the_o two_o figure_n note_v two_o and_o iii_o you_o may_v plain_o see_v 15._o all_o angle_n plain_a and_o spherical_a be_v either_o acute_a right_o or_o obtuce_n 17._o a_o right_a angle_n be_v alwaâ—â—â—_n just_a 90_o degree_n as_o you_o may_v 〈◊〉_d in_o the_o figure_n two_o and_o iii_o by_o the_o angle_n at_o a_o in_o both_o of_o they_o 18._o a_o obtuce_n angle_n be_v always_o more_o than_o 90_o degree_n as_o the_o angle_n at_o d_o in_o both_o figure_n show_v 19_o a_o parallel_n line_n be_v any_o line_n draw_v by_o another_o line_n in_o such_o a_o way_n that_o though_o it_o be_v infinite_o produce_v yet_o they_o will_v never_o meet_v or_o cross_a one_o another_o as_o the_o line_n ab_fw-la cd_o 20._o a_o perpendicule_a line_n be_v when_o one_o line_n so_o fall_v on_o another_o line_n that_o the_o angle_n on_o each_o side_n be_v equal_a as_o ca_n fall_v of_o the_o line_n basilius_n figure_n vi._n 21._o all_o triangle_n be_v either_o with_o three_o equal_a side_n as_o figure_n four_o 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that_o shall_v be_v the_o altitude_n require_v be_v measure_v on_o the_o same_o scale_n example_n let_v abgd_n represent_v the_o board_n end_n or_o trencher_n and_o on_o that_o let_v ab_fw-la be_v one_o straight_a line_n and_o agnostus_n another_o perpendicule_a to_o it_o in_o the_o point_n a_o knock_v in_o one_o pin_n and_o in_o b_o or_o any_o where_o towards_o the_o end_n another_o on_o the_o pin_n at_o a_o hang_v a_o thread_n and_o plummet_n and_o stand_v at_o ay_o any_o convenient_a station_n look_v up_o by_o the_o two_o pin_n at_o b_o and_o a_o till_o they_o bourn_n in_o a_o right_a line_n with_o the_o point_n h_n the_o object_n who_o height_n be_v to_o be_v measure_v then_o the_o plummet_n play_v well_o and_o even_o make_v a_o point_n just_a therein_o and_o draw_v the_o line_n ad_fw-la as_o the_o thread_n show_v then_o have_v measure_v the_o distance_n from_o g_o the_o foot_n of_o the_o object_n to_o i_o the_o station_n take_v it_o from_o any_o first_o scale_n and_o lay_v it_o 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the_o measure_n from_o your_o foot_n to_o the_o glass_n be_v to_o ab_fw-la the_o height_n from_o your_o eye_n to_o the_o ground_n at_o your_o foot_n so_o be_v the_o measure_n from_o c_o to_o d_o to_o the_o height_n de._n see_v figure_n vi._n again_o to_o find_v a_o distance_n by_o the_o square_n that_o be_v not_o overlong_a let_v c_o represent_v the_o upper-corner_n of_o a_o square_a hang_v on_o a_o staff_n at_o f_o then_o the_o one_o part_n of_o the_o square_n direct_v to_o e_o and_o the_o other_o to_o a._n the_o proportion_n will_v hold_v by_o the_o line_n of_o number_n as_o favorina_n 11-37_a to_o fc_v 50_o so_o be_v fc_fw-mi 50_o to_o fe_o 220._o that_o be_v so_o many_o time_n as_o you_o find_v of_o in_o fc_n so_o many_o time_n be_v fc_a in_o fe_o and_o the_o like_a note_v that_o you_o must_v conceive_v afe_a to_o be_v the_o ground_n or_o base-line_n in_o this_o operation_n by_o the_o square_a c_o be_v the_o top_n of_o a_o upright_a staff_n 5_o foot_n long_o call_v 50_o for_o 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any_o two_o station_n iu._n let_v ab_fw-la be_v two_o mark_n as_o two_o corner_n of_o a_o house_n or_o wall_n and_o let_v the_o breadth_n between_o they_o be_v demand_v and_o their_o distance_n from_o c_o and_o d_o the_o two_o station_n first_o set_v up_o two_o mark_n at_o the_o two_o station_n then_o set_v up_o the_o instrument_n at_o c_o set_v the_o fiducial_a line_n on_o the_o rule_n to_o d_o the_o other_z mark_n then_o direct_v the_o sight_n exact_o to_o b_o and_o to_o a_o observe_v the_o angle_n dcb_n 45_o and_o dca_n 113-0_a as_o in_o the_o figure_n second_o remove_v the_o instrument_n to_o d_o the_o other_z station_n and_o set_v the_o fiducial-line_n of_o the_o quadrant_n viz._n the_o line_n of_o line_n and_o sines_n direct_o to_o c_z then_o fix_v it_o there_o and_o remove_v the_o index_n and_o sight_n to_o a_o and_o to_z b_o to_o get_v the_o angle_n cda_n 42-30_a and_o cdb_v 109-0_a then_o observe_v that_o the_o 3_o angle_n of_o every_o triangle_n be_v equal_a to_o 180_o degree_n have_v get_v the_o angle_n at_o c_o 113_o and_o the_o angle_n at_o d_o 42-30_a by_o consequence_n as_o you_o take_v 155_o the_o sum_n of_o the_o angle_n at_o c_o and_o d_o out_z of_o 180_o then_o there_o remain_v 24-30_a the_o angle_n at_o a._n so_o also_o take_v 109_o and_o 45_o from_o 180_o rest_v 26_o the_o angle_n at_o b_o then_o also_o take_v 45_o the_o angle_n bcd_v out_o of_o 113_o the_o angle_n dca_n rest_v 68_o degree_n the_o angle_n bca_n in_o like_a manner_n take_v 42-30_a from_o 109_o the_o angle_n at_o d_o rest_v
every_o long_a stroke_n be_v 100_o and_o every_o short_a cut_n be_v ten_o and_o every_o single_a integer_fw-la be_v as_o near_o as_o can_v be_v estimate_v by_o any_o ordinary_a instrument_n now_o our_o present_a example_n will_v proper_o come_v under_o the_o three_o rule_n by_o conceive_v the_o whole_a line_n to_o represent_v 100_o then_o the_o first_o long_a stroke_n by_o 1_o be_v 10_o than_o the_o next_o short_a be_v for_o 11_o the_o next_o 12_o etc._n etc._n to_o 15_o which_o be_v cut_v up_o a_o little_a above_o the_o line_n for_o the_o more_o ready_a reckon_n without_o tell_v the_o part_n which_o 15_o be_v the_o point_n require_v to_o be_v find_v example_n the_o second_o to_o find_v out_o 1550_o on_o the_o line_n this_o will_v come_v under_o the_o notion_n of_o the_o 5_o rule_n wherein_o the_o whole_a line_n be_v conceive_v to_o represent_v 10000_o then_o the_o first_o 1_o be_v for_o the_o 1_o thousand_o than_o the_o five_o long_o stroke_v next_o be_v for_o the_o 500_o and_o last_o the_o middle_a between_o the_o 500_o stroke_n and_o the_o 6000_o stroke_n be_v for_o the_o 50_o be_v a_o little_a beyond_o the_o point_n for_o 15_o in_o the_o first_o example_n a_o three_o example_n of_o 5025._o this_o three_o example_n may_v suffice_v for_o this_o work_n be_v so_o plain_a after_o a_o little_a due_a consideration_n for_o first_o the_o whole_a line_n be_v conceive_v to_o represent_v 10000_o than_o the_o long_a stroke_n by_o 5_o be_v for_o 5000_o then_o there_o be_v no_o hundred_o therefore_o the_o point_n require_v must_v be_v short_a of_o the_o next_o long_o stroke_v which_o signify_v hundred_o and_o be_v it_o be_v just_a 25_o which_o be_v ¼_n of_o a_o hundred_o the_o true_a point_n ready_o show_v itself_o if_o you_o require_v a_o more_o plain_a and_o large_a word_v oâ—_n this_o matter_n i_o refer_v you_o to_o the_o three_o chapter_n of_o mr._n windgates_n rule_n of_o proportion_n 〈◊〉_d the_o first_o chapter_n of_o the_o carpenter_n rule_n bâ—_n i._o brown_n last_o in_o name_a of_o any_o point_n find_v out_o on_o the_o line_n great_a care_n and_o respect_n musâ—_n be_v have_v as_o to_o the_o true_a value_n of_o the_o number_n according_a to_o the_o rate_n of_o the_o question_n propound_v for_o the_o same_o point_n that_o represent_v 15_o do_v represent_v 150_o and_o also_o 1500_o or_o 15000_o increase_v above_o the_o bound_n before_o mention_v also_o it_o signify_v one_o and_o a_o half_a or_o 15_o of_o one_o hundred_o which_o be_v usual_o express_v thus_o in_o a_o decimal_n fraction_n 15_o 100_o or_o more_o ready_o 0.15_o also_o if_o it_o shall_v be_v a_o number_n with_o a_o digit_fw-la two_o cipher_n and_o another_o digit_fw-la as_o 2.005_o this_o number_n will_v be_v find_v close_o to_o the_o long_a stroke_n by_o the_o figure_n 2_o and_o may_v represent_v either_o two_o thousand_o and_o 5_o of_o 1000_o more_o or_o 20_o and_o 5_o of_o a_o hundred_o or_o 2_o hundred_o and_o 5_o of_o another_o 10_o more_o or_o plain_o as_o it_o be_v set_v down_o two_o thousand_o no_o hundred_o but_o five_o thus_o you_o see_v the_o manner_n of_o express_v whole_a number_n or_o whole_a number_n and_o decimal_a fraction_n which_o on_o the_o line_n be_v one_o and_o the_o same_o thing_n and_o thus_o all_o decimal_n scales_n be_v to_o be_v account_v and_o in_o the_o same_o manner_n be_v the_o line_n of_o number_n to_o be_v read_v as_o you_o may_v see_v more_o at_o large_a in_o the_o two_o book_n before_o mention_v sect_n ii_o but_o for_o numerateon_n on_o all_o circular_a line_n it_o be_v much_o easy_a for_o first_o very_o few_o instrument_n unless_o at_o one_o part_n of_o the_o line_n can_v express_v near_o than_o minute_n of_o a_o degree_n second_o the_o whole_a radius_fw-la or_o line_n of_o line_n be_v but_o 90_o degree_n or_o but_o 45_o of_o the_o tangent_n or_o 60_o of_o the_o chord_n or_o secant_v so_o that_o in_o instrument_n of_o 12_o or_o 18_o inch_n radius_fw-la you_o may_v express_v very_o well_o every_o ten_o minute_n to_o 60_o on_o the_o line_n of_o sines_n and_o every_o half_a degree_n to_o 75_o and_o whole_a degree_n to_o 90._o and_o on_o the_o tangent_n or_o chord_n every_o 10_o minute_n quite_o through_o and_o the_o secant_v as_o the_o sin_n so_o that_o any_o degree_n or_o minute_n be_v name_v tâ—_n find_v the_o same_o on_o the_o respective_a line_n count_v thus_o first_o every_o 10_o degree_n be_v note_v with_o a_o long_a stroke_n and_o figure_n set_v thereunto_o second_o every_o whole_a degree_n be_v cut_v between_o two_o or_o three_o line_n and_o sometime_o with_o a_o point_n or_o mark_n on_o the_o end_n of_o the_o stroke_n and_o every_o 5_o degree_n cut_v up_o high_o than_o the_o rest_n and_o sometime_o with_o three_o point_n on_o the_o end_n of_o the_o line_n or_o some_o other_o convenient_a distinction_n for_o readiness_n sake_n and_o every_o 10_o 15_o or_o 30_o minute_n be_v cut_v only_o between_o two_o line_n and_o no_o more_o as_o will_v appear_v very_o plain_a with_o a_o little_a practice_n example_n to_o find_v the_o sine_fw-la of_o the_o latitude_n be_v at_o london_n 51_o degree_n 32_o minute_n 1._o first_o look_v on_o any_o line_n of_o sines_n on_o the_o quadrantal_n or_o sector_n side_n according_a aâ—_n you_o have_v occasion_n till_o you_o see_v 50_o which_o iâ—_n 50_o degree_n then_o one_o degree_n forward_o towards_o 60_o be_v 51_o degree_n then_o count_v threâ—_n 10th_n of_o minute_n more_o for_o 30_o minute_n and_o then_o for_o the_o odd_a two_o minute_n estimate_n onâ—_n five_o part_n of_o the_o next_o 10_o minute_n forward_a and_o that_o be_v the_o precise_a point_n for_o the_o sine_fw-la 〈◊〉_d 51_o degree_n 32_o minute_n the_o latitude_n of_o london_n where_o sometime_o be_v set_v a_o braâ—s_n center-pin_n example_n the_o second_o 2._o to_o find_v the_o cousin_n of_o the_o latitude_n there_o be_v two_o way_n to_o count_v the_o compleâ—ment_n of_o any_o ark_n or_o angle_n first_o by_o substract_v the_o ark_n or_o angle_n out_o of_o 90_o by_o the_o pen_n and_o count_v the_o residue_n from_o the_o begin_n of_o the_o line_n of_o sines_n and_o that_o shall_v be_v the_o sine_fw-la compliment_n of_o the_o latitude_n require_v example_n 51_o 32_o take_v from_o 90_o the_o remainder_n be_v 38_o 28_o now_o if_o you_o count_v so_o much_o from_o the_o begin_n of_o the_o line_n of_o sines_n according_a to_o the_o last_o rule_n that_o shall_v be_v the_o point_n for_o the_o sine_fw-la of_o 38_o 28_o the_o compliment_n of_o 51_o 32_o or_o the_o sine_fw-la compliment_n of_o the_o latitude_n or_o second_o if_o you_o count_v 51_o 32_o from_o 90_o call_v 80_o 10_o and_o 70_o 20_o and_o 60_o 30_o 50_o 40_o 40_o 50_o etc._n etc._n wheresoever_o the_o number_n who_o compliment_n you_o will_v have_v shall_v end_v that_o be_v the_o sine_fw-la compliment_n require_v which_o will_v be_v at_o 38_o 28_o from_o the_o centre_n or_o beginning_n for_o the_o cousin_a of_o 51_o 32_o the_o like_a work_n serve_v for_o any_o other_o number_n or_o on_o any_o other_o line_n as_o on_o the_o degree_n tangent_n or_o secant_v natural_a or_o artificial_a as_o by_o practice_n will_v more_o plain_o appear_v to_o the_o willing_a practitioner_n sect_n iii_o to_o find_v the_o verse_v sine_fw-la of_o a_o ark_n or_o angle_n or_o the_o sine_fw-la of_o a_o ark_n or_o angle_n above_o 90_o degree_n or_o the_o chord_n above_o 180_o degree_n observe_v these_o rule_n 1._o first_o a_o right_a sine_fw-la be_v the_o measure_n on_o the_o line_n of_o sines_n from_o the_o centre_n or_o beginning_n of_o that_o line_n to_o the_o point_n that_o do_v represent_v the_o ark_n or_o angle_n require_v 2._o the_o right_a sine_fw-la of_o a_o ark_n or_o angle_n above_o 90_o degree_n be_v equal_a to_o the_o right_n sine_fw-la of_o the_o compliment_n thereof_o to_o 180_o degree_n be_v ready_o account_v thus_o count_v the_o excess_n above_o 90_o backward_o from_o 90_o towards_o the_o centre_n then_o the_o measure_n or_o distance_n from_o the_o end_n of_o the_o account_n to_o the_o centre_n be_v the_o sine_fw-la of_o the_o ark_n above_o 90_o require_v example_n let_v the_o sine_fw-la of_o 130_o be_v require_v first_o if_o you_o take_v 130_o from_o 180_o the_o remainder_n be_v 50_o then_o i_o say_v that_o the_o right_a sine_fw-la of_o 50_o be_v also_o right_a sine_fw-la of_o 130_o for_o if_o you_o count_v backward_o from_o 90_o call_v 80_o 100_o and_o 70_o 110_o and_o 60_o 120_o and_o 50_o 130_o the_o measure_n from_o thence_o to_o 00_o or_o the_o centre_n be_v the_o right_a sine_fw-la of_o 130_o degree_n 3._o the_o verse_v sine_fw-la of_o a_o ark_n or_o angle_n be_v the_o measure_n on_o the_o line_n of_o sines_n from_o 90_o towards_o the_o centre_n count_v backward_o as_o the_o small_a
