of_o 20_o deg_n and_o so_o of_o the_o rest_n as_o in_o the_o table_n follow_v de_fw-fr chord_n 1_o 17_o 2_o 35_o 3_o 52_o 4_o 70_o 5_o 87_o 6_o 105_o 7_o 122_o 8_o 139_o 9_o 157_o 10_o 175_o 11_o 192_o 12_o 209_o 13_o 226_o 14_o 244_o 15_o 261_o 16_o 278_o 17_o 296_o 18_o 313_o 19_o 330_o 20_o
347_o 21_o 364_o 22_o 382_o 23_o 398_o 24_o 416_o 25_o 432_o 26_o 450_o 27_o 466_o 28_o 384_o 29_o 501_o 30_o â—18_n 31_o 534_o 32_o 551_o 33_o 568_o 34_o 585_o 35_o 601_o 36_o 618_o 37_o 635_o 38_o 651_o 39_o 668_o 40_o 684_o 41_o 700_o 42_o 717_o 43_o 733_o 44_o 749_o 45_o 765_o 44_o 781_o 47_o 797_o 48_o 813_o 49_o 830_o 50_o 845_o 51_o 861_o 52_o 876_o 53_o 892_o 54_o 908_o 55_o 923_o 56_o 939_o 57_o 954_o 58_o 970_o 59_o 984_o 60_o 1000_o 61_o 1015_o 62_o 1030_o 63_o 1045_o 64_o 1060_o 65_o 1074_o 66_o 1089_o 67_o 1104_o 68_o 1118_o 69_o 1133_o 70_o 1147_o 71_o 1161_o 72_o 1176_o 73_o 1190_o 74_o 1204_o 75_o 1217_o 76_o 1231_o 77_o 1245_o 78_o 1259_o 79_o 1273_o 80_o 1286_o 81_o 1299_o 82_o 1312_o 83_o 1325_o 84_o 1338_o 85_o 1351_o 86_o 1364_o 87_o 1377_o 88_o 1389_o 89_o 1402_o 90_o 1414_o this_o do_v proportion_n the_o radius_fw-la of_o a_o circle_n to_o what_o extent_n you_o please_v make_fw-mi ab_fw-la equal_v thereto_o in_o the_o middle_n whereof_o as_o in_o c_o erect_v the_o perpendicular_a cd_o and_o draw_v the_o line_n ad_fw-la and_o bd_o equal_a in_o length_n to_o your_o line_n of_o equal_a part_n so_o have_v you_o make_v a_o equiangle_v triangle_n by_o help_n whereof_o and_o the_o table_n aforesaid_a the_o chord_n of_o any_o arch_a proportionable_a to_o this_o radius_fw-la may_v speedy_o be_v obtain_v as_o for_o example_n let_v there_o be_v require_v the_o chord_n of_o 30_o deg_n the_o number_n in_o the_o table_n answer_v to_o this_o ark_n be_v 518_o or_o in_o proportion_n to_o this_o scale_n 52_o almost_o i_o take_v therefore_o 52_o from_o the_o scale_n of_o equal_a part_n and_o set_v they_o from_o d_o to_z e_z and_o f_o and_o draw_v the_o line_n of_o which_o be_v the_o chord_n desire_v thus_o may_v you_o find_v the_o chord_n of_o any_o other_o arch_a agreeable_a to_o this_o radius_fw-la or_o if_o your_o radius_fw-la be_v either_o of_o a_o great_a or_o lesser_a extent_n if_o you_o make_v the_o base_a of_o your_o triangle_n ab_fw-la equal_v 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therefore_o with_o my_o compass_n 69_o from_o my_o scale_n of_o equal_a part_n and_o set_v they_o from_o d_o to_z g_z and_o h_n so_o be_v the_o line_n gh_o the_o sine_fw-la of_o 44_o degree_n where_o the_o radius_fw-la of_o the_o circle_n be_v ab_fw-la again_o if_o there_o be_v require_v the_o tangent_fw-la of_o 44_o degree_n the_o number_n in_o the_o table_n be_v 965_o and_o therefore_o 96_o set_v from_o d_o to_o k_o and_o l_o shall_v give_v the_o tangent_fw-la require_v and_o so_o for_o any_o other_o your_o scale_n be_v thus_o prepare_v for_o the_o mechanical_a part_n we_o will_v now_o show_v you_o how_o to_o project_v the_o sphere_n in_o plano_fw-la and_o so_o proceed_v to_o the_o arithmetical_a work_n probl._n 2._o the_o explanation_n and_o make_n of_o the_o fundamental_a diagram_n this_o scheme_n represent_v to_o the_o eye_n the_o true_a and_o natural_a situation_n of_o those_o circle_n of_o the_o sphere_n whereof_o we_o shall_v have_v use_n in_o the_o description_n of_o such_o 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material_a sphere_n be_v the_o pole_n of_o the_o horizon_n and_o z_o in_o