the_o point_n e_o draw_v the_o line_n ed_z of_o which_o line_n find_v the_o middle_n between_o e_z and_o d_o viz._n the_o point_n c_o then_o the_o extent_n ce_fw-fr or_o cd_o keep_v one_o point_n in_o c_o shall_v cross_v the_o ground-line_n in_o the_o point_n b_o by_o which_o and_o e_o you_o may_v draw_v the_o perpendicule_a line_n ebb_v which_o be_v but_o the_o converse_n of_o the_o former_a 6._o to_o draw_v a_o line_n parallel_n to_o another_o at_o any_o distance_n to_o the_o line_n ab_fw-la i_o will_v have_v another_o parallel_n thereto_o to_o the_o distance_n of_o ai_fw-fr take_z a_z between_o your_o compass_n and_o set_v one_o point_n in_o one_o end_n of_o the_o line_n as_o at_o a_o sweep_v the_o ark_n eif_fw-mi then_o set_v the_o compass_n in_o the_o other_o end_n as_o at_z b_o and_o sweep_v the_o ark_n gdh_n then_o just_a by_o the_o round-side_n of_o those_o ark_n draw_v a_o line_n which_o shall_v be_v the_o parallel-line_n require_v or_o thus_o take_v bc_n the_o measure_n from_o the_o point_n that_o be_v to_o cut_v the_o parallel-line_n and_o one_o end_n of_o the_o given-line_n viz._n b_o with_o this_o distance_n set_v one_o foot_n in_o a_o at_o the_o other_o end_n of_o the_o given-line_n and_o draw_v the_o arch_n at_o k_o then_o take_v all_o ab_fw-la the_o given-line_n and_o set_v one_o point_n in_o c_o cross_v the_o ark_n at_o k_o then_o c_z and_o k_o shall_v be_v point_n to_o draw_v the_o parallel-line_n by_o note_v the_o figure_n 4._o 7._o to_o make_v one_o angle_n equal_a to_o another_o the_o angle_n bac_n be_v give_v and_o i_o will_v have_v another_o angle_n equal_a unto_o it_o set_v one_o point_n of_o the_o compass_n in_o a_o and_o draw_v the_o arch_n cb_n then_o on_o the_o line_n de_fw-fr from_o the_o point_n d_o draw_v the_o like_a ark_n of_o then_o in_o that_o ark_n make_v of_o equal_a to_o cb_n then_o draw_v the_o line_n df_n it_o shall_v make_v the_o angle_n edf_n equal_a to_o the_o angle_n bac_n which_o be_v require_v 8._o to_o divide_v a_o line_n into_o any_o number_n of_o part_n let_v ab_fw-la represent_v a_o line_n to_o be_v divide_v into_o eight_o part_n on_o one_o end_n viz._n a_o draw_v a_o line_n as_o ad_fw-la to_o any_o angle_n and_o from_o the_o other_o end_n b_o draw_v another_o line_n parallel_n to_o ad_fw-la as_o be_v then_o open_a the_o compass_n to_o any_o convenient_a distance_n and_o from_o a_o and_o b_o divide_v the_o line_n ad_fw-la and_o be_v into_o eight_o part_n then_o line_n draw_v by_o a_o ruler_n lay_v to_o every_o division_n in_o the_o line_n ad_fw-la and_o be_v shall_v divide_v the_o line_n ab_fw-la in_o the_o part_n require_v note_v the_o figure_n mark_v vi_o this_o proposition_n be_v much_o easy_o wrought_v by_o the_o line_n of_o line_n on_o the_o sector_n thus_o take_v ab_fw-la between_o your_o compass_n and_o fit_v it_o over_o parral_o in_o 8_o and_o 8_o of_o the_o line_n of_o line_n then_o the_o parallel_n distance_n between_o 1_o and_o 1_o shall_v divide_v ab_fw-la into_o 8_o part_n require_v 9_o any_o three_o point_n give_v to_o bring_v they_o into_o a_o circle_n let_v abc_n be_v three_o point_n to_o be_v bring_v into_o a_o circle_n first_o set_v one_o point_v on_o a_o and_o open_v the_o other_o above_o halfway_n to_o c_o and_o sweep_v the_o part_n of_o a_o circle_n above_o and_o below_o the_o point_n a_o as_o the_o two_o arch_n at_o d_o and_o e_o not_o move_v the_o compass_n do_v the_o like_a on_o c_o as_o the_o ark_n f_o and_o g_o then_o set_v the_o compass-point_n in_o b_o and_o cross_v those_o ark_n in_o def_n and_o g_o then_o a_o rule_n lay_v from_o d_o to_z e_z and_o from_o f_o to_o g_z and_o line_n draw_v do_v inter-sect_n at_o h_n the_o true_a centre_n to_o bring_v abc_n into_o a_o circle_n 10._o any_o two_o point_n give_v in_o a_o circle_n to_o draw_v part_n of_o a_o circle_n which_o shall_v cut_v they_o and_o the_o circumference_n first_o give_v into_o two_o equal_a part_n let_v a_o and_o b_o be_v two_o point_n in_o a_o circle_n by_o which_o two_o point_n i_o will_v draw_v a_o arch_n which_o shall_v cut_v the_o whole_a circumference_n into_o two_o equal_a part_n first_o draw_v a_o line_n from_o a_o the_o point_n remote_a from_o the_o centre_n through_o the_o centre_n and_o beyond_o the_o circumference_n as_o ad_fw-la then_o draw_v another_o line_n from_o a_o to_o a_o point_n in_o the_o circumference_n perpendicule_a to_o ad_fw-la and_o cut_v the_o centre_n c_o as_o the_o line_n ae_n then_o on_o the_o point_n e_o draw_v another_o line_n perpendicule_a to_o the_o line_n ae_n till_o it_o inter-sect_n ad_fw-la at_o d_o then_o these_o three_o point_n abdella_n bring_v into_o a_o circle_n or_o arch_n by_o the_o last_o rule_n shall_v divide_v the_o circumference_n into_o two_o equal_a part_n note_v the_o figure_n 8_o where_o the_o first_o circle_n be_v cut_v into_o two_o equal_a part_n at_o f_o and_o g_o by_o part_n of_o a_o circle_n pass_v through_o the_o point_n a_o and_o b._n 11._o any_o segment_n of_o a_o circle_n give_v to_o find_v the_o diameter_n and_o centre_n of_o the_o circle_n belong_v to_o it_o let_v abc_n be_v the_o segment_n of_o a_o circle_n to_o which_o i_o will_v find_v a_o centre_n any_o where_o about_o the_o midst_n of_o the_o segment_n set_v one_o point_n of_o the_o compass_n at_o pleasure_n as_o at_o b_o on_o the_o point_n b_o at_o any_o meet_a distance_n describe_v a_o circle_n and_o note_v where_o the_o circle_n do_v cross_v the_o segment_n as_o at_o d_o and_o e_o then_o not_o stir_v the_o compass_n set_v one_o point_n in_o d_o and_o cross_v the_o circle_n twice_o as_o at_o f_o and_o i_o and_o again_o set_v one_o point_n in_o e_o and_o cross_v the_o circle_n twice_o in_o g_z and_o h_n last_o by_o the_o point_n gh_o and_o fi_n draw_v two_o line_n which_o will_v meet_v in_o the_o point_n o_o the_o centre_n require_v 12._o or_o else_o to_o find_v the_o diameter_n thus_o multiply_v the_o chord_n or_o flat-side_n of_o the_o half-segment_a viz_o ak_v 12_o by_o itself_o which_o be_v call_v square_n which_o make_v 144_o then_o divide_v that_o product_n 144_o by_o 8_o the_o line_n kb_n call_v a_o sine_fw-la the_o quotient_a which_o come_v out_o will_v be_v find_v to_o be_v 18_o then_o if_o you_o add_v 8_o the_o sine_fw-la and_o 18_o the_o quotient_a together_o it_o shall_v make_v 26_o for_o the_o diameter_n require_v to_o be_v find_v 13._o any_o segment_n of_o a_o circle_n give_v to_o find_v the_o length_n of_o the_o arch_n of_o the_o segment_n lay_v the_o chord_n of_o the_o whole_a segment_n and_o twice_o the_o chord_n of_o half_a the_o segment_n from_o one_o point_n several_o and_o to_o the_o great_a extent_n add_v one_o three_o part_n of_o the_o difference_n between_o the_o extent_n and_o that_o sum_n of_o extent_n shall_v be_v equal_a to_o the_o archippus_n example_n 14._o to_o draw_v a_o helical_a line_n from_o any_o three_o point_n to_o several_a radiuss_n without_o much_o gibbiosity_n useful_a for_o architect_n shipwright_n and_o other_o let_v abcde_v be_v five_o point_n to_o be_v bring_v into_o a_o helical-line_n smooth_o and_o even_o without_o gibbiosity_n or_o bunch_n as_o the_o under-side_n of_o a_o arch_n or_o the_o bend_n of_o a_o ship_n or_o the_o like_a first_o between_o the_o two_o remote_a point_n of_o 3_o as_o a_o and_o c_o draw_v the_o line_n ac_fw-la then_o let_v fall_v a_o perpendicule_a from_o b_o to_o cut_v the_o line_n ac_fw-la at_o right_a angle_n and_o produce_v it_o to_o f_o draw_v the_o like_a perpendiculer-line_n from_o the_o point_n d_o to_o cut_v the_o line_n ce_fw-fr at_o right-angle_n produce_v to_o f._n i_o say_v the_o centre_n both_o for_o the_o arch_n ab_fw-la the_o lesser_a and_o bc_n the_o great_a will_v be_v find_v to_o be_v in_o the_o line_n bf_n the_o like_a on_o the_o other-side_n for_o de_fw-fr and_o cd_o the_o helical-circle_n or_o arch_n require_v but_o if_o you_o divide_v the_o arch_n abcde_v into_o 24_o or_o more_o part_n the_o several_a centre_n of_o the_o splay-line_n be_v thus_o find_v take_v the_o measure_n agnostus_n and_o lay_v it_o from_o b_o or_o d_o or_o c_o on_o the_o line_n gf_n and_o those_o point_n on_o gf_n shall_v be_v the_o several_a point_n to_o draw_v the_o splay-line_n of_o the_o arch_n and_o key-stone_n by_o chap._n iu._n of_o the_o explanation_n of_o certain_a term_n use_v in_o this_o follow_a book_n 1._o radius_fw-la or_o sine_fw-la of_o 90_o radius_fw-la or_o tangent_fw-la of_o 45_o or_o secant_fw-la of_o 00_o be_v all_o one_o and_o the_o same_o thing_n yet_o take_v respective_o in_o their_o proper_a place_n and_o be_v the_o whole_a line_n of_o sines_n or_o tangent_n to_o 45_o or_o more_o particulary_a
that_o point_n at_o the_o end_n of_o the_o natural-sines_n on_o the_o sector-side_n and_o at_o 90_o and_o 45_o on_o the_o edge_n of_o the_o rule_n for_o the_o artificial_a sin_n and_o tangent_n or_o 10_o on_o the_o line_n of_o number_n and_o 10_o and_o 90_o on_o the_o line_n of_o line_n and_o sines_n on_o the_o quadrantal-side_n of_o the_o instrument_n right-sine_a 2._o a_o right_a sine_fw-la of_o any_o ark_n or_o angle_n be_v the_o measure_n from_o the_o begin_n of_o the_o line_n of_o natural-sines_n to_o that_o point_n on_o that_o line_n of_o sines_n which_o represent_v the_o degree_n and_o minute_n contain_v in_o that_o ark_n or_o angle_n require_v but_o on_o the_o artificial-sines_n we_o respect_v not_o any_o measure_n but_o the_o point_n only_o tangent_fw-la 3._o the_o same_o account_n be_v use_v both_o for_o the_o right-tangent_a and_o secant_fw-la also_o the_o natural-tangent_a take_v from_o the_o begin_n to_o the_o degree_n and_o minute_n require_v the_o artificial_a respect_v the_o point_n only_o chord_n 4._o in_o the_o same_o manner_n count_v for_o the_o secant_v and_o chord_n line_n or_o verse_v sin_n cousin_n 5._o a_o cousin_n or_o sine_fw-la compliment_n of_o any_o ark_n or_o angle_n be_v the_o measure_n from_o the_o point_n represent_v the_o ark_n or_o angle_n count_v from_o 90_o to_o the_o begin_n of_o the_o line_n of_o sines_n be_v in_o effect_n the_o right-sine_a of_o the_o cousin_n of_o the_o ark_n or_o angle_n require_v as_o for_o example_n i_o will_v take_v out_o the_o cousin_n of_o the_o latitude_n of_o london_n which_o be_v 51_o 32_o count_v 51_o 32_o from_o 90_o towards_o the_o beginning_n and_o you_o shall_v find_v your_o account_n to_o end_v at_o the_o right-sine_a of_o 38_o 28_o which_o be_v the_o compliment_n of_o 51_o 32_o for_o both_o put_v together_o make_v 90_o the_o whole_a sine_fw-la or_o radius_fw-la but_o on_o the_o artificial-line_n count_v backward_o to_o the_o point_n require_v without_o mind_v any_o distance_n or_o measure_n till_o you_o come_v to_o proportion_n 6._o a_o lateral_a sine_fw-la tangent_fw-la or_o secant_fw-la lateral-sine_a or_o scale_n of_o equal_a part_n be_v any_o sine_fw-la tangent_fw-la or_o secant_fw-la take_v along_o the_o length_n of_o any_o line_n from_o the_o beginning_n onward_o be_v a_o term_n use_v only_o in_o operation_n with_o a_o sector_n or_o one_o line_n and_o a_o thread_n and_o oppose_v to_o a_o parallel-sine_a tangent_fw-la or_o secant_fw-la the_o thing_n next_o to_o be_v explain_v 7._o a_o parallel_n sine_fw-la tangent_fw-la or_o secant_fw-la parallel_n be_v any_o sine_fw-la tangent_fw-la or_o secant_fw-la take_v across_o from_o one_o leg_n to_o the_o other_o of_o a_o sector_n or_o from_o any_o degree_n and_o minute_n on_o one_o line_n to_o a_o thread_n draw_v straight_o with_o the_o other_o hand_n or_o any_o other_o fix_a line_n whatsoever_o at_o the_o near_a distance_n 8._o the_o near_o distance_n to_o any_o line_n nearest-distance_n be_v thus_o take_v when_o one_o point_n of_o the_o compass_n stand_v in_o any_o one_o point_n and_o the_o line_n be_v lay_v i_o open_v or_o close_o my_o compass_n till_o the_o other_o moveable-foot_n be_v turn_v about_o will_v but_o just_a touch_n or_o cleave_v the_o thread_n but_o if_o you_o be_v to_o lay_v the_o thread_n to_o the_o near_a distance_n than_o one_o point_v of_o the_o compass_n be_v set_v fast_o the_o other_o be_v to_o be_v turn_v about_o and_o the_o thread_n also_o slip_v to_o and_o fro_o till_o the_o compass-point_n shall_v just_o cleave_v the_o thread_n in_o the_o midst_n line_n 9_o to_o add_v one_o sine_fw-la or_o tangent_fw-la to_o a_o sine_fw-la or_o tangent_fw-la be_v to_o take_v the_o right-sine_a or_o tangent_fw-la of_o any_o ark_n or_o angle_n between_o your_o compass_n and_o set_v one_o point_n of_o the_o compass_n in_o the_o point_n of_o the_o other_o number_n and_o then_o to_o see_v how_o far_o the_o other_o point_n will_v extend_v lateral_o example_n to_o add_v the_o sine_fw-la of_o 20_o to_o the_o sine_fw-la of_o 30_o take_v the_o sine_fw-la of_o 20_o between_o your_o compass_n and_o then_o put_v one_o point_n in_o 30_o the_o other_o shall_v reach_v to_o the_o sine_fw-la of_o 51_o 21_o therefore_o the_o distance_n from_o the_o begin_n to_o 51_o 21_o be_v the_o sum_n of_o the_o sine_fw-la of_o 30_o and_o 20_o add_v together_o the_o like_a way_n be_v to_o add_v tangent_n line_n 10._o to_o subtract_v a_o sine_fw-la from_o a_o sine_fw-la or_o a_o tangent_fw-la from_o a_o tangent_fw-la be_v but_o to_o take_v the_o lateral_a lest_o right-sine_a or_o tangent_fw-la between_o your_o compass_n and_o set_v one_o point_n in_o the_o term_n of_o the_o great_a turn_n the_o other_o towards_o the_o beginning_n and_o note_v the_o degree_n and_o minute_n that_o the_o other_o point_n stay_v in_o for_o that_o be_v the_o difference_n or_o remainder_n example_n suppose_v i_o will_v take_v the_o sine_fw-la of_o 10_o degree_n from_o 25_o take_v the_o distance_n 10_o between_o your_o compass_n and_o set_v one_o foot_n in_o 25_o and_o the_o other_o turn_v towards_o the_o beginning_n shall_v reach_v to_o 14_o 23_o the_o residue_n or_o difference_n require_v or_o you_o may_v sometime_o take_v the_o distance_n between_o the_o great_a and_o the_o less_o and_o lay_v this_o from_o the_o beginning_n shall_v give_v the_o remainder_n in_o distance_n on_o the_o sin_n as_o before_o 11._o the_o rectifying-point_n point_n be_v a_o point_n or_o hole_n on_o the_o head_n of_o the_o triangule_a quadrant_n in_o the_o inter-secting_a of_o the_o hour_n and_o azimuth-line_n and_o the_o common_a line_n to_o the_o line_n and_o sin_n on_o the_o head-leg_n in_o which_o point_n you_o be_v when_o the_o rule_n be_v open_a to_o stick_v a_o small_a pin_n to_o look_v to_o the_o object_n who_o altitude_n above_o the_o horizon_n you_o will_v have_v in_o degree_n and_o minute_n of_o term_n use_v in_o dial_v plain_o be_v that_o board_n glass_n or_o flat_a superficies_n you_o intend_v to_o draw_v the_o dial_n upon_o either_o single_a of_o itself_o plain_a or_o join_v to_o some_o other_o pole_n of_o the_o plain_a plain_a be_v a_o imaginary_a point_n in_o the_o horizon_n for_o all_o upright_a dial_n direct_o opposite_a to_o the_o plain_a or_o in_o all_o plain_n a_o point_n every_o way_n 90_o degree_n from_o the_o plain_a declination_n declination_n of_o a_o plain_a be_v only_o the_o number_n of_o degree_n and_o minute_n that_o the_o pole-point_n of_o the_o plain_n be_v distant_a from_o the_o north_n and_o south-point_n of_o the_o horizon_n plain_a the_o perdendiculer-line_n on_o the_o plain_a be_v a_o line_n square_n to_o a_o horizontal-line_n be_v part_n of_o a_o circle_n pass_v through_o the_o zenith_n and_o nadir_n and_o pole-point_n of_o the_o plain_a horizontal-line_n the_o horizontal-line_n be_v a_o line_n draw_v on_o any_o plain_a exact_o parallel_v to_o the_o true_a horizon_n of_o the_o place_n you_o dwell_v in_o inclination_n reclination_n be_v when_o a_o plain_a behold_v the_o zenith-point_n over_o our_o head_n but_o inclination_n be_v when_o a_o plain_a behold_v the_o nadier_n as_o in_o a_o roof_n of_o a_o house_n the_o tiled-part_n recline_v and_o the_o celid-part_n incline_v meridian-line_n the_o meridian-line_n on_o all_o plain_n be_v the_o hour-line_n of_o 12_o but_o the_o meridian_n of_o the_o plain_a be_v the_o great_a circle_n of_o azimuth_n perpendicule_a to_o the_o plain_a be_v the_o same_o with_o the_o perpendiculer-line_n on_o the_o plain_a pass_v through_o the_o point_n of_o declination_n substile_n the_o substile-line_n on_o all_o dial_n be_v that_o line_n wherein_o the_o style_n gnomon_n or_o cock_n of_o the_o dial_n do_v stand_v usual_o count_v from_o 12_o the_o meridian-line_n or_o from_o the_o perpendiculer-line_n which_o in_o all_o erect_a dial_n be_v 12._o the_o style_n of_o a_o dial_n be_v the_o angle_n style_n between_o the_o common_a axis_n of_o the_o world_n and_o the_o plain_a upon_o the_o substile-line_n on_o the_o plain_a on_o all_o dial_n the_o angle_n between_o 12_o and_o 6_o 6._o be_v only_o the_o number_n of_o degree_n and_o minute_n contain_v between_o the_o hour-line_n of_o 12_o and_o the_o hour-line_n of_o 6_o a_o clock_n on_o any_o kind_n of_o plain_a especial_o those_o have_v centre_n the_o inclination_n of_o meridian_n meridian_n be_v the_o number_n of_o degree_n and_o minute_n count_v on_o the_o aequinoctial_a between_o the_o meridian_n or_o hour-line_n of_o 12_o and_o the_o substile_a be_v the_o distance_n between_o the_o meridian_n of_o the_o place_n viz._n 12_o a_o clock_n and_o the_o meridian_n of_o the_o plain_a but_o count_v on_o the_o aequinoctial_a and_o do_v serve_v to_o make_v the_o table_n of_o hour-arks_a at_o the_o pole_n and_o to_o prove_v your_o work_n the_o line_n parallel_v to_o 12_o parallel_n be_v two_o line_n peculiar_a to_o this_o way_n of_o dial_v by_o the_o sector_n and_o be_v only_o
a_o foot_n shut_v be_v draw_v usual_o just_a 5_o degree_n asunder_o or_o rather_o the_o two_o innermost_a line_n on_o each_o leg_n be_v always_o just_a one_o degree_n from_o the_o inside_n so_o that_o if_o you_o put_v a_o center-pin_n in_o the_o line_n of_o tangent_n just_a against_o the_o sine_fw-la of_o 30_o it_o make_v the_o two_o innermost_a line_n that_o come_v from_o the_o centre_n just_a 2_o degree_n asunder_o which_o be_v easy_a to_o remember_v either_o in_o add_v or_o substract_v as_o follow_v two_o way_n 1._o take_v the_o latteral_a sine_fw-la of_o 30_o viz._n the_o measure_n from_o the_o centre_n to_o 30_o the_o compass_n so_o set_v set_v one_o point_n in_o the_o center-pin_n in_o the_o tangent_n just_a against_o 30_o and_o turn_v the_o other_o till_o it_o cut_v the_o common_a line_n in_o the_o line_n of_o sines_n on_o the_o other_o leg_n and_o there_o it_o shall_v show_v what_o angle_n the_o two_o innermost-line_n make_v count_v from_o the_o end_n towards_o the_o head_n and_o two_o degree_n less_o be_v the_o angle_n the_o sector_n stand_v at_o both_o on_o the_o inside_n and_o outside_n the_o leg_n be_v parallel_n which_o number_n must_v near_o agree_v with_o what_o the_o inside_n of_o the_o leg_n cut_v on_o the_o head-semicircle_n or_o there_o be_v a_o mistake_n as_o thus_o for_o example_n suppose_v i_o open_v the_o rule_n at_o all_o adventure_n and_o take_a the_o latteral_a sine_fw-la of_o 30_o from_o the_o sin_n on_o the_o sector-side_n and_o put_v one_o point_n of_o the_o compass_n in_o the_o centre_n on_o the_o tangent_n right_a against_o the_o sine_fw-la of_o 30_o on_o the_o other_o leg_n or_o the_o begin_n of_o the_o secant_v on_o the_o same_o leg_n and_o turn_v the_o other_o point_n to_o the_o line_n of_o sines_n on_o the_o other_o leg_n it_o cut_v the_o sine_fw-la of_o 60_o on_o the_o innermost_a line_n that_o come_v from_o the_o centre_n then_o i_o say_v that_o the_o line_n of_o sines_n and_o tangent_n be_v just_a 30_o degree_n asunder_o and_o the_o inside_n or_o outside_n of_o the_o leg_n but_o 28_o viz._n two_o degree_n less_o as_o a_o glance_n of_o your_o eye_n to_o the_o head_n will_v plain_o show_v 2._o this_o way_n will_v serve_v very_o well_o for_o all_o angle_n above_o 20_o and_o under_o 80_o but_o for_o all_o under_o 20_o and_o above_o 80_o to_o 120_o this_o be_v a_o better_a way_n open_v the_o rule_n to_o any_o angle_n at_o pleasure_n and_o take_v the_o distance_n parallelly_n that_o be_v across_o from_o one_o leg_n to_o the_o other_o between_o the_o center-pin_n at_o 30_o in_o the_o sin_n and_o that_o in_o the_o tangent_n right_a against_o it_o and_o measure_v it_o latteral_o from_o the_o centre_n and_o it_o shall_v show_v the_o sine_fw-la of_o half_a the_o angle_n the_o sin_n and_o tangent_n stand_v at_o and_o one_o degree_n less_o be_v the_o sine_fw-la of_o half_a the_o angle_n the_o sector_n stand_v at_o example_n suppose_v that_o open_v the_o sector_n at_o adventure_n or_o to_o the_o level_n of_o any_o thing_n i_o will_v know_v the_o angle_n it_o stand_v at_o i_o take_v the_o parallel_n distance_n between_o the_o two_o centre_n and_o measure_v it_o latteral_o from_o the_o centre_n i_o find_v it_o give_v i_o the_o sine_fw-la of_o 51_o degree_n viz._n the_o half_a angle_n the_o line_n stand_v at_o or_o 50_o the_o angle_n the_o rule_n stand_v at_o which_o double_v be_v 102_o for_o the_o line_n or_o 100_o for_o the_o leg_n of_o the_o sector_n as_o a_o glance_n of_o the_o eye_n present_o resolve_v by_o the_o inner-edge_n of_o the_o moving-leg_n and_o the_o divide_a semicircle_n 3._o on_o the_o contrary_a will_v you_o set_v the_o leg_n or_o line_n to_o any_o angle_n take_v the_o half_a thereof_o latteral_o or_o one_o degree_n less_o in_o the_o half_a for_o the_o leg_n and_o make_v it_o a_o parallel_n in_o the_o two_o centre_n and_o the_o sector_n be_v so_o set_v according_o example_n i_o will_v set_v the_o leg_n to_o 90_o degree_n or_o a_o just_a square_n take_v out_o the_o latteral_a sine_fw-la of_o 44_o one_o degree_n less_o than_o 45_o the_o half_a of_o 90_o and_o make_v it_o a_o parallel_n in_o the_o two_o centre_n abovesaid_a and_o you_o shall_v find_v the_o leg_n set_v just_a to_o a_o square_a or_o right-angle_n as_o by_o look_v to_o the_o head_n you_o may_v near_o see_v at_o the_o same_o time_n if_o you_o take_v latteral_a 30_o and_o lay_v it_o from_o the_o centre_n according_a to_o the_o first_o rule_n you_o shall_v see_v a_o great_a deficiency_n therein_o as_o above_o be_v hint_v use_v v._o the_o day_n of_o the_o month_n be_v give_v to_o find_v the_o sun_n declination_n true_a place_n in_o the_o zodiac_n right_a ascention_n ascentional_a difference_n or_o rise_v and_o set_v 1._o lay_v the_o thread_n to_o the_o day_n of_o the_o month_n in_o the_o upper_a line_n of_o month_n where_o the_o length_n of_o the_o day_n be_v increase_v or_o in_o the_o lower-line_n when_o the_o day_n be_v decrease_v according_a to_o the_o time_n of_o the_o year_n then_o in_o the_o line_n of_o degree_n you_o have_v his_o declination_n wherein_o note_n that_o if_o the_o thread_n lie_v on_o the_o right_a hand_n of_o 600_o than_o the_o sun_n declination_n be_v northwards_o the_o contrary-way_n be_v southwards_o also_o on_o the_o line_n of_o the_o sun_n be_v right_a ascension_n you_o have_v his_o right_a ascension_n in_o degree_n and_o hour_n count_v one_o hour_n for_o 15_o degree_n as_o the_o month_n proceed_v from_o march_n the_o 10_o or_o equinoctial_a the_o right_a ascention_n be_v then_o 00_o and_o so_o forward_o to_o 24_o hour_n or_o 360_o degree_n as_o the_o month_n and_o day_n proceed_v again_o on_o the_o line_n oâ—_n the_o sun_n be_v true_a place_n you_o have_v the_o sign_n and_o degree_n of_o his_o place_n in_o the_o ecliptic_a aries_n or_o the_o equinoctial-point_n be_v the_o place_n to_o begin_v and_o then_o proceed_v forward_o as_o the_o month_n and_o day_n go_v last_o on_o the_o hour-line_n you_o have_v the_o ascentional-difference_n in_o degree_n and_o minute_n count_v from_o 6_o or_o the_o sun_n rise_v count_v as_o the_o morning_n hour_n proceed_v or_o his_o set_v count_v as_o the_o afternoon_n hour_n proceed_v of_o all_o which_o take_v two_o or_o three_o example_n 1._o for_o march_v the_o 12_o lay_v the_o thread_n to_o the_o day_n and_o extend_v it_o straight_o then_o on_o the_o line_n of_o degree_n it_o show_v near_o 1_o degree_n or_o 54_o minute_n northward_o 2._o the_o sun_n right_a ascension_n be_v in_o time_n 8_o minute_n and_o better_a or_o in_o degree_n 2_o deg_n 5_o minute_n 3._o the_o sun_n place_n be_v 2_o degree_n and_o 16_o minute_n in_o aries_n ♈_o 4._o the_o ascentional_a difference_n be_v 1_o degree_n and_o 10_o minute_n or_o the_o sun_n rise_v 4_o minute_n before_o and_o set_v 4_o minute_n after_o 6._o again_o for_o may_v the_o 10_o the_o thread_n lay_v thereon_o cut_v in_o the_o degree_n 20_o deg_n 9_o min._n for_o northern_a declination_n and_o 57_o deg_n 24_o min._n or_o 3_o hour_n 52_o min._n right_a ascension_n and_o 29_o 37_o in_o ♉_o taurus_n for_o his_o true_a place_n and_o 27_o 12_o for_o difference_n of_o ascension_n or_o rise_v 11_o minute_n after_o 4_o and_o set_v 49_o minute_n after_o 7._o again_o on_o the_o last_o of_o october_n or_o the_o 21_o of_o january_n near_o the_o declination_n be_v 17_o 22_o southwards_o the_o right_a ascension_n for_o october_n 31_o be_v 225_o 53_o for_o january_n 21_o 314_o 21_o the_o true_a place_n for_o october_n 31_o be_v â™_o scorpio_n 18_o deg_n 22_o min_fw-mi but_o for_o january_n 21_o â™’_o aquarius_n 11_o 52_o according_a as_o the_o month_n go_v to_o the_o end_n at_o ♑_o and_o then_o back_o again_o but_o the_o ascentional_a difference_n and_o rise_v and_o set_v be_v very_o near_o the_o same_o at_o both_o time_n viz._n 23_o 10_o and_o rise_v 32_o minute_n and_o more_o after_z 7_o and_o set_v 28_o minute_n less_o after_o 4._o use_v vi_o the_o declination_n of_o the_o sun_n or_o a_o star_n give_v to_o find_v his_o amplitude_n take_v the_o declination_n be_v count_v on_o the_o particular_a scale_n of_o altitude_n between_o your_o compass_n and_o with_o this_o distance_n set_v one_o foot_n in_o 90_o on_o the_o azimuth-line_n the_o other_o point_v apply_v to_o the_o same_o line_n shall_v give_v the_o amplitude_n count_v from_o 90._o example_n the_o declination_n be_v 12_o north_n the_o amplitude_n be_v 19_o deg_n 15_o min._n northwards_o or_o the_o declination_n be_v 20_o south_n the_o amplitude_n be_v 34_o deg_n 10_o min._n southwards_o use_v vii_o the_o right_a ascention_n and_o ascentional-difference_n be_v give_v to_o find_v his_o oblique-ascention_n when_o the_o declination_n be_v north_n than_o the_o difference_n between_o the_o
the_o sphere_n inscribe_v and_o to_o the_o circle_n circumscribe_v be_v measure_v by_o compass_n calliper_n and_o line_n of_o line_n very_o careful_o and_o exact_o then_o divide_v the_o circumference_n of_o the_o two_o end_n of_o the_o cillender_n into_o 10_o equal_a part_n and_o draw_v line_n perpendicule_a from_o end_n to_o end_n and_o plain_a all_o away_o between_o the_o line_n flat_a and_o smooth_a so_o that_o the_o two_o plain_n on_o both_o end_n will_v become_v a_o regular_a ten-sided_n figure_n then_o make_v the_o whole_a diameter_n abovesaid_a 10000_o in_o the_o line_n of_o line_n take_v out_o 0-309_a and_o with_o this_o measure_n as_o a_o radius_fw-la on_o the_o centre_n at_o both_o end_n describe_v a_o circle_n and_o if_o you_o draw_v line_n from_o every_o opposite_a line_n of_o the_o 10_o first_fw-mi draw_v you_o shall_v have_v point_n in_o the_o last_o describe_v circle_n to_o draw_v a_o pentagon_n by_o which_o be_v the_o base_a of_o one_o of_o the_o 12_o pentagonal_a pyramid_n contain_v in_o the_o body_n this_o work_n be_v to_o be_v do_v at_o both_o end_n but_o be_v sure_a that_o the_o angle_n of_o the_o pentagon_n at_o one_o end_n be_v opposite_a to_o a_o side_n of_o the_o pentagon_n at_o the_o other_o end_n then_o these_o line_n draw_v the_o two_o end_n be_v full_o mark_v then_o to_o mark_v the_o 10_o side_n do_v thus_o count_v the_o first_o length_n 1000_o viz._n the_o measure_n from_o the_o top_n to_o the_o bottom_n or_o from_o centre_n to_o centre_n and_o fit_a this_o length_n in_o 10_o and_o 10_o of_o the_o line_n of_o line_n the_o sector_n so_o set_v take_v out_o 0-3821_a and_o lay_v it_o from_o the_o two_o end_n and_o either_o draw_v or_o gage_z line_n round_o about_o from_o each_o end_n and_o in_o the_o midst_n between_o the_o two_o line_n will_v remain_v 0-2358_a then_o line_n draw_v diagonal_o on_o the_o 10_o side_n will_v guide_v to_o the_o true_a cut_n of_o the_o dodecahedron_n if_o you_o set_v a_o bevel_n to_o 116_o deg_n 33_o min._n 54_o sec._n and_o apply_v it_o from_o the_o two_o end_n you_o may_v try_v the_o truth_n of_o your_o work_n the_o declination_n and_o reclination_n of_o all_o the_o 10_o pantagonal_a plain_n be_v as_o follow_v first_o you_o have_v 1_o north_n recline_v 26_o deg_n 34_o min_fw-mi and_o 1_o south_n incline_v as_o much_o second_o you_o have_v 2_o north_n decline_v 72_o and_o recline_v 26_o 34_o and_o 2_o south_n decline_v 72_o and_o incline_v 26_o 34._o three_o you_o have_v 2_o north_n decline_v 36_o and_o incline_v 26_o 34_o and_o 2_o south_n decline_v 36_o and_o recline_v 26_o 34_o and_o 1_o horizontal_a plain_n and_o his_o opposite_a base_a to_o stand_v on_o as_o for_o the_o measure_n of_o this_o body_n the_o plain_a and_o natural_a way_n be_v thus_o first_o find_v the_o superficial_a content_n of_o the_o base_a of_o one_o of_o the_o pentagon_n by_o multiply_v the_o measure_n from_o the_o centre_n to_o the_o middle_n of_o one_o of_o the_o side_n which_o be_v the_o contain_v circle_n semi-diameter_n and_o half_o the_o sum_n of_o the_o measure_n of_o all_o the_o side_n put_v together_o and_o then_o to_o multiply_v this_o product_n by_o one_o three_o part_n of_o half_a the_o altitude_n of_o the_o body_n and_o the_o product_n shall_v be_v the_o content_a of_o one_o pentagonal_a pyramid_n be_v one_o twelve_o part_n of_o the_o dodecahedron_n and_o this_o last_o multiply_v by_o 12_o gives_z the_o solid_a content_n of_o the_o dodecahedron_n or_o 12_o time_n the_o superficial_a content_n of_o one_o side_n be_v the_o superficial_a content_a thereof_o example_n suppose_v the_o side_n of_o a_o dodecahedron_n be_v 6_o than_o the_o sum_n of_o the_o side_n measure_v be_v 30_o the_o contain_v circle_n semi-diameter_n be_v 4-12_a then_o 15_o the_o half_a of_o 30_o and_o 4-12_a multiply_v together_o make_v 61-80_a and_o 12_o time_n this_o make_v 741-60_a for_o the_o superficial_a content_n of_o the_o dodecahedron_n then_o for_o the_o solid_a content_a multiply_v 61-80_a the_o superficial_a content_n of_o one_o side_n by_o 2-233_a one_o 6_o part_n of_o 13-392_a the_o whole_a altitude_n of_o the_o body_n the_o product_n be_v 137-99940_a again_o this_o multiply_v by_o 12_o