the_o scheme_n be_v the_o centre_n of_o the_o limb_n represent_v the_o same_o from_o which_o point_n the_o distance_n of_o each_o circle_n be_v give_v both_o way_n as_o it_o lie_v in_o the_o sphere_n and_o set_v upon_o the_o azimuth_n or_o straight_a line_n of_o the_o scheme_n proper_a thereunto_o you_o may_v by_o help_n of_o the_o natural_a tangent_n of_o half_a their_o arch_n give_v three_o point_n to_o draw_v each_o circle_n by_o for_o if_o the_o natural_a tangent_n of_o both_o distance_n from_o the_o zenith_n be_v add_v together_o the_o half_a thereof_o shall_v be_v the_o semidiameter_n of_o those_o circle_n desire_v the_o reason_n why_o the_o natural_a tangent_fw-la of_o half_a the_o arch_n be_v here_o take_v may_v be_v make_v plain_a by_o this_o diagram_n follow_v wherein_o make_v ez_n the_o radius_fw-la szn_n be_v a_o tangent_fw-la line_n thereunto_o upon_o which_o if_o you_o will_v project_v the_o whole_a semicircle_n sw_v it_o be_v manifest_a by_o the_o work_n that_o every_o part_n of_o the_o line_n zn_n or_o zs_n can_v be_v no_o more_o than_o the_o tangent_fw-la of_o half_a the_o arch_n desire_v because_o the_o whole_a line_n zn_n or_o zs_n be_v the_o tang_n of_o no_o more_o than_o half_a the_o quadrant_n that_o be_v of_o 45_o degree_n by_o the_o 19_o of_o the_o second_o chapter_n of_o the_o first_o part_n and_o therefore_o uvea_o be_v but_o half_a the_o angle_n wza_n and_o web_n be_v but_o half_a the_o angle_n wzb._n now_o than_o if_o ez_n or_o radius_fw-la of_o the_o fundamental_a scheme_n be_v 1000_o zp_n shall_v be_v 349_o the_o natural_a tangent_fw-la of_o 19_o degree_n 23_o minute_n 50_o second_n the_o half_a of_o 38_o degree_n 47_o
r_o of_o the_o substile_a both_o way_n set_v off_o the_o hour_n distance_n by_o help_n of_o the_o chord_n for_o 8_o of_o the_o clock_n 2_o degr_n 91_o min._n and_o so_o of_o the_o rest_n and_o draw_v straight_a line_n from_o the_o centre_n g_o through_o those_o point_n which_o shall_v be_v the_o true_a hour_n line_n desire_v last_o of_o all_o the_o height_n of_o the_o stile_n pr._n 26_o degr_n 69_o min._n be_v set_v from_o r_o to_o p_o draw_v the_o straight_a line_n gp_n for_o the_o axis_fw-la of_o the_o stile_n which_o must_v give_v the_o shadow_n on_o the_o dial_n erect_v gp_n at_o the_o angle_n rgp_v perpendicular_o over_o the_o substile_a line_n graccus_n and_o let_v the_o point_n p_o be_v direct_v to_o the_o north_n pole_n go12_n place_v in_o the_o meridian_n the_o centre_n g_z represent_v the_o south_n and_o the_o plane_n at_o of_o elevate_v above_o the_o horizon_n 55_o degree_n so_o have_v you_o finish_v this_o dial_n for_o use_n only_o remember_v because_o the_o sun_n rise_v but_o a_o little_a before_o 4_o and_o set_v a_o little_a after_o 8_o to_o leave_v out_o the_o hour_n of_o 3_o and_o 9_o and_o put_v on_o all_o the_o rest_n and_o thus_o you_o have_v the_o projection_n of_o four_o dial_n in_o one_o for_o that_o which_o be_v the_o west_n recliner_n be_v also_o the_o east_n incliner_n 〈◊〉_d you_o take_v the_o compliment_n of_o the_o recliner_n â—ours_n unto_o 12_o and_o that_o but_o from_o 3_o in_o the_o afternoon_n till_o 8_o at_o night_n again_o if_o you_o draw_v the_o same_o line_n on_o the_o other_o side_n of_o your_o â—â—per_n and_o change_v the_o hour_n of_o 8_o 7_o 6_o etc._n etc._n into_o 4_o 5_o 6_o etc._n etc._n you_o have_v the_o east_n recliner_n and_o the_o compliment_n of_o the_o east_n recliner_n hour_n from_o 3_o to_o 8_o be_v