the_o number_n of_o pyramid_n make_v 16â—â—_n 9928_o the_o solid_a content_a as_o near_o as_o may_v be_v in_o such_o a_o decimal_n way_n of_o computation_n 5._o for_o the_o icosahedron_n which_o be_v a_o regular_a solid_a body_n make_v up_o of_o or_o contain_v under_o 20_o triangule_a pyramid_n who_o base_a or_o one_o of_o who_o side_n be_v a_o equilatteral_a triangle_n and_o the_o perpendicule_a altitude_n of_o one_o of_o these_o 20_o pyramid_n be_v equal_a to_o half_a the_o perpendicule_a altitude_n of_o the_o icosahedron_n from_o any_o one_o side_n to_o his_o opposite_a side_n or_o equal_a to_o the_o semi-diameter_n of_o the_o inscribe_v sphere_n to_o cut_v this_o body_n take_v any_o round_a piece_n and_o if_o the_o diameter_n thereof_o be_v 10000_o let_v the_o length_n thereof_o be_v turn_v flat_a and_o even_o to_o 8075_o or_o if_o the_o true_a round_n and_o cillendrical_a form_n in_o diameter_n be_v 4910_o let_v the_o true_a length_n when_o the_o end_n be_v plain_a and_o flat_a be_v 3964_o then_o divide_v the_o cillendrical_a part_n into_o 6_o equal_a part_n and_o plain_a away_o all_o to_o the_o line_n so_o that_o the_o two_o end_n may_v be_v two_o 6-sided-figure_n then_o make_v 5000_o the_o former_a semi-diameter_n 1000_o in_o the_o line_n of_o line_n take_v out_o 616_o and_o on_o the_o centre_n at_o each_o end_n describe_v a_o circle_n and_o by_o draw_v line_n to_o each_o opposite_a point_n make_v a_o triangle_n who_o circumscribe_v circle_n may_v be_v the_o circle_n draw_v at_o each_o end_n but_o be_v sure_a to_o mark_v the_o side_n of_o one_o triangle_n opposite_a to_o the_o point_n of_o the_o other_o triangle_n at_o the_o other_o end_n as_o before_z in_o the_o dodecahedron_n thus_o both_o the_o end_n shall_v be_v full_o and_o true_o mark_v then_o make_v the_o length_n a_o parallel_n in_o 1000_o of_o the_o line_n of_o line_n take_v out_o 379_o and_o 095_o and_o prick_v those_o two_o measure_n from_o each_o end_n and_o by_o those_o point_n draw_n or_o gage_n line_n round_a about_o on_o the_o 6_o side_n then_o diagonal_a line_n draw_v from_o point_n to_o line_n and_o from_o line_n to_o point_v round_o about_o show_v how_o to_o cut_v the_o body_n at_o 12_o cut_n note_v that_o if_o you_o set_v a_o bevel_n to_o 138-11-23_a and_o apply_v it_o from_o each_o end_n it_o will_v guide_v you_o in_o the_o true_a plain_v of_o the_o side_n of_o the_o icosahedron_n and_o a_o bevel_n set_v to_o 100_o degree_n will_v fit_v be_v apply_v from_o the_o midst_n of_o one_o side_n to_o the_o meeting_n of_o two_o side_n the_o reclination_n of_o the_o three_o triangle_n who_o upper_a side_n be_v adjacent_a or_o next_o to_o the_o three_o side_n of_o the_o upper_a horizontal-triangle_n be_v 48_o 11.23_o from_o the_o perpendicule_a or_o 41_o 48_o 37_o from_o the_o horizontal_a and_o when_o one_o corner_n stand_v south_n the_o declination_n of_o one_o of_o these_o 3_o viz._n that_o opposite_a to_o the_o south-corner_n a_o direct_a north_n the_o other_a two_o decline_v 60_o degree_n one_o southeast_n the_o other_o south-west_n the_o other_o 6_o about_o the_o corner_n of_o the_o horizontal-plain_a do_v all_o recline_v 19_o deg_n 28_o min._n 16_o sec._n the_o two_o that_o behold_v the_o south_n decline_v 22_o deg_n 14_o min._n 29_o sec._n and_o those_o two_o that_o behold_v the_o north_n decline_v 37_o deg_n 45_o min._n 51_o sec._n towards_o the_o east_n and_o west_n the_o other_o two_o remain_v recline_v as_o before_o and_o decline_v one_o north-east_n and_o the_o other_o northwest_n 82_o deg_n 14_o min._n 19_o sec._n the_o other_o nine_o under-plain_n opposite_a to_o every_o one_o of_o these_o decline_v and_o incline_v as_o much_o as_o the_o opposite_a do_v recline_v and_o decline_v as_o by_o due_a consideration_n will_v plain_o appear_v for_o the_o measure_n of_o this_o body_n do_v as_o you_o do_v by_o the_o dodecahedron_n find_v the_o area_n of_o one_o triangle_n and_o multiply_v it_o by_o 20_o give_v the_o superficial_a content_a and_o the_o area_n of_o one_o triangle_n multiply_v by_o one_o six_o part_n of_o the_o altitude_n of_o the_o body_n give_v the_o solid_a content_n of_o the_o triangule_a pyramid_n and_o that_o product_v multiply_v by_o 20_o the_o number_n of_o pyramid_n give_v the_o whole_a solid_a content_n of_o the_o icosahedron_n example_n suppose_v the_o side_n of_o a_o icosahedron_n be_v 12_o first_o square_v one_o side_n viz._n 12_o which_o make_v 144_o then_o multiply_v that_o square_n by_o 13_o and_o then_o divide_v the_o product_n by_o 30_o the_o quotient_a and_o his_o remainder_n be_v the_o superficial_a
as_o to_o the_o precept_n in_o work_v 12._o in_o use_v the_o logarithm_n great_a regard_n be_v to_o be_v have_v to_o the_o index_n or_o charracteristick_a to_o rule_v in_o the_o number_n of_o place_n the_o characteristic_a be_v one_o unite_v less_o than_o the_o number_n of_o place_n to_o express_v that_o number_n thus_o the_o characteristic_a of_o 3-5932861_a the_o logarithm_n of_o 3920_o be_v 3_o be_v one_o less_o than_o the_o number_n of_o place_n in_o 3920_o which_o consist_v of_o 4_o figure_n chap._n xv._n the_o use_n of_o the_o triangule_a quadrant_n in_o geometry_n and_o astronomy_n use_v i._o the_o radius_fw-la of_o a_o circle_n or_o line_n be_v give_v to_o find_v ready_o any_o require_v sine_fw-la tangent_fw-la or_o secant_fw-la or_o chord_n to_o that_o radius_fw-la and_o first_o trianguler-quadrant_n to_o do_v it_o by_o the_o quadrantal-side_n first_o if_o your_o radius_fw-la happen_v to_o be_v equal_a to_o the_o great_a scale_n of_o altitude_n or_o sines_n issue_v from_o the_o centre_n sines_n than_o the_o measure_n of_o any_o degree_n or_o minute_n from_o the_o centre_n towards_o the_o head_n shall_v be_v the_o sine_fw-la the_o measure_n from_o the_o center-point_n at_o 600_o on_o the_o degree_n to_o any_o degree_n and_o minuit_fw-la require_v shall_v be_v the_o tangent_fw-la to_o the_o same_o radius_fw-la tangent_fw-la and_o the_o measure_n from_o the_o tangent_fw-la to_o the_o centre_n shall_v be_v the_o secant_fw-la secant_fw-la to_o the_o same_o radius_fw-la and_o if_o you_o have_v a_o index_n or_o a_o bead_n upon_o your_o thread_n and_o set_v the_o bead_n when_o the_o thread_n be_v draw_v straight_o to_o the_o centre_n at_o 600_o on_o the_o degree_n or_o tangent_n or_o to_o the_o sine_fw-la of_o 90_o chord_n then_o if_o you_o lay_v the_o thread_n to_o any_o number_n of_o degree_n and_o minute_n count_v from_o 90_o and_o there_o keep_v it_o then_o the_o extent_n from_o the_o sine_fw-la of_o 90_o to_o the_o bead_n shall_v be_v the_o chord_n of_o the_o angle_n the_o thread_n be_v lay_v to_o to_o the_o radius_fw-la of_o the_o great_a scale_n of_o sines_n issue_v from_o the_o centre_n but_o if_o this_o happen_v to_o be_v too_o large_a than_o the_o other_o lesser_a line_n of_o sines_n issue_v upward_o from_o the_o centre_n be_v about_o one_o three_o part_n of_o the_o other_o have_v first_o itself_o for_o sin_n sines_n second_o the_o degree_n on_o the_o loose-piece_n for_o tangent_n count_v from_o the_o centre_n at_o 60_o tangent_n three_o the_o measure_n from_o the_o tangent_fw-la secant_v to_o the_o centre_n for_o a_o secant_fw-la four_a the_o bead_n and_o thread_n for_o a_o chord_n chord_n as_o before_o all_o at_o once_o to_o one_o radius_fw-la clear_o and_o distinct_o without_o any_o interruption_n to_o 75_o degree_n of_o the_o tangent_fw-la or_o secant_fw-la but_o if_o any_o other_o radius_fw-la be_v give_v than_o they_o will_v not_o be_v have_v so_o ready_o altogether_o but_o thus_o in_o order_n one_o after_o another_o and_o 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out_o ready_o how_o many_o of_o they_o part_n will_v go_v to_o make_v the_o sine_fw-la tangent_fw-la or_o secant_fw-la to_o any_o number_n of_o degree_n and_o minute_n as_o thus_o sines_n take_v the_o distance_n from_o the_o sine_fw-la of_o 90_o on_o the_o artificial_a sin_n to_o the_o sine_fw-la of_o any_o degree_n and_o minuit_fw-la require_v and_o set_v the_o same_o distance_n the_o same_o way_n from_o 10_o on_o the_o line_n of_o number_n read_v it_o as_o a_o scale_n of_o equal_a part_n and_o that_o shall_v be_v the_o natural_a sine_fw-la of_o the_o degree_n and_o minuit_fw-la require_v or_o if_o you_o lay_v a_o square_a to_o the_o sine_fw-la give_v on_o the_o number_n it_o cut_v the_o natural_a sine_fw-la require_v example_n right_o against_o the_o artificial_a sine_fw-la of_o 30_o on_o the_o line_n of_o number_n you_o find_v 5000_o which_o be_v the_o natural_a number_n thereof_o but_o if_o you_o measure_v this_o distance_n from_o 10_o in_o the_o line_n of_o line_n it_o will_v give_v the_o logarithmal_n sine_fw-la thereof_o viz._n 69897._o tangent_fw-la and_o the_o like_a for_o the_o tangent_fw-la also_o under_o 45_o in_o the_o same_o manner_n but_o for_o the_o secant_v and_o degree_n for_o the_o tangent_n above_o 45_o you_o must_v count_v thus_o measure_v as_o before_o from_o 90_o to_o the_o cousin_a of_o the_o angle_n require_v for_o a_o secant_fw-la and_o from_o 45_o to_o the_o co-tangent_a of_o 45_o for_o a_o tangent_fw-la this_o extent_n lay_v the_o contrary_a way_n from_o 1_o in_o the_o number_n show_v how_o many_o radiuss_n and_o also_o how_o much_o above_o radius_fw-la you_o must_v have_v to_o make_v up_o the_o natural_a tangent_fw-la or_o secant_fw-la require_v in_o number_n example_n the_o secant_fw-la of_o 50_o degree_n and_o the_o tangent_fw-la of_o 57_o degree_n 16_o minute_n artificial-logarithms_a be_v near_o alike_o be_v 1_o radius_fw-la 5556_o for_o the_o natural_a number_n thereof_o and_o this_o distance_n measure_v on_o the_o line_n of_o 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tangent_fw-la of_o 45_o on_o the_o tangent_fw-la line_n the_o measure_n from_o the_o centre_n to_o the_o rectifying-point_n on_o the_o head_n at_o the_o meeting_n of_o the_o line_n for_o the_o hour_n and_o azimuth_n and_o the_o line_n for_o the_o sin_n and_o line_n be_v equal_a to_o the_o tangent_fw-la of_o twice_o 30_o on_o that_o piece_n again_o the_o measure_n from_o the_o centre_n to_o the_o rectifying-point_n on_o the_o end_n of_o the_o head-leg_n shall_v reach_v from_o thence_o to_o 30_o on_o the_o loose-piece_n and_o be_v turn_v twice_o reach_v to_o 060_o on_o the_o loose-piece_n also_o the_o radius_fw-la or_o tangent_fw-la of_o 45_o turn_v twice_o from_o 060_o on_o the_o loose-piece_n shall_v reach_v to_o 75_o as_o by_o compare_v the_o natural_a number_n together_o will_v most_o exact_o appear_v though_o perhaps_o without_o this_o hint_n it_o may_v not_o have_v beeen_n observe_v by_o a_o ordinary_a eye_n have_v be_v so_o
large_a and_o plain_a in_o this_o first_o use_n i_o shall_v be_v i_o hope_v as_o plain_a though_o far_o more_o brief_a in_o all_o the_o rest_n for_o if_o you_o look_v back_o to_o chapt._n vi_o probl._n i._o sect._n 3_o you_o shall_v there_o see_v the_o full_a explain_v of_o latteral_a and_o parallel_n and_o nearest-distance_n and_o how_o to_o take_v they_o the_o mark_n for_o latteral_a be_v thus_o the_o mark_n for_o parallel_v thus_o =_o nearest-distance_n thus_o nd_o etc._n etc._n use_v ii_o the_o sine_fw-la of_o any_o ark_n or_o angle_n give_v to_o find_v the_o radius_fw-la to_o it_o take_v the_o sine_fw-la between_o your_o compass_n quadrant_n and_o set_v one_o foot_n of_o the_o compass_n in_o the_o give_v sine_fw-la and_o with_o the_o other_o point_n lie_v the_o thread_n to_o the_o nearest-distance_n and_o there_o keep_v it_o then_o the_o nearest-distance_n from_o the_o sine_fw-la of_o 90_o to_o the_o thread_n shall_v be_v the_o radius_fw-la require_v make_v the_o give_v sine_fw-la 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tangent_fw-la require_v to_o the_o thread_n shall_v be_v the_o tangent_fw-la require_v and_o the_o nd_o from_o 90_o to_o the_o thread_n shall_v be_v the_o secant_fw-la require_v make_v the_o give_v radius_fw-la a_o =_o in_o the_o cousin_a of_o the_o tangent_fw-la require_v sector_n then_o the_o =_o sine_fw-la of_o the_o inquire_v ark_n or_o angle_n shall_v be_v the_o tangent_fw-la require_v and_o =_o 90_o shall_v be_v the_o secant_fw-la require_v to_o that_o radius_fw-la use_v v._n any_o tangent_fw-la or_o secant_fw-la be_v give_v to_o find_v the_o answerable_a radius_fw-la and_o then_o any_o other_o proportionable_a tangent_fw-la or_o secant_fw-la by_o sines_n only_o first_o if_o it_o be_v a_o tangent_fw-la that_o be_v give_v quadr._n take_v it_o between_o your_o compass_n and_o set_v one_o foot_n in_o the_o sine_fw-la thereof_o lay_v the_o thread_n to_o nd_o than_o the_o =_o cousin_a thereof_o shall_v be_v radius_fw-la but_o if_o it_o be_v a_o secant_fw-la take_v it_o between_o 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more_o ready_a set_n it_o again_o note_n what_o degree_n and_o minuit_fw-la the_o thread_n do_v stay_v at_o on_o the_o degree_n and_o there_o keep_v it_o then_o the_o nd_o from_o the_o sine_fw-la of_o half_a the_o angle_n you_o will_v have_v shall_v be_v the_o chord_n of_o the_o angle_n require_v sector_n take_v the_o give_v radius_fw-la and_o make_v it_o always_o a_o =_o in_o 30_o and_o 30_o of_o sin_n the_o =_o sine_fw-la of_o half_a the_o chord_n shall_v be_v the_o chord_n require_v use_v vii_o to_o lay_v down_o any_o chord_n to_o the_o radius_fw-la of_o the_o whole_a line_n of_o sines_n quadr._n take_v the_o radius_fw-la between_o your_o compass_n and_o set_v one_o point_n in_o 90_o of_o the_o sin_n lay_v the_o thread_n to_o the_o nd_o observe_v the_o place_n there_o keep_v it_o then_o take_v the_o =_o sine_fw-la of_o the_o angle_n require_v with_o it_o set_v one_o point_n in_o the_o line_n to_o which_o you_o will_v draw_v the_o angle_n as_o far_o from_o the_o centre_n as_o the_o radius_fw-la be_v then_o draw_v the_o convexity_n of_o a_o ark_n and_o by_o that_o convexity_n and_o the_o centre_n draw_v the_o line_n for_o the_o angle_n require_v example_n let_v ab_fw-la be_v a_o radius_fw-la of_o any_o length_n under_o or_o equal_v to_o the_o whole_a line_n of_o sines_n take_v ab_fw-la between_o your_o compass_n and_o set_v one_o point_n in_o 90_o lay_v the_o thread_n to_o nd_o then_o take_v out_o the_o =_o sine_fw-la of_o 38_o or_o any_o other_o number_n you_o please_v and_o set_v one_o point_n in_o b_o the_o end_n of_o the_o radius_fw-la from_o a_o the_o centre_n and_o trace_v the_o ark_n dc_o by_o the_o convexity_n of_o which_o ark_n draw_v the_o line_n ac_fw-la for_o the_o angle_n require_v take_v the_o give_v radius_fw-la ab_fw-la sector_n make_v it_o a_o =_o in_o 90_o and_o 90_o of_o sin_n then_o take_v out_o =_o 38_o and_o set_v one_o foot_n in_o b_o draw_v the_o ark_n dc_o and_o draw_v ac_fw-la for_o the_o angle_n require_v or_o else_o work_v thus_o take_v 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the_o secant_fw-la of_o the_o angle_n as_o the_o sector_n stand_v and_o lay_v it_o twice_o in_o the_o line_n ae_n it_o will_v reach_v just_a to_o c_o the_o point_n require_v also_o note_v that_o if_o you_o want_v a_o angle_n above_o 45_o degree_n as_o the_o sector_n stand_v take_v the_o same_o from_o the_o small_a tangent_fw-la that_o proceed_v to_o 75_o and_o turn_v that_o extent_n 4_o time_n from_o b_o and_o it_o shall_v give_v the_o point_n require_v in_o the_o line_n bc._n use_v ix_o to_o lay_v down_o or_o protract_v any_o angle_n by_o the_o tangent_fw-la of_o 45_o only_a first_o make_v a_o geometrical_a square_n a_o abcd_a and_o let_v a_o be_v the_o angule_a point_n then_o make_v ab_fw-la radius_fw-la make_v ab_fw-la a_o =_o cousin_a of_o the_o angle_n you_o will_v have_v and_o lay_v the_o thread_n to_o
four_o ark_n be_v take_v from_o the_o sun_n distance_n from_o the_o elevate_v pole_n than_o the_o residue_n be_v the_o 5_o ark_n but_o for_o hour_n before_o and_o after_o 6_o add_v the_o four_o ark_n and_o the_o sun_n distance_n from_o the_o pole_n together_o to_o make_v a_o 5_o ark._n then_o say_v as_o the_o cousin_a of_o the_o four_o ark_n to_o the_o sine_fw-la of_o the_o latitude_n so_o be_v the_o cousin_a of_o the_o residue_n or_o sum_n be_v the_o 5_o ark_n to_o the_o sine_fw-la of_o the_o sun_n altitude_n at_o that_o hour_n use_v xxxviii_o the_o latitude_n sun_n azimuth_n and_o declination_n give_v to_o find_v the_o altitude_n or_o height_n thereof_o first_o to_o find_v the_o sun_n altitude_n at_o all_o azimuth_n in_o the_o equinoctial_a t._n as_o sine_fw-la 90_o to_o co-tangent_a latitude_n so_o be_v the_o cousin_a azimuth_n from_o south_n to_o the_o tangent_fw-la of_o the_o sun_n altitude_n in_o aries_n or_o as_o sine_fw-la 90_o to_o the_o cousin_a of_o the_o azimuth_n from_o south_n so_o be_v co-tangent_a lat._n to_o the_o tangent_fw-la of_o the_o sun_n altitude_n at_o that_o azimuth_n in_o the_o equinoctial_a which_o you_o must_v gather_v into_o a_o table_n for_o every_o single_a degree_n then_o as_o the_o sine_fw-la lat._n to_o the_o sine_fw-la of_o the_o sun_n declination_n so_o be_v the_o cousin_a of_o the_o sun_n altitude_n in_o equinoctial_a to_o the_o sine_fw-la of_o a_o four_o ark._n then_o when_o the_o latitude_n and_o declination_n be_v alike_o as_o both_o north_n or_o south_n then_o add_v the_o four_o ark_n and_o the_o altitude_n in_o the_o equator_fw-la together_o and_o the_o sum_n be_v the_o altitude_n require_v but_o in_o wintertime_n when_o the_o latitude_n and_o declination_n be_v unlike_a take_v the_o four_o ark_n out_o of_o the_o reciprocal_a altitude_n in_o the_o equator_fw-la and_o the_o residue_n be_v the_o sun_n altitude_n require_v also_o in_o all_o azimuth_n from_o east_n and_o west_n northwards_o in_o summertime_n also_o you_o must_v use_v substraction_n also_o and_o not_o addition_n as_o the_o rule_n before-going_a suggest_v by_o the_o particular_a quadrant_n work_v thus_o take_v the_o sun_n or_o star_n declination_n from_o the_o particular_a scale_n and_o set_v one_o point_n in_o the_o sun_n azimuth_n on_o the_o azimuth_n line_n and_o with_o the_o other_o lie_v the_o thread_n to_o the_o nd_o the_o right_a way_n and_o on_o the_o degree_n the_o thread_n cut_v the_o altitude_n require_v by_o the_o general_n quadrant_n as_o the_o co-tangent_a latitude_n take_v from_o the_o moving-leg_n or_o loose-piece_n to_o =_o sine_fw-la of_o 90_o lay_v the_o thread_n to_o nd_o so_o be_v the_o =_o cousin_a of_o the_o sun_n azimuth_n from_o south_n to_o the_o tangent_fw-la of_o the_o sun_n altitude_n in_o the_o equator_fw-la at_o that_o reciprocal_a azimuth_n which_o be_v remember_v or_o gather_v into_o a_o table_n together_o then_o say_v as_o the_o cousin_a of_o the_o sun_n altitude_n in_o the_o equator_fw-la to_o the_o =_o sine_fw-la of_o the_o latitude_n lay_v the_o thread_n to_o the_o nd_o so_o be_v the_o =_o sine_fw-la of_o the_o sun_n declination_n to_o the_o sine_fw-la of_o the_o four_o ark._n which_o four_o ark_n be_v to_o be_v add_v or_o substract_v as_o immediate_o before_o be_v direct_v and_o the_o sum_n or_o residue_n shall_v be_v the_o true_a altitude_n require_v example_n at_o 60_o degree_n of_o azimuth_n from_o south_n the_o equinoctial_a altitude_n will_v be_v find_v to_o be_v 21-40_a for_o london_n latitude_n of_o 51-32_a and_o the_o four_o ark_n in_o ♋_o or_o ♑_o be_v 28-16_a then_o 21-40_a the_o sun_n altitude_n at_o 60_o in_o ♈_o and_o 28-16_a the_o reciprocal_a four_o ark_n in_o ♋_o add_v make_v 49-56_a the_o sun_n altitude_n at_o 60_o degree_n from_o the_o south_n in_o ♋_o the_o same_o way_n of_o work_v serve_v for_o the_o sector_n as_o be_v use_v for_o the_o general_n quadrant_n sector_n only_o observe_v to_o set_v the_o sector_n instead_o of_o lay_v the_o thread_n to_o the_o near_a distance_n as_o the_o ingenious_a will_n soon_o perceive_v use_v thirty-nine_o have_v the_o latitude_n declination_n azimuth_n and_o altitude_n to_o find_v the_o hour_n as_o the_o cousin_a of_o the_o sun_n altitude_n sector_n to_o =_o cousin_a of_o the_o sun_n declination_n so_o be_v the_o =_o sine_fw-la of_o the_o sun_n azimuth_n to_o sine_z of_o the_o hour_n or_o else_o thus_o first_o find_v the_o altitude_n at_o that_o azimuth_n and_o then_o at_o that_o altitude_n and_o declination_n quadrant_n the_o hour_n as_o cousin_a of_o declination_n to_o =_o sine_fw-la of_o the_o azimuth_n so_o be_v the_o cousin_a altitude_n to_o =_o sine_fw-la of_o the_o hour_n tangent_n as_o cousin_a declination_n to_o the_o sine_fw-la of_o the_o azimuth_n so_o be_v cousin_a altitude_n to_o sine_fw-la of_o the_o hour_n use_v xl._o have_v the_o latitude_n declination_n hour_n and_o altitude_n to_o find_v the_o azimuth_n general-quadr_a as_o cousin_a of_o declination_n to_o =_o cousin_a of_o the_o sun_n altitude_n so_o be_v =_o sine_fw-la of_o the_o hour_n to_o sine_z of_o the_o azimuth_n quadrant_n first_o find_v the_o altitude_n at_o that_o hour_n and_o then_o the_o azimuth_n at_o that_o altitude_n as_o before_o t._n as_o cousin_a of_o altitude_n to_o sine_fw-la of_o the_o hour_n so_o be_v cousin_a declination_n to_o sine_fw-la of_o the_o azimuth_n from_o south_n or_o north_n as_o the_o hour_n be_v count_v that_o be_v to_o say_v from_o south_n if_o the_o hour_n be_v between_o 6_o at_o morning_n and_o 6_o at_o night_n and_o from_o the_o north_n if_o the_o contrary_a that_o be_v to_o say_v between_z 6_o at_o night_n and_o 6_o next_o morning_n or_o next_o to_o midnight_n use_v xli_o have_v the_o latitude_n and_o the_o sun_n declination_n to_o find_v the_o sun_n azimuth_n at_o 6._o as_o the_o sine_fw-la of_o 90_o t._n to_o the_o cousin_a of_o the_o latitude_n so_o be_v the_o tangent_fw-la of_o the_o sun_n declination_n to_o co-tangent_a of_o the_o sun_n azimuth_n from_o the_o north_n at_o the_o hour_n of_o 6._o first_o find_v the_o sun_n height_n at_o 6_o and_o then_o the_o sun_n azimuth_n at_o that_o altitude_n q._n make_v the_o tangent_fw-la of_o the_o declination_n a_o =_o sine_fw-la of_o 90_o quad_n lay_v the_o thread_n to_o nd_o than_o the_o =_o cousin_a latitude_n shall_v be_v the_o co-tangent_a of_o the_o sun_n azimuth_n from_o the_o north_n at_o 6._o use_v xlii_o to_o find_v the_o amplitude_n azimuth_n rise_v set_v and_o sothe_v of_o the_o fix_a star_n have_v the_o latitude_n altitude_n and_o declination_n or_o time_n of_o the_o year_n give_v first_o for_o the_o amplitude_n quadrant_n take_v the_o star_n declination_n out_o of_o the_o particular_a scale_n of_o altitude_n and_o measure_v it_o from_o 90_o in_o the_o azimuth-line_n and_o count_v the_o same_o way_n and_o the_o other_o point_n shall_v show_v the_o star_n amplitude_n require_v example_n the_o declination_n of_o the_o bull_n eye_n be_v 15-48_a if_o you_o take_v 15-48_a from_o the_o particular_a scale_n and_o lay_v it_o from_o 90_o in_o the_o azimuth-line_n it_o will_v reach_v to_o 26_o degree_n count_v from_o 90_o towards_o either_o end_n the_o same_z as_o for_o the_o sun_n in_o use_n 16._o but_o in_o other_o latitude_n work_v as_o you_o do_v for_o the_o sun_n by_o the_o rule_n in_o the_o 16_o use_v abovesaid_a for_o a_o star_n azimuth_n the_o work_n here_o be_v the_o same_o as_o for_o the_o sun_n thus_o take_v the_o star_n declination_n from_o the_o particular_a scale_n of_o altitude_n or_o sines_n between_o your_o compass_n and_o lay_v the_o thread_n to_o the_o star_n altitude_n count_v from_o 600_o towards_o the_o loose-piece_n then_o carry_v the_o compass_n or_o the_o rightside_n of_o the_o thread_n for_o northern-star_n and_o on_o the_o left-side_n for_o southern-star_n along_o the_o azimuth-line_n till_o the_o other_o foot_n be_v turn_v about_o will_v but_o just_a touch_n the_o thread_n then_o the_o fix_a point_n on_o the_o azimuth-line_n shall_v show_v the_o star_n azimuth_n from_o the_o south_n require_v example_n the_o bull_n eye_n be_v 30_o degree_n high_a shall_v have_v 77_o degree_n and_o 10_o minute_n of_o azimuth_n from_o the_o south_n if_o you_o be_v in_o other_o latitude_n use_v the_o general_a way_n as_o for_o the_o sun_n in_o all_o respect_n have_v the_o same_o declination_n that_o the_o star_n have_v north_n or_o south_n to_o find_v the_o star_n rise_v or_o set_v count_n the_o star_n declination_n on_o the_o degree_n as_o you_o count_v the_o sun_n north_n or_o south_n and_o there_o lie_v the_o thread_n and_o in_o the_o line_n of_o hour_n be_v the_o star_n rise_v or_o set_v when_o the_o star_n right_a ascention_n and_o declination_n be_v equal_a but_o at_o other_o time_n you_o must_v reckon_v thus_o first_o find_v the_o sun_n right_a ascension_n by_o use_v 14_o and_o set_v down_o
or_o suchlike_a but_o on_o large_a instrument_n a_o meridian_n line_n to_o one_o inch_n or_o half_o a_o inch_n more_o or_o less_o for_o one_o degree_n of_o the_o aequinoctial_a for_o the_o draw_v of_o chart_n according_a to_o mercator_n or_o any_o other_o more_o useful_a line_n you_o shall_v appoint_v for_o your_o particular_a purpose_n 3._o three_o in_o decribe_v the_o line_n on_o the_o two_o side_n first_o i_o shall_v speak_v to_o the_o sector-side_n where_o the_o middle_a line_n all_o meet_a at_o the_o centre_n at_o the_o head_n where_o the_o joint_n be_v the_o order_n of_o which_o go_v the_o head_n or_o joint_v end_n lie_v towards_o your_o left_a hand_n the_o sector_n be_v shut_v and_o the_o sector-side_n upermost_a be_v thus_o 1._o the_o first_o pair_n of_o line_n and_o lie_v next_o to_o you_o be_v the_o line_n of_o sines_n and_o line_n of_o line_n note_v at_o the_o end_n with_o saint_n and_o l_o for_o sines_n and_o line_n the_o middle_a line_n between_o they_o that_o run_v up_o to_o the_o centre_n and_o wherein_o the_o brass_n centre_n prick_v be_v be_v common_a both_o to_o the_o sin_n and_o line_n in_o all_o parallel_n use_n or_o entrance_n 2._o the_o line_n next_o these_o and_o count_v from_o you_o be_v the_o line_n of_o secant_v beginning_n at_o the_o middle_n of_o the_o rule_n and_o proceed_v to_o 60_o at_o the_o end_n and_o note_v also_o with_o se._n for_o secant_v one_o of_o which_o marginal_a line_n continue_v will_v run_v to_o the_o centre_n as_o the_o other_o do_v 3._o the_o next_o line_n forward_o and_o next_o the_o inner-edge_n on_o the_o moving-leg_n be_v the_o line_n of_o tangent_n the_o first_o of_o which_o and_o next_o to_o you_o be_v the_o tangent_fw-la of_o 45_o be_v the_o large_a radius_fw-la as_o to_o the_o length_n of_o the_o rule_n the_o other_o be_v another_o tangent_fw-la to_o one_o four_o part_n of_o the_o length_n of_o the_o other_o and_o proceed_v to_o 76_o degree_n a_o little_a beyond_o the_o other_o 45_o the_o middle_a line_n of_o these_o also_o be_v common_a to_o both_o in_o which_o the_o centre_n prick_v must_v be_v at_o the_o end_n of_o these_o line_n be_v usual_o set_v t._n t._n for_o tangent_n 4._o on_o the_o other_o leg_n of_o the_o sector_n be_v the_o same_o line_n again_o in_o the_o same_o order_n count_v from_o you_o wherein_o you_o may_v note_v that_o as_o the_o line_n of_o sines_n and_o line_n on_o one_o leg_n be_v next_o the_o outward-edge_n on_o the_o other_o leg_n they_o be_v next_o the_o inward-edge_n so_o that_o at_o every_o or_o any_o angle_n whatsoever_o the_o sector_n stand_v at_o you_o have_v line_n sines_n and_o tangent_n to_o the_o same_o radius_fw-la and_o the_o secant_v to_o just_a half_a the_o radius_fw-la and_o consequent_o to_o the_o same_o radius_fw-la by_o turn_v the_o compass_n twice_o also_o any_o tangent_fw-la to_o the_o great_a radius_fw-la above_o 45_o and_o under_o 76_o by_o turn_v the_o compass_n four_o time_n as_o afterward_o will_n more_o appear_v which_o contrivance_n be_v of_o excellent_a convenience_n to_o avoid_v trouble_n and_o save_v time_n and_o happy_o make_v use_n of_o in_o this_o contrary_a manner_n to_o the_o former_a way_n of_o order_v they_o 5._o five_o without_o or_o beyond_o yet_o next_o to_o the_o great_a line_n of_o tangent_n on_o the_o head-leg_n be_v place_v the_o first_o 45_o degree_n of_o the_o lesser_a tangent_n which_o begin_v from_o the_o centre_n at_o 45_o degree_n because_o of_o the_o straitness_n of_o the_o room_n next_o the_o centre_n where_o they_o meet_v in_o a_o point_n yet_o this_o be_v almost_o of_o as_o good_a use_n as_o if_o it_o have_v go_v quite_o to_o the_o centre_n by_o take_v any_o parallel_n tangent_fw-la from_o the_o middle_a or_o common_a line_n on_o the_o great_a tangent_n right_a against_o the_o requisite_a number_n count_v on_o the_o small_a tangent_fw-la under_o 45._o 6._o six_o next_o to_o this_o will_v not_o be_v amiss_o to_o add_v a_o line_n of_o sines_n to_o the_o same_o radius_fw-la of_o the_o small_a tangent_fw-la last_v mention_v and_o figure_v both_o way_n for_o sine_fw-la and_o cousin_a or_o sometime_o verse_v sines_n 7._o seven_o next_o to_o this_o a_o line_n of_o equal_a part_n and_o chord_n and_o the_o secant_v in_o a_o prick_a line_n beyond_o the_o little_a tangent_fw-la of_o 45_o all_o to_o one_o radius_fw-la to_o which_o if_o you_o please_v may_v be_v add_v mr._n foster_n line_n soll_z and_o his_o line_n of_o latitude_n but_o these_o at_o pleasure_n 8._o eight_o on_o the_o outermost-part_n of_o both_o leg_n next_o the_o outside_n in_o rule_n of_o half_a a_o inch_n thick_a and_o under_o be_v set_v the_o line_n of_o artificial_a verse_v sin_n lay_v next_o to_o the_o line_n of_o artificial_a sin_n on_o the_o