the_o west_n incliner_n only_o remember_v that_o as_o the_o stile_n in_o the_o west_n recliner_n behold_v the_o north_n and_o the_o plane_n the_o zenith_n so_o in_o the_o east_n incliner_n the_o stile_n must_v behold_v the_o south_n and_o the_o plane_n the_o nadir_n probl._n 9_o to_o draw_v the_o hour-line_n upon_o any_o direct_a south_n recline_v or_o incline_v plane_n as_o the_o base_a of_o east_n and_o west_n recline_v or_o incline_v plane_n do_v always_o lie_v in_o the_o meridian_n of_o the_o place_n or_o parallel_v thereunto_o and_o the_o pole_n in_o the_o prime_n vertical_a so_o do_v the_o base_a of_o south_n and_o north_n recline_v or_o incline_v plane_n lie_v in_o the_o prime_n vertical_a or_o azimuth_n of_o east_n and_o west_n and_o their_o pole_n consequent_o in_o the_o meridian_n now_o if_o you_o suppose_v the_o circle_n of_o position_n which_o astrological_o take_v be_v fix_v in_o the_o intersection_n of_o the_o meridian_n and_o horizon_n to_o move_v about_o upon_o the_o horizon_n till_o it_o come_v into_o the_o plane_n of_o the_o prime_n vertical_a and_o be_v fix_v in_o the_o intersection_n thereof_o with_o the_o horizon_n to_o be_v let_v fall_v either_o way_n from_o the_o zenith_n upon_o the_o meridian_n it_o shall_v true_o represent_v all_o the_o south_n and_o north_n recline_v and_o incline_v plane_n also_o of_o which_o there_o be_v six_o variety_n three_o of_o south_n and_o three_o of_o north_n recline_v for_o either_o the_o south_n plane_n do_v recline_v just_a to_o the_o pole_n and_o then_o it_o become_v a_o equinoctial_a because_o the_o pole_n of_o this_o plane_n do_v then_o lie_v in_o the_o equinoctial_a some_o call_v it_o a_o polar_n plane_n or_o else_o it_o recline_v more_o and_o less_o than_o the_o pole_n and_o consequent_o the_o pole_n of_o the_o plane_n above_o and_o under_o the_o equinoctial_a somewhat_o differ_v from_o the_o former_a in_o like_a manner_n the_o north_n plane_n recline_v just_a to_o the_o equinoctial_a and_o then_o become_v a_o polar_n plane_n because_o the_o pole_n of_o that_o plane_n lie_v in_o the_o pole_n of_o the_o world_n some_o term_n it_o a_o equinoctial_a plane_n or_o else_o it_o recline_v more_o or_o less_o than_o the_o equinoctial_a and_o consequent_o the_o pole_n of_o the_o plane_n above_o and_o under_o the_o pole_n of_o the_o world_n somewhat_o differ_v from_o the_o former_a of_o the_o equinoctial_a plane_n and_o because_o this_o dial_n be_v no_o other_o but_o the_o very_a horizontall_a of_o a_o right_a sphere_n where_o the_o equinoctial_a be_v zenith_n and_o the_o pole_n of_o the_o world_n in_o the_o horizon_n therefore_o it_o be_v not_o capable_a of_o the_o six_o of_o clock_n hour_n no_o more_o than_o the_o east_n and_o west_n be_v of_o the_o 12_o a_o clock_n hour_n which_o vanish_v upon_o the_o plane_n unto_o which_o they_o be_v parallel_v and_o the_o twelve_o a_o clock_n hour_n be_v the_o middle_a line_n of_o this_o dial_n because_o the_o meridian_n cut_v the_o plane_n of_o six_o a_o clock_n at_o right_a angle_n which_o the_o sun_n attain_v not_o till_o he_o be_v perpendicular_a to_o the_o plain_a and_o this_o in_o my_o opinion_n beside_o the_o respect_n of_o the_o pole_n be_v reason_n enough_o to_o call_v it_o a_o equinoctial_a dial_n see_v it_o be_v the_o dial_n proper_a to_o they_o that_o live_v under_o the_o equinoctial_a this_o dial_n be_v to_o be_v make_v in_o all_o respect_n as_o the_o east_n and_o west_n be_v be_v indeed_o the_o very_a