outer-edge_n but_o if_o the_o rule_n be_v thick_a enough_o to_o bear_v four_o line_n then_o in_o this_o place_n may_v be_v set_v the_o meridian_n line_n according_a to_o mr._n gunter_n count_v the_o line_n of_o line_n as_o a_o scale_n of_o equal_a part_n thus_o much_o as_o for_o the_o sector-side_n of_o the_o instrument_n 4._o four_o the_o last_o side_n to_o be_v describe_v be_v the_o quadrantal-side_n of_o the_o instrument_n wherein_o it_o chief_o be_v new_a therefore_o i_o shall_v be_v as_o plain_a as_o i_o can_v herein_o to_o that_o purpose_n i_o shall_v in_o the_o description_n thereof_o imagine_v the_o loose_a piece_n or_o third_z piece_n to_o be_v put_v into_o the_o two_o mortisehole_n which_o position_n make_v it_o in_o form_n of_o a_o aequilateral_a triangle_n according_a to_o the_o figure_n annex_v note_v with_o abcd_n where_o in_o ab_fw-la be_v for_o brevity_n and_o plainness_n sake_n call_v the_o moveable-leg_n db_v the_o head_n or_o fixed-leg_n da_n the_o loose-piece_n b_o the_o head_n a_o and_o d_o the_o end_n c_o the_o leg-center_n at_o the_o begin_n of_o the_o general_a scale_n the_o centre_n at_o b_o the_o head-center_n use_v only_o in_o large_a instrument_n and_o when_o you_o please_v on_o any_o ooth_a for_o the_o line_n graduate_v on_o this_o side_n first_o on_o the_o outer-edge_n of_o the_o moveable-leg_n and_o loose-piece_n be_v graduate_v the_o 180_o degree_n of_o a_o semicircle_n c_o be_v the_o centre_n thereof_o and_o these_o degree_n be_v number_v from_o 060_o on_o the_o loose-piece_n towards_o both_o end_n with_o 10_o 20_o 30_o 40_o etc._n etc._n and_o about_o on_o the_o moveable-leg_n with_o 20_o 30_o 40_o 50_o 60_o 70_o 80_o and_o 90_o at_o the_o head_n also_o it_o be_v number_v from_o 600_o on_o the_o moveable-leg_n with_o 10_o 20_o towards_o the_o head_n and_o the_o other_o way_n with_o 10_o 20_o 30_o 40_o 50_o 60_o on_o the_o loose-piece_n and_o sometime_o also_o from_o the_o head_n along_o the_o moveable-leg_n with_o 10_o 20_o 30_o etc._n etc._n to_o â—0_n on_o the_o loose-piece_n and_o the_o like_a also_o from_o the_o end_n of_o the_o head-leg_n and_o sometime_o from_o 60_o on_o the_o loose-piece_n both_o way_n as_o your_o use_n and_o occasion_n shall_v require_v second_o on_o the_o quadrantal-side_n of_o the_o loose-piece_n but_o next_o the_o inward-edge_n be_v graduate_v 60_o degree_n or_o the_o tangent_fw-la of_o twice_o 30_o degree_n who_o centre_n be_v the_o center-hole_n or_o pin_n at_o b_o on_o the_o head_n or_o joint_n of_o the_o sector_n which_o degree_n be_v number_v three_o way_n viz._n first_o from_o d_o to_o a_o for_o forward_a observation_n and_o from_o the_o middle_n at_o 30_o to_o a_o the_o end_n of_o the_o moving-leg_n with_o 10_o 20_o 30_o and_o again_o from_o d_o the_o end_n of_o the_o head-leg_n to_o a_o with_o 40_o 50_o 60_o 70_o 80_o 90_o for_o observation_n with_o thread_n and_o plummet_n three_o next_o to_o these_o degree_n on_o the_o moving-leg_n be_v the_o line_n of_o the_o sun_n right_a ascension_n number_v from_o 600_o on_o the_o degree_n with_o 1_o 2_o 3_o 4_o 5_o 6_o towards_o the_o head_n and_o then_o back_o again_o with_o 7_o 8_o 9_o 10_o 11_o 12_o etc._n etc._n 1_o 2_o 3_o 4_o 5_o on_o the_o other_o side_n of_o the_o line_n as_o the_o figure_n annex_v show_v the_o division_n on_o this_o line_n be_v for_o the_o most_o part_n whole_a degree_n or_o every_o four_o minute_n of_o time_n four_o next_o above_o this_o be_v the_o line_n of_o the_o sun_n place_n in_o the_o zodiac_n note_v with_o ♈_o ♉_o ♊_o ♋_o towards_o the_o head_n then_o back_o again_o with_o ♌_o â™_o ♎_o over_o 600_o in_o the_o degree_n and_o 12_o and_o 24_o in_o the_o line_n of_o the_o sun_n right_a ascension_n then_o towards_o the_o end_n with_o â™_o â™_o ♑_o then_o back_o again_o with_o â™’_o and_o ♓_o be_v the_o character_n of_o the_o 12_o sign_n of_o the_o zodiac_n wherein_o you_o have_v express_v every_o whole_a degree_n as_o the_o number_n of_o they_o do_v show_v
there_o be_v 30_o degree_n in_o one_o sign_n five_o next_o above_o this_o be_v a_o calendar_n of_o month_n and_o day_n every_o single_a day_n be_v express_v and_o three_o or_o more_o letter_n of_o the_o name_n of_o every_o month_n be_v set_v in_o the_o month_n and_o also_o at_o the_o begin_n of_o each_o month_n and_o every_o 10_o day_n note_v with_o a_o prick_n on_o the_o top_n of_o the_o line_n represent_v it_o as_o be_v usual_a in_o such_o work_n six_o next_o over_o the_o month_n be_v the_o line_n to_o find_v the_o hour_n and_o azimuth_n in_o a_o particular_a latitude_n put_v always_o on_o small_a instrument_n and_o very_o rare_o on_o large_a triangular_a quadrant_n for_o sea_n observation_n the_o low_a margin_n whereof_o and_o next_o the_o month_n be_v number_v from_o the_o end_n towards_o the_o head_n with_o 10_o 20_o 30_o 40_o 50_o 60_o 70_o 80_o 90_o 100_o 110_o 120_o 130_o near_o the_o head_n centre_n for_o the_o semi-diurnal_a ark_n of_o the_o sun_n azimuth_n and_o in_o the_o margin_n next_o above_o this_o with_o 4_o 5_o 6_o 7_o 8_o 9_o 10_o 11_o 12_o near_o the_o end_n for_o the_o morning_n hour_n then_o the_o other_o way_n viz._n towards_o the_o head_n on_o the_o other-side_n the_o hour_n line_n with_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o for_o the_o afternoon_n hour_n seven_o on_o the_o same_o quadrantal-side_n and_o moveable-leg_n on_o the_o spare_a place_n beyond_o the_o month_n towards_o the_o end_n be_v set_v a_o almanac_n and_o the_o name_n of_o 12_o or_o more_o star_n to_o find_v the_o hour_n of_o the_o night_n which_o 12_o star_n be_v note_v with_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o 10_o 11_o 12._o among_o the_o degree_n in_o small_a figure_n as_o in_o the_o figure_n eight_o next_o of_o all_o to_o the_o inside_n be_v the_o line_n of_o natural_a verse_v sin_n draw_v to_o the_o centre_n with_o his_o correspondent_a line_n on_o the_o other_o or_o head-leg_n express_v sometime_o in_o a_o prick_a line_n for_o want_v of_o room_n nine_o on_o the_o head-leg_n and_o next_o to_o the_o verse_v sines_n last_o mention_v be_v first_o the_o line_n of_o equal_a part_n or_o line_n of_o line_n and_o on_o the_o same_o common_a line_n wherein_o be_v the_o centre_n be_v the_o line_n of_o natural_a sin_n who_o length_n be_v equal_a to_o the_o measure_n from_o the_o centre_n at_o c_o too_o 600_o on_o the_o moveable-leg_n so_o that_o the_o line_n of_o degree_n be_v a_o tangent_fw-la and_o the_o measure_n from_o c_o to_o any_o tangent_fw-la a_o secant_fw-la to_o the_o same_o radius_fw-la of_o the_o natural_a line_n of_o sines_n and_o line_n also_o beyond_o the_o centre_n c_o on_o the_o same_o common_a middle_a line_n be_v another_o small_a line_n of_o natural_a sin_n who_o length_n be_v equal_a to_o the_o measure_n from_o c_o to_o 60_o on_o the_o loose-piece_n then_o if_o you_o count_v from_o the_o centre_n pin_n at_o 60_o on_o the_o loose-piece_n towards_o the_o end_n of_o the_o movable-leg_n they_o shall_v be_v tangent_n to_o the_o same_o radius_fw-la and_o the_o measure_n from_o the_o centre_n c_o to_o those_o tangent_n shall_v be_v secant_v to_o the_o same_o radius_fw-la which_o may_v be_v well_o to_o be_v order_v to_o a_o three_o or_o four_o part_n of_o the_o former_a from_o the_o centre_n downward_o these_o two_o line_n of_o sines_n be_v best_a figure_v with_o their_o sin_n and_o cosines_n the_o other_o way_n with_o a_o small_a figure_n and_o the_o line_n of_o line_n from_o the_o centre_n downward_o from_o 1_o to_o 10_o where_o 90_o be_v which_o line_n of_o sines_n may_v be_v call_v a_o general_a scale_n for_o all_o latitude_n ten_o next_o to_o this_o towards_o the_o outer-edge_n be_v another_o line_n of_o natural_a sin_n fit_v to_o the_o particular_a line_n of_o hour_n and_o azimuth_n for_o one_o particular_a latitude_n note_v pert._n scale_n of_o altitude_n or_o sines_n eleven_o next_o to_o this_o be_v the_o line_n of_o 29_o ½_o for_o so_o many_o day_n of_o the_o moon_n be_v age_n in_o short_a rule_n of_o the_o whole_a length_n but_o in_o long_a not_o be_v easy_o know_v by_o the_o single_a stroke_n and_o figure_n annex_v to_o those_o stroke_n twelve_o next_o the_o outer-edge_n be_v a_o line_n of_o 24_o hour_n 360_o degree_n or_o 12_o sign_n or_o in_o most_o rule_n inch_n also_o use_v together_o with_o the_o former_a line_n of_o 29_o ½_o and_o as_o a_o theory_n of_o the_o sun_n and_o moon_n and_o ready_a way_n of_o find_v the_o hour_n by_o the_o moon_n or_o fix_a star_n thirteen_o to_o this_o instrument_n also_o belong_v a_o thread_n and_o plummet_n and_o sight_n as_o to_o other_o quadrant_n and_o a_o pair_n of_o compass_n as_o to_o other_o sector_n a_o staff_n and_o ball_n socket_n also_o if_o you_o will_v be_v curious_a and_o accurate_a and_o for_o large_a instrument_n for_o sea_n a_o square_a and_o a_o index_n which_o make_v it_o a_o perfect_a sinical_a quadrant_n and_o two_o slide_a sight_n also_o which_o make_v it_o a_o fore_n and_o back-staff_n and_o bow_n as_o will_v appear_v more_o at_o large_a afterward_o some_o use_v of_o the_o triangule_a quadrant_n for_o land_n and_o sea_n observation_n and_o operation_n chap._n i._o numeration_n on_o the_o line_n graduate_v on_o the_o instrument_n in_o the_o first_o place_n it_o will_v not_o be_v amiss_o to_o hint_n a_o few_o word_n as_o to_o the_o read_v the_o line_n or_o more_o proper_o numeration_n on_o the_o line_n wherein_o take_v notice_n that_o all_o line_n of_o equal_a part_n or_o line_n applicable_a to_o arithmetic_n as_o the_o line_n of_o line_n the_o line_n of_o number_n the_o line_n of_o foot-measure_n and_o the_o like_a wherein_o fraction_n of_o number_n be_v requisite_a they_o be_v most_o common_o account_v in_o a_o decimal_n way_n and_o as_o much_o as_o may_v be_v the_o small_a division_n be_v number_v and_o count_v according_o but_o in_o the_o line_n of_o sines_n tangent_n secant_v and_o chord_n be_v line_n belong_v proper_o to_o a_o circle_n in_o regard_n that_o the_o sexagenary_a fraction_n be_v still_o in_o use_n the_o intermediate_v division_n be_v as_o much_o as_o may_v be_v fit_v to_o that_o way_n of_o account_n viz._n by_o whole_a degree_n where_o they_o come_v close_o together_o or_o the_o line_n of_o no_o great_a use_n and_o if_o more_o room_n be_v to_o half_a degree_n or_o 30_o minute_n and_o sometime_o to_o quarter_n of_o degree_n or_o 15_o minute_n but_o towards_o the_o begin_n of_o the_o line_n of_o natural_a sin_n or_o the_o end_n of_o the_o natural_a tangent_n and_o secant_v where_o the_o degree_n be_v large_a they_o be_v divide_v to_o every_o 10_o minute_n in_o all_o large_a rule_n as_o by_o consider_v and_o account_v you_o may_v plain_o perceive_v take_v two_o or_o three_o example_n of_o each_o kind_n 1._o first_o on_o the_o line_n of_o line_n to_o find_v the_o point_n that_o represent_v 15._o in_o the_o do_n of_o this_o or_o any_o the_o like_a you_o must_v consider_v your_o whole_a scale_n radius_fw-la or_o length_n of_o the_o line_n may_v be_v account_v as_o 1_o as_o 10_o as_o 100_o as_o 1000_o or_o as_o 10000_o and_o no_o further_o can_v be_v applicable_a to_o any_o ordinary_a instrument_n wherein_o observe_v that_o if_o the_o whole_a line_n be_v one_o than_o the_o long_a stroke_n by_o every_o figure_n do_v represent_v one_o ten_o of_o that_o integer_fw-la and_o the_o next_o short_a without_o figure_n be_v hundred_o part_n of_o that_o one_o integer_fw-la and_o a_o 1000th_o part_n be_v estimate_v in_o small_a instrument_n and_o sometime_o express_v in_o large_a but_o the_o hundred_o thousand_o part_n be_v always_o to_o be_v estimate_v by_o the_o eye_n in_o all_o instrument_n whatsoever_o 2._o but_o if_o the_o whole_a line_n of_o line_n shall_v represent_v 10_o as_o it_o usual_o do_v and_o as_o it_o be_v figure_v than_o the_o long_a stroke_n at_o every_o figure_n be_v 1_o and_o the_o next_o long_o be_v ten_o and_o the_o short_a be_v hundred_o part_n and_o the_o thousand_o part_n as_o near_o as_o can_v be_v estimate_v 3._o but_o if_o the_o whole_a line_n represent_v a_o hundred_o as_o here_o in_o our_o present_a example_n than_o the_o long_a stroke_n by_o every_o figure_n represent_v 10_o and_o every_o short_a stroke_n be_v one_o and_o the_o short_a stroke_n be_v ten_o and_o the_o hundred_o part_n as_o much_o as_o can_v be_v estimate_v 4._o but_o if_o the_o whole_a line_n shall_v represent_v a_o 1000_o than_o the_o long_a stroke_n by_o the_o figure_n shall_v represent_v a_o hundred_o and_o every_o short_a 10_o and_o every_o of_o the_o short_a stroke_n be_v one_o integer_fw-la and_o a_o 10_o part_n as_o near_o as_o can_v be_v estimate_v 5._o but_o last_o if_o the_o whole_a line_n represent_v 10000_o than_o
right_a ascension_n and_o the_o ascentional-difference_n be_v the_o oblique-ascention_n but_o in_o southern_a declination_n the_o sum_n of_o the_o right_a ascension_n and_o difference_n of_o asscention_n be_v the_o oblique_a ascension_n example_n on_o or_o between_o the_o 25_o and_o 26_o of_o july_n the_o oblique-ascention_n be_v by_o substraction_n 112_o 15_o on_o the_o 30_o of_o october_n the_o oblique-ascention_n be_v 337_o 45_o by_o addition_n use_v viii_o the_o day_n of_o the_o month_n or_o sun_n declination_n and_o altitude_n be_v give_v to_o find_v the_o hour_n of_o the_o day_n take_v the_o sun_n altitude_n from_o the_o particular_a scale_n of_o altitude_n set_v one_o point_n of_o the_o compass_n in_o the_o centre_n at_o the_o begin_n of_o that_o line_n and_o open_v the_o other_o to_o the_o degree_n and_o minute_n of_o the_o sun_n altitude_n count_v on_o that_o line_n then_o lay_v the_o thread_n on_o the_o day_n of_o the_o month_n or_o declination_n and_o there_o keep_v it_o then_o carry_v the_o compass_n set_v at_o the_o former_a distance_n along_o the_o line_n of_o hour_n perpendicule_a to_o the_o thread_n till_o the_o other_o point_n be_v turn_v about_o will_v but_o just_a touch_n the_o thread_n the_o compass_n stand_v between_o the_o thread_n and_o the_o hour_n 12_o than_o the_o fix_a point_n in_o the_o hour-line_n shall_v show_v the_o hour_n and_o minute_n require_v but_o whether_o it_o be_v the_o fore_n or_o afternoon_n your_o judgement_n or_o a_o second_o observation_n must_v determine_v example_n on_o the_o first_o of_o august_n in_o the_o morning_n at_o 20_o degree_n of_o altitude_n you_o shall_v find_v it_o to_o be_v just_a 52_o minute_n past_o 6_o but_o at_o the_o same_o altitude_n in_o the_o afternoon_n it_o be_v 7_o minute_n past_o 5_o at_o night_n in_o the_o latitude_n of_o 51_o 32_o for_o london_n use_v ix_o the_o sun_n declination_n and_o altitude_n give_v to_o find_v the_o sun_n azimuth_n from_o the_o south-part_n of_o the_o horizon_n first_o by_o the_o four_o use_n find_v the_o sun_n declination_n count_v the_o same_o on_o the_o particular_a scale_n and_o take_v the_o distance_n between_o your_o compass_n then_o lay_v the_o thread_n to_o the_o sun_n altitude_n count_v the_o same_o way_n as_o the_o southern-declination_n be_v from_o 600_o towards_o the_o loose-piece_n and_o when_o need_n require_v on_o the_o loose-piece_n then_o carry_v the_o compass_n along_o the_o azimuth-line_n on_o the_o rightside_n of_o the_o thread_n that_o be_v between_o the_o thread_n and_o the_o head_n when_o the_o declination_n be_v northward_o and_o on_o the_o left-side_n of_o the_o thread_n that_o be_v between_o the_o thread_n and_o the_o end_n when_o the_o declination_n be_v southward_o so_o as_o the_o compass_n set_v to_o the_o declination_n as_o before_o and_o one_o point_v stay_v on_o the_o azimuth-line_n and_o the_o other_o turn_v about_o shall_v but_o just_a touch_n the_o thread_n at_o the_o near_a distance_n then_o i_o say_v the_o fixed-point_n shall_v in_o the_o azimuth-line_n show_v the_o suns-azimuth_n require_v example_n 1._o the_o sun_n be_v in_o the_o equinoctial_a and_o have_v no_o declination_n you_o have_v nothing_o to_o take_v with_o your_o compass_n but_o only_o lay_v the_o thread_n to_o the_o altitude_n count_v from_o 600_o towards_o the_o loose-piece_n and_o in_o the_o azimuth-line_n it_o cut_v the_o azimuth_n require_v example_n at_o 25_o degree_n high_a you_o shall_v find_v the_o sun_n azimuth_n to_o be_v 54_o 10_o at_o 32_o degree_n high_a you_o shall_v find_v 38_o 20_o the_o azimuth_n again_o at_o 20_o degree_n of_o declination_n take_v 20_o from_o the_o particular_a scale_n and_o at_o 10_o degree_n of_o altitude_n lay_v the_o thread_n to_o 10_o count_v as_o before_o then_o if_o you_o carry_v the_o compass_n on_o the_o rightside_n for_o north-declination_n you_o shall_v find_v 109_o 30_o from_o south_n but_o if_o you_o carry_v they_o on_o the_o left-side_n for_o south-declination_n you_o shall_v find_v 38_o 30_o from_o south_n the_o rest_n of_o the_o use_v you_o shall_v have_v more_o ample_o afterward_o chap._n vi_o the_o use_n of_o the_o line_n of_o number_n on_o the_o edge_n and_o the_o line_n of_o line_n on_o the_o quadrantal-side_n or_o on_o the_o sector-side_n be_v all_o as_o one_o have_v show_v the_o way_n of_o numeration_n on_o the_o line_n as_o in_o chapter_n the_o first_o also_o to_o add_v or_o subtract_v one_o line_n or_o number_n to_o or_o from_o one_o another_o as_o in_o chapter_n four_o explanation_n the_o 9th_o i_o come_v now_o to_o work_v the_o rule_n of_o multiplication_n and_o division_n and_o the_o rule_n of_o three_o direct_a and_o reverse_v both_o by_o the_o artificial_a and_o natural-line_n and_o first_o by_o the_o artificial_a be_v the_o most_o easy_a and_o then_o by_o the_o natural-line_n both_o on_o the_o sector_n and_o triangule_a quadrant_n be_v alike_o and_o i_o work_v they_o together_o first_o because_o i_o will_v avoid_v tautology_n second_o because_o thereby_o be_v better_o see_v the_o harmony_n between_o they_o and_o which_o be_v best_a and_o speedy_a three_o because_o it_o be_v a_o way_n not_o yet_o as_o i_o know_v of_o go_v by_o any_o other_o and_o last_o of_o all_o because_o one_o may_v explain_v the_o other_o the_o geometrical_a figure_n be_v the_o same_o with_o the_o instrumental-work_n by_o the_o natural_a way_n sect._n i._o to_o multiply_v one_o number_n by_o another_o 1._o by_o the_o line_n of_o number_n on_o the_o edge_n artificial_o thus_o extend_v the_o compass_n from_o 1_o to_o the_o multiplicator_n the_o same_o extent_n apply_v the_o same_o way_n from_o the_o multiplicand_a will_v cause_v the_o other_o point_n to_o fall_v on_o the_o product_n require_v example_n let_v 8_o be_v give_v to_o be_v multiply_v by_o 6_o if_o you_o set_v one_o point_n of_o the_o compass_n in_o 1_o either_o at_o the_o beginning_n or_o at_o the_o middle_n or_o at_o the_o end_n it_o matter_n not_o which_o yet_o the_o middle_n 1_o on_o the_o head-leg_n be_v for_o the_o most_o part_n the_o most_o convenient_a and_o open_v the_o other_o to_o 6_o or_o 8_o it_o matter_n not_o which_o for_o 6_o time_n 8_o and_o 8_o time_n 6_o be_v alike_o but_o yet_o you_o may_v mind_v the_o precept_n if_o you_o will_v the_o same_o extent_n lay_v the_o same_o way_n from_o 8_o shall_v reach_v to_o 48_o the_o product_v require_v which_o without_o these_o parenthesis_n be_v thus_o the_o extent_n from_o 1_o to_o 6_o shall_v reach_v the_o same_o way_n from_o 8_o to_o 48._o or_o the_o extent_n from_o 1_o to_o 8_o shall_v reach_v the_o same_o way_n from_o 6_o to_o 48._o the_o product_v require_v by_o the_o natural-line_n on_o the_o sector-side_n or_o triangule_a quadrant_n with_o a_o thread_n and_o compass_n the_o work_n be_v thus_o 1._o for_o the_o most_o part_n it_o be_v wrought_v by_o change_v the_o term_n from_o the_o artificial_a way_n as_o thus_o the_o former_a way_n be_v as_o 1_o to_o 6_o so_o be_v 8_o to_o 48_o or_o as_o 1_o to_o 8_o so_o be_v 6_o to_o 48_o but_o by_o the_o sector_n it_o be_v thus_o as_o the_o latteral_a 6_o take_v from_o the_o centre_n towards_o the_o end_n be_v to_o the_o parallel_n 10_o &_o 10_o set_v over_o from_o 10_o to_o 10_o at_o the_o end_n count_v as_o 1_o so_o be_v the_o parallel-distance_n between_o 8_o &_o 8_o on_o the_o line_n of_o line_n take_v across_o from_o one_o leg_n to_o the_o other_o to_o the_o latteral-distance_n from_o the_o centre_n to_o 48_o the_o product_v require_v or_o short_a thus_o as_o the_o latteral_a 8_o to_o the_o parallel_n 10_o so_o be_v the_o parallel_n 6_o to_o the_o latteral_a 48._o see_v figure_n i._n 2._o another_o way_n may_v you_o work_v without_o alter_v the_o term_n from_o the_o artificial_a way_n as_o thus_o by_o a_o double_a radius_fw-la take_v the_o latteral-extent_n from_o the_o centre_n to_o 1_o or_o from_o 10_o to_o 9_o if_o the_o beginning_n be_v defective_a make_v this_o a_o parallel_n in_o 6_o &_o 6_o than_o the_o latteral-extent_n from_o the_o centre_n to_o 8_o of_o the_o 10_o part_n between_o figure_n and_o figure_n shall_v reach_v across_o from_o 48_o to_o 48_o as_o before_o see_v fig._n ii_o the_o same_o work_n as_o be_v do_v by_o the_o sector_n be_v do_v by_o the_o line_n of_o line_n and_o thread_n on_o the_o quadrant-side_n that_o if_o your_o sector_n be_v put_v together_o as_o a_o triangule_a quadrant_n you_o may_v work_v any_o thing_n by_o it_o as_o well_o as_o by_o the_o sector_n in_o this_o manner_n or_o by_o the_o scale_n and_o compass_n as_o in_o the_o figure_n i._o and_o first_o as_o above_z sector-wise_a take_v the_o extent_n from_o the_o centre_n to_o 6_o latteral_o between_o your_o compass_n set_v one_o point_n in_o 10_o and_o with_o the_o other_o lie_v the_o thread_n in_o the_o near_a distance_n
like_a to_o the_o first_o superfecy_n give_v first_o find_v a_o mean_a proportion_n between_o the_o unequal_a side_n of_o the_o give_v superfecy_n that_o you_o be_v to_o make_v one_o like_a and_o find_v the_o mean_a proportion_n also_o between_o the_o unequal_a side_n of_o the_o figure_n that_o you_o be_v to_o make_v one_o equal_a to_o as_o thus_o for_o example_n i_o have_v a_o romboide_n who_o base_a be_v 5_o and_o perpendicule_a be_v 3_o and_o side_n be_v 3-55_a the_o mean_a proportion_n between_o be_v 3-866_a also_o i_o have_v a_o triangle_n who_o half-base_a be_v 8_o and_o the_o perpendicule_a 4_o the_o mean_a proportional_a be_v 5-6552_a and_o i_o will_v make_v another_o romboide_v as_o big_a as_o the_o triangle_n give_v who_o be_v area_n be_v 32_o then_o by_o the_o line_n of_o number_n say_v as_o 3-866_a the_o one_o mean_v proportion_n be_v to_o 5-6552_a the_o other_o mean_a proportion_n so_o be_v the_o side_n of_o the_o romboide_n who_o like_a i_o be_o to_o make_v to_o the_o side_n and_o perpendicule_a of_o the_o romboide_v require_v to_o make_v a_o romboide_v equal_a to_o a_o triangle_n give_v and_o like_a to_o another_o romboide_n first_o give_v as_o thus_o for_o example_n as_o 3-866_a be_v to_o 5-6552_a so_o be_v 5_o the_o base_a of_o the_o romboide_v give_v to_o 7-30_a the_o base_a of_o the_o romboide_v require_v and_o so_o be_v 3_o the_o give_v perpendicule_a of_o the_o romboide_n to_o 4-38_a the_o perpendicule_a of_o the_o romboide_v require_v so_o also_o be_v 3-55_a the_o side_n of_o the_o romboide_v give_v to_o 5-19_a the_o side_n of_o the_o romboide_v require_v for_o if_o you_o multiply_v 7-30_a the_o base_a thus_o find_v by_o 4-38_a the_o perpendicule_a now_o find_v it_o will_v make_v a_o romboide_n who_o area_n be_v equal_a to_o 32_o the_o area_n of_o the_o triangle_n that_o i_o be_v to_o make_v the_o romboide_v equal_a to_o and_o make_v the_o side_n to_o be_v 3-55_a it_o will_v be_v like_o the_o first_o romboide_v propound_v if_o it_o have_v be_v a_o trapesia_n or_o other_o form_a figure_n it_o may_v have_v be_v resolve_v into_o triangle_n and_o then_o bring_v into_o square_n as_o before_o then_o all_o they_o square_n add_v into_o one_o sum_n who_o square-root_n be_v the_o mean_a proportional_a or_o side_n of_o a_o square_a equal_a to_o that_o many-sided_n figure_n who_o like_a or_o equal_a be_v desire_v to_o be_v make_v and_o produce_v 5._o one_o diameter_n and_o content_a of_o a_o circle_n give_v to_o find_v the_o content_a of_o another_o circle_n by_o have_v the_o diameter_n thereof_o only_o give_v the_o extent_n from_o one_o diameter_n to_o the_o other_o be_v twice_o repeat_v the_o right-way_n from_o the_o give_v area_n shall_v reach_v to_o the_o area_n require_v if_o the_o area_n of_o two_o circle_n be_v give_v and_o the_o diameter_n require_v then_o the_o half-distance_n on_o the_o number_n between_o the_o two_o area_n shall_v reach_v from_o the_o one_o diameter_n to_o the_o other_o sect._n viii_o to_o find_v the_o square-root_n of_o a_o number_n to_o do_v this_o by_o the_o line_n of_o number_n you_o must_v first_o consider_v whether_o the_o figure_n whereby_o the_o number_n who_o root_n you_o will_v have_v be_v express_v be_v even_o or_o odd_a figure_n that_o be_v consist_v of_o 2_o 4_o 6_o 8_o or_o 10_o or_o 1_o 3_o 5_o 7_o or_o 9_o figure_n for_o if_o it_o be_v of_o even_a figure_n than_o you_o must_v count_v the_o 10_o at_o the_o end_n for_o the_o unite_v and_o the_o root_n and_o square_n be_v backward_o towards_o 1._o but_o if_o it_o consist_v of_o odd_a figure_n than_o the_o 1_o in_o the_o middle_n of_o the_o line_n be_v the_o unite_v and_o the_o root_n and_o square_n be_v forward_o towards_o 10_o for_o the_o square-root_n of_o any_o number_n be_v always_o the_o mean_a proportional_a or_o middle_a space_n between_o 1_o and_o the_o number_n propound_v count_v the_o unite_v according_a to_o the_o rule_n abovesaid_a so_o that_o the_o square-root_n of_o 1728_o consist_v of_o four_o figure_n it_o be_v at_o 41_o and_o â—_o â—_o count_v 10_o for_o the_o unite_v for_o the_o number_n 42_o &_o â—_o â—â—_o be_v just_a in_o the_o midst_n between_o 1728_o and_o 10._o and_o to_o find_v the_o square-root_n of_o 144_o consist_v of_o three_o figure_n divide_v the_o space_n between_o the_o middle_n 1_o and_o 144_o count_v forward_o into_o two_o equal_a part_n and_o the_o point_n shall_v rest_v at_o 12_o the_o square-root_n require_v to_o do_v this_o by_o the_o line_n of_o line_n or_o sector_n first_o find_v out_o a_o number_n that_o may_v part_v the_o number_n give_v even_o or_o as_o even_o as_o may_v be_v then_o the_o divisor_n shall_v be_v one_o extreme_a and_o the_o quotient_a another_o extreme_a the_o mean_a proportional_a between_o which_o two_o shall_v be_v the_o square-root_n require_v work_v by_o the_o last_o rule_n example_n to_o find_v the_o square-root_n of_o 144._o if_o you_o divide_v 144_o by_o 9_o you_o shall_v find_v 16_o in_o the_o quotient_a now_o a_o mean_a proportion_n between_o 9_o the_o divisor_n and_o 16_o the_o quotient_a be_v 12_o the_o root_n find_v by_o the_o last_o rule_n viz._n the_o seven_o sect._n ix_o to_o find_v the_o cubick-root_n of_o a_o number_n the_o cubick-root_n of_o a_o number_n be_v always_o the_o first_o of_o two_o mean_a proportional_n between_o 1_o and_o the_o number_n give_v count_v the_o unite_v with_o the_o follow_a caution_n set_v the_o number_n down_o and_o put_v a_o point_n under_o the_o one_a the_o four_o the_o seven_o and_o the_o 10_o figure_n and_o look_v how_o many_o point_v you_o have_v so_o many_o figure_n shall_v you_o have_v in_o the_o root_n then_o if_o the_o last_o point_n fall_v on_o the_o last_o figure_n than_o the_o middle_a 1_o must_v be_v the_o unite_v and_o the_o root_n the_o square_a and_o cube_n will_v fall_v forward_o towards_o 10._o but_o if_o the_o last_o point_n fall_v on_o the_o last_o but_o one_o than_o the_o unite_v may_v be_v place_v at_o either_o end_n viz._n at_o 1_o at_o the_o beginning_n or_o at_o 10_o at_o the_o end_n and_o then_o the_o cube_n will_v be_v one_o radius_fw-la beyond_o the_o unite_v either_o forward_o or_o backward_o but_o if_o it_o fall_v on_o the_o last_o but_o two_o then_z 10_o at_o the_o end_n of_o the_o line_n must_v be_v the_o unite_v and_o the_o root_n the_o square_a and_o cube_n will_v always_o be_v in_o the_o same_o radius_fw-la that_o be_v between_o 10_o at_o the_o end_n and_o the_o middle_n 1._o so_o that_o by_o these_o rule_n the_o cubick-root_n of_o 8_o be_v 2_o for_o put_v a_o point_n under_o 8_o be_v but_o one_o figure_n it_o have_v but_o one_o point_n therefore_o but_o one_o figure_n in_o the_o root_n second_o the_o point_n be_v under_o the_o last_o figure_n the_o middle_a 1_o be_v the_o unite_v then_o divide_v the_o space_n between_o 1_o and_o 8_o into_o three_o equal_a part_n the_o first_o part_n end_v at_o 2_o the_o root_n require_v so_o likewise_o in_o 1331_o there_o be_v two_o point_n therefore_o two_o figure_n in_o the_o root_n and_o the_o last_o point_n be_v under_o the_o last_o figure_n the_o middle_a 1_o be_v the_o unite_v and_o the_o space_n between_o 1_o and_o 1331_o be_v divide_v into_o three_o equal_a part_n the_o first_o part_n do_v end_n at_o 11_o the_o cubick-root_n of_o 1331._o again_o for_o 64_o there_o be_v one_o point_n and_o it_o fall_v on_o the_o last_o figure_n but_o one_o therefore_o the_o root_n contain_v but_o one_o figure_n and_o 1_o at_o the_o beginning_n or_o 10_o at_o the_o end_n which_o you_o please_v may_v be_v the_o unite_v but_o yet_o with_o this_o caution_n that_o the_o cube_n must_v be_v in_o the_o next_o radius_fw-la beyond_o that_o which_o belong_v to_o the_o unite_v so_o that_o divide_v the_o space_n between_o 10_o and_o 64_o beyond_o the_o middle_n 1_o towards_o the_o beginning_n into_o three_o equal_a part_n the_o first_o part_n fall_v on_o 4_o the_o cubick-root_n require_v or_o if_o you_o divide_v the_o space_n between_o 1_o and_o 64_o near_o the_o 10_o into_o three_o equal_a part_n the_o first_o part_n fall_v on_o 4_o also_o again_o for_o 729_o there_o be_v but_o one_o point_n therefore_o but_o one_o figure_n again_o it_o fall_v on_o the_o last_o but_o 2_o therefore_o 10_o at_o the_o end_n be_v the_o unite_v and_o between_o 10_o and_o the_o middle_a 1_o backward_o you_o shall_v have_v both_o root_n square_n and_o cube_n for_o the_o number_n require_v which_o will_v be_v at_o 9_o for_o if_o you_o divide_v the_o space_n between_o 10_o and_o 729_o into_o three_o equal_a part_n the_o first_o part_n will_v stay_v at_o 9_o the_o cubick-root_n require_v note_v if_o it_o be_v a_o surd_a number_n that_o can_v be_v cube_v exact_o yet_o the_o