same_o with_o they_o only_o change_v the_o number_n of_o the_o hour_n for_o see_v the_o six_o of_o clock_n hour_n in_o which_o this_o plane_n lie_v cross_v the_o twelve_o of_o clock_n hour_n at_o right_a angle_n in_o which_o the_o east_n and_o west_n plane_n lie_v the_o rest_n of_o the_o hour-line_n will_v have_v equal_a respect_n unto_o they_o both_o so_o that_o the_o five_o hour_n from_o six_o of_o the_o clock_n be_v equal_a to_o the_o five_o hour_n from_o twelve_o the_o four_o to_o the_o four_o and_o so_o of_o the_o â—est_n these_o analogy_n hold_v the_o hour_n distance_n from_o six_o be_v to_o be_v set_v off_o by_o the_o natural_a tangent_n in_o these_o dial_n as_o they_o be_v from_o twelve_o in_o the_o east_n and_o west_n dial_n the_o geometrical_a projection_n draw_v the_o tangent_fw-la line_n dsk_n parallel_n to_o the_o line_n ezw_n in_o the_o scheme_n cross_v it_o at_o right_a angle_n with_o msa_n the_o meridian_n line_n make_v sa_o the_o radius_fw-la to_o that_o tangent_fw-la line_n on_o which_o prick_v down_o the_o hour_n and_o that_o there_o may_v be_v as_o many_o hour_n upon_o the_o plane_n as_o it_o be_v capable_a of_o you_o must_v proportion_v the_o stile_n to_o the_o plane_n as_o in_o the_o five_o problem_n after_o this_o manner_n let_v the_o length_n of_o the_o plane_n from_o a_o be_v give_v in_o know_a part_n then_o because_o the_o extreme_a hour_n upon_o this_o plane_n be_v 5_o or_o 7_o reckon_v 15_o degree_n to_o every_o hour_n from_o 12_o the_o arch_n of_o the_o equator_fw-la will_v be_v 75_o degree_n and_o therefore_o in_o the_o right_n angle_v plain_a triangle_n sa_n ♎_o we_o have_v give_v the_o base_a a_o ♎_o the_o length_n of_o the_o plane_n from_o a_o and_o the_o angle_n as_o ♎_o 75_o degree_n to_o find_v the_o perpendicular_a sa_o for_o which_o as_o in_o the_o five_o chapter_n i_o say_v as_o the_o radius_fw-la 90_o 10.000000_o be_v to_o the_o base_a a_o ♎_o 3.50_o 2.544068_o so_o be_v the_o tangent_fw-la of_o a_o ♎_o s_o 15_o 9,428052_o  _fw-fr  _fw-fr to_o the_o perpendicular_a as_o 94_o 1.972120_o at_o which_o height_n a_o stile_n be_v erect_v over_o the_o 12_o a_o clock_n hour_n line_n and_o the_o hour_n from_o 12_o draw_v parallel_n thereunto_o through_o the_o point_v make_v in_o the_o tangent_fw-la line_n by_o set_v off_o the_o natural_a tangent_n thereon_o and_o then_o the_o dial_n be_v finish_v let_v sa_n 12_o be_v place_v in_o the_o meridian_n and_o the_o whole_a plane_n at_o saint_n raise_v to_o the_o height_n of_o the_o pole_n 51_o degr_n 53_o min._n then_o will_v the_o stile_n show_v the_o hour_n true_o and_o the_o dial_n stand_v in_o its_o due_a position_n 2._o of_o south_n recline_v less_o than_o the_o pole_n this_o plane_n be_v represent_v by_o the_o prick_a circle_n in_o the_o fundamental_a diagram_n ecw_o and_o be_v intersect_v by_o the_o hour_n circle_n from_o the_o pole_n p_o as_o by_o the_o scheme_n appear_v and_o therefore_o the_o dial_n proper_a to_o this_o plane_n must_v have_v a_o centre_n above_o which_o the_o south_n pole_n be_v elevate_v and_o therefore_o the_o stile_n must_v look_v downward_o as_o in_o south_n direct_a plane_n to_o calculate_v which_o dial_n there_n must_v be_v give_v the_o pole_n elevation_n and_o the_o quantity_n of_o reclination_n by_o which_o to_o find_v the_o hour_n distance_n from_o the_o meridian_n and_o thus_o in_o the_o triangle_n pc_n 1_o have_v the_o pole_n elevation_n 51_o degr_n 53_o min._n and_o the_o reclination_n