particular_a product_n of_o every_o double_a number_n for_o the_o three_o term_n one_o after_o another_o this_o do_v repeat_v the_o rule_n of_o three_o as_o often_o as_o there_o be_v double_a number_n the_o four_o term_v produce_v from_o those_o operation_n shall_v be_v answer_n to_o the_o question_n require_v viz._n the_o quantity_n of_o each_o man_n gain_n or_o loss_n example_n 25_o &_o 30_o a_n ox_n and_o time_n of_o feed_v multiply_v be_v 750_o 15_o &_o 40_o b_n ox_n and_o time_n of_o feed_v multiply_v be_v 600_o 20_o &_o 40_o c_n ox_n and_o day_n of_o feed_v multiply_v be_v 800_o the_o sum_n 2150_o as_o 215_o to_o 50_o so_o be_v 750_o a_n stock_n to_o 17-9_a a_n rent_n  _fw-fr 600_o b_n 13-19_a b's_n  _fw-fr 800_o c_n 18-12_a c_n 9_o to_o work_v by_o the_o line_n of_o number_n the_o extent_n of_o the_o compass_n from_o 1_o to_o 25_o shall_v reach_v the_o same_o way_n from_o 30_o to_o 750_o the_o first_o product_n of_o a_o a'_v double_a number_n or_o stock_n and_o as_o 1_o to_o 15_o so_o be_v 40_o to_o 600_o the_o product_n of_o b_n double_a number_n and_o stock_n and_o as_o 1_o to_o 20_o so_o be_v 40_o to_o 800_o the_o product_n of_o c_n double_a number_n and_o stock_n which_o three_o product_n add_v make_v 2150_o the_o first_o term_n and_o 50_o be_v the_o second_o term_n and_o 750_o 600_o and_o 800_o the_o three_o product_n several_o the_o three_o term_n then_o the_o extent_n from_o 2150_o to_o 50_o shall_v reach_v the_o same_o way_n from_o 750_o to_o 17-45_a or_o 17_o l._n 9_o s._n and_o from_o 600_o to_o 13-95_a or_o 13_o l._n 19_o s._n and_o from_o 800_o to_o 18-60_a or_o 18_o l._n =_o 12_o s._n the_o several_a answer_n require_v which_o be_v add_v together_o make_v up_o 50_o l._n the_o whole_a rent_n to_o be_v pay_v among_o they_o there_o be_v other_o rule_n of_o arithmetic_n as_o the_o rule_n call_v allegation_n medial_a and_o alternate_a and_o the_o rule_n of_o position_n or_o falsehood_n in_o the_o work_n of_o which_o be_v so_o many_o caution_n in_o order_v the_o number_n before_o you_o come_v to_o the_o proportional_a work_n that_o it_o will_v make_v the_o book_n more_o bulky_a than_o useful_a therefore_o i_o shall_v wave_v it_o and_o refer_v you_o to_o the_o particular_a book_n of_o arithmetic_n as_o that_o of_o mr._n record_n dee_n and_o mellis_n or_o that_o of_o mr._n wingate_n natural_a and_o artificial_a have_v in_o it_o plenty_n of_o example_n and_o other_o also_o as_o johnfâ—us_o jaggers_o or_o moor_n arithmetic_n any_o of_o which_o exceed_v the_o bound_n i_o intend_v for_o this_o whole_a discourse_n i_o shall_v therefore_o pass_v on_o to_o the_o rule_n of_o practice_n in_o several_a kind_n as_o measure_v superfecy_n and_o solid_n and_o rule_n of_o double_a and_o treble_a proportion_n and_o question_n of_o interest_n which_o be_v tedious_a by_o the_o pen_n without_o the_o help_n of_o particular_a table_n and_o very_o easy_a by_o the_o line_n of_o number_n as_o will_v full_o appear_v in_o the_o next_o chapter_n chap._n vii_o the_o use_n of_o the_o line_n of_o number_n in_o measure_v of_o any_o kind_n of_o superficial_a measure_n the_o measure_n that_o be_v common_o use_v in_o this_o work_n be_v a_o foot-rule_n divide_v into_o 100_o part_n or_o else_o into_o 12_o inch_n and_o those_o inch_n into_o half_n and_o quarter_n or_o 8_o part_n or_o inch_n and_o 10_o part_n but_o in_o regard_n that_o the_o number_n do_v most_o fit_o agree_v to_o the_o 100_o part_n of_o a_o foot_n it_o will_v be_v convenient_a here_o to_o show_v how_o to_o reduce_v they_o or_o any_o other_o fraction_n from_o 12_o s._n to_o 10_o s._n or_o any_o other_o whatsoever_o from_o one_o fraction_n to_o the_o other_o which_o by_o the_o line_n of_o number_n be_v quick_o do_v as_o thus_o from_o 12_o to_o 10_o reduction_n extend_v the_o compass_n from_o one_o denominator_fw-la to_o the_o other_o the_o same_o extent_n shall_v reach_v the_o same_o way_n from_o one_o numerator_n to_o the_o other_o example_n as_o 12_o to_o 10_o so_o be_v 6_o half_a of_o 12_o to_z 5_o half_n of_o 10._o again_o as_o 120_o to_o 100_o so_o be_v 30_o a_o four_o of_o 120_o to_o 25_o a_o four_o of_o 100_o which_o two_o line_n of_o inch_n and_o foot-measure_n be_v usual_o set_v together_o on_o rule_n for_o the_o ready_a way_n of_o reduction_n by_o ocular_a inspection_n only_o in_o this_o manner_n as_o in_o the_o figure_n and_o the_o like_a may_v be_v for_o any_o thing_n whatsoever_o as_o mr._n edmond_n windgate_n have_v large_o show_v in_o his_o arithmetic_n which_o line_n be_v next_o to_o the_o line_n of_o number_n on_o your_o rule_n will_v be_v very_o plain_a and_o ready_a in_o the_o use_n of_o the_o line_n of_o number_n for_o foot_n and_o inch_n or_o shilling_n and_o penny_n and_o the_o same_o rule_n of_o reduction_n serve_v for_o all_o manner_n of_o fraction_n for_o as_o the_o denominator_fw-la of_o one_o fraction_n be_v to_o the_o denominator_fw-la of_o the_o other_o which_o in_o the_o decimal_n work_n be_v always_o a_o unite_v with_o one_o two_o or_o more_o cipher_n so_o be_v the_o numerator_n of_o one_o to_o the_o numerator_n of_o the_o other_o and_o note_v that_o the_o operation_n of_o decimal_a number_n and_o their_o fraction_n be_v no_o other_o than_o whole_a number_n except_o only_o the_o cut_n off_o so_o many_o figure_n as_o there_o be_v fraction_n in_o the_o multiplicator_n and_o multiplicand_a after_o any_o multiplication_n as_o in_o the_o follow_a example_n will_v appear_v this_o be_v premise_v i_o come_v next_o to_o the_o work_n problem_n i._o the_o breadth_n of_o a_o oblong_a superficies_n give_v in_o foot_n measure_n to_o find_v how_o much_o in_o length_n make_v one_o foot_n the_o extent_n of_o the_o compass_n from_o the_o breadth_n to_o 1_o shall_v reach_v the_o same_o way_n from_o 1_o to_o the_o length_n require_v example_n at_o 7_o 10_o broad_a as_o 7_o to_o 1_o so_o be_v 1_o to_o 1_o foot_n and_o 43_o part_n the_o breadth_n give_v in_o inch_n to_o find_v how_o much_o make_v a_o foot_n as_o the_o breadth_n in_o inch_n to_o 12_o so_o be_v 12_o to_o the_o length_n of_o a_o foot_n in_o inch_n and_o 10_o part_n example_n at_o 8_o inch_n broad_a you_o must_v have_v 18_o inch_n to_o make_v a_o foot_n for_o the_o extent_n from_o 8_o to_o 12_o shall_v reach_v the_o same_o way_n from_o 12_o to_o 18._o to_o work_v these_o two_o by_o the_o line_n of_o line_n by_o inches_n as_o 1_o to_o =_o 7_o so_o be_v =_o 1_o to_o 1_o 43_o the_o length_n in_o foot-measure_n by_o inch_n as_o 12_o to_o =_o 8_o so_o be_v =_o 12_o to_o 18_o or_o else_o by_o inch_n as_o 8_o to_o =_o 12_o so_o be_v 12_o to_o =_o 18_o the_o length_n in_o inch_n problem_n ii_o have_v the_o breadth_n of_o a_o oblong_a superficies_n give_v in_o foot-measure_n to_o find_v how_o much_o be_v in_o a_o foot_n long_o this_o be_v soon_o wrought_v for_o in_o every_o foot_n long_o there_o be_v just_a as_o much_o as_o the_o breadth_n be_v either_o in_o foot-measure_n or_o inch_n for_o a_o piece_n of_o board_n half_o a_o foot_n broad_a and_o a_o foot_n long_o be_v just_o half_a a_o foot_n problem_n iii_o have_v the_o length_n and_o breadth_n in_o foot-measure_n to_o find_v the_o content_a in_o foot_n the_o extent_n from_o 1_o to_o the_o length_n shall_v reach_v the_o same_o way_n from_o the_o length_n to_o the_o content_a in_o foot_n example_n as_o 1_o to_o 1_o foot_n 50_o the_o breadth_n so_o be_v 11_o foot_n 10_o part_n the_o length_n to_o 16_o foot_n and_o 65_o part_n the_o comment_n require_v the_o breadth_n give_v in_o inch_n and_o the_o length_n in_o foot_n to_o find_v the_o content_a in_o foot_n as_o 12_o to_o the_o breadth_n in_o inch_n so_o be_v the_o length_n in_o foot_n to_o the_o content_a in_o foot_n require_v example_n at_o 9_o inch_n broad_a and_o 11_o foot_n long_o the_o extent_n from_o 12_o to_o 9_o shall_v reach_v the_o same_o way_n from_o 11_o to_o 8_o foot_n 3_o inch_n or_o ¼_n by_o the_o line_n of_o line_n as_o 11_o to_z =_o 12_o so_o be_v 9_o to_z 8_o ¼_n but_o note_v that_o in_o work_v this_o and_o many_o suchlike_a it_o will_v be_v convenient_a to_o double_v your_o scale_n in_o account_n call_v 10_o at_o the_o end_n 20_o and_o every_o single_a figure_n as_o much_o more_o as_o to_o call_v 12_o and_o 24_o etc._n etc._n so_o that_o in_o this_o operation_n the_o work_n run_v thus_o as_o 11_o take_v from_o the_o line_n of_o line_n count_v 1_o for_o 10_o as_o usual_o to_o =_o 6_o the_o half_a of_o 12_o reckon_v double_a for_o 12_o so_o be_v =_o 4_o ½_o count_v for_o 9_o to_z 8_o ¼_n between_o the_o centre_n and_o 1._o
of_o one_o side_n and_o the_o half_a sum_n of_o the_o measure_n of_o all_o the_o side_n multiply_v together_o shall_v be_v the_o true_a area_n or_o content_v thereof_o all_o other_o figure_n whatsoever_o of_o how_o many_o side_n soever_o they_o be_v may_v be_v reduce_v to_o triangle_n or_o to_o trapeziaes_n and_o measure_v as_o before_o which_o kind_n of_o figure_n surveyor_n and_o builder_n oftentimes_o meet_v withal_o in_o their_o operation_n problem_n xi_o for_o the_o measure_n of_o on_o oval_n the_o best_a way_n oval_n be_v to_o reduce_v it_o to_o a_o circle_n thus_o divide_v the_o distance_n on_o the_o line_n of_o number_n between_o the_o length_n and_o the_o breadth_n of_o the_o oval_a into_o two_o equal_a part_n and_o the_o middle_a point_n where_o the_o compass_n stay_v on_o shall_v be_v the_o diameter_n of_o a_o circle_n equal_a in_o area_n to_o the_o oval_a give_v example_n suppose_v a_o oval_a be_v 10_o foot_n long_o transverse_n and_o 8_o foot_n broad_a conjugate_v the_o mean_a proportion_n between_o 10_o and_o 8_o be_v 8-95_a i_o say_v that_o a_o circle_n who_o diameter_n be_v 8-95_a be_v equal_a to_o a_o oval_a of_o 8_o broad_a and_o 10_o long_a and_o how_o to_o measure_v the_o circle_n be_v show_v before_o of_o these_o figure_n if_o the_o content_a be_v 100_o than_o the_o side_n of_o these_o regular_a figure_n be_v 〈…〉_z and_o also_o so_o in_o proportion_n be_v any_o other_o quantity_n of_o content_a require_v perpendiculer-triangle_n 13._o 123._o trianguler-side_n 15._o 2._o square_n its_o side_n 10._o 0._o pantagon_n of_o five_o side_n 7._o 62._o hexagon_n of_o six_o 6._o 02._o heptagon_n of_o seven_o 5._o 26._o octagon_n of_o eight_o 4._o 55._o nonagon_n of_o nine_o 4._o 03._o decagon_n of_o ten_o 3._o 06._o half_a diameter_n or_o radius_fw-la 5._o 64._o example_n as_o thus_o i_o will_v have_v a_o triangle_n to_o contain_v 200_o what_o must_v the_o side_n be_v the_o half_a distance_n on_o the_o number_n between_o 100_o and_o 200_o shall_v reach_v from_o 15-2_a to_o 21-5_a the_o side_n require_v and_o from_o 13-123_a the_o fix_a perpendicule_a for_o a_o triangle_n who_o area_n be_v 100_o to_o 18-6_a the_o perpendicule_a of_o a_o equilatteral_a triangle_n who_o area_n be_v 200._o but_o if_o the_o side_n be_v give_v and_o you_o will_v find_v the_o area_n work_v thus_o the_o extent_n from_o the_o fixed-side_n to_o the_o given-side_n shall_v reach_v at_o two_o turn_n from_o the_o fix_a area_n to_o the_o area_n require_v the_o extent_n from_o 15-2_a to_o 21-5_a shall_v reach_v at_o twice_o repeat_v from_o 100_o to_o 200._o problem_n xii_o to_o make_v a_o oval_a equal_a to_o a_o circle_n have_v the_o diameter_n of_o the_o circle_n and_o the_o length_n or_o breadth_n of_o the_o oval_a give_v sâ—t_a one_o point_n of_o the_o compass_n in_o the_o diameter_n of_o the_o circle_n find_v out_o on_o the_o line_n of_o number_n and_o the_o other_o point_n to_o the_o oval_n length_n then_o turn_v that_o distance_n the_o contrary_a way_n from_o the_o same_o diameter-point_n and_o it_o shall_v reach_v to_o the_o breadth_n of_o the_o oval_a require_v example_n let_v the_o diameter_n of_o a_o circle_n be_v 10_o foot_n i_o will_v have_v a_o oval_a to_o contain_v as_o much_o as_o the_o circle_n and_o be_v 12_o foot_n long_o the_o query_n be_v how_o broad_a must_v it_o be_v set_a one_o point_n in_o 10_o and_o the_o other_o in_o 12_o that_o extent_n turn_v the_o other_o way_n from_o 10_o shall_v reach_v to_o 8-34_a the_o breadth_n of_o the_o oval_a require_v if_o you_o please_v to_o alter_v the_o breadth_n or_o length_n you_o shall_v soon_o find_v the_o length_n or_o breadth_n according_o to_o work_v this_o by_o the_o line_n of_o line_n you_o must_v work_v by_o the_o direction_n in_o the_o seven_o section_n of_o the_o 6_o chapter_n as_o thus_o first_o to_o find_v the_o content_a of_o the_o oval_a join_v the_o length_n and_o breadth_n in_o one_o sum_n to_o get_v the_o sum_n the_o half_a sum_n and_o difference_n and_o half_a difference_n then_o open_a the_o sector_n or_o lay_v the_o thread_n on_o 600_o to_o a_o right_a angle_n then_o count_v half_a the_o difference_n from_o the_o centre_n downward_o and_o note_v the_o place_n then_o take_v half_a the_o sum_n between_o your_o compass_n and_o set_v one_o point_n in_o the_o half-difference_n and_o extend_v the_o other_o to_o the_o other_o leg_n or_o perpendicule_a line_n and_o it_o shall_v show_v a_o point_n who_o distance_n from_o the_o centre_n be_v the_o mean_a proportional_a require_v which_o be_v the_o diameter_n of_o a_o circle_n equal_a in_o area_n to_o the_o oval_a or_o elipsis_n give_v to_o be_v measure_v as_o before_o be_v show_v to_o make_v a_o oval_a equal_a to_o a_o circle_n take_v the_o guess_v half-sum_n of_o the_o length_n and_o breadth_n of_o the_o oval_a and_o set_v one_o point_n in_o the_o diameter_n of_o the_o circle_n and_o on_o the_o other_o leg_n set_v at_o a_o right_a angle_n the_o other_o point_n shall_v show_v half_o the_o difference_n between_o the_o length_n and_o breadth_n of_o the_o oval_a then_o if_o the_o mean_v proportional_a between_o they_o be_v equal_a to_o the_o diameter_n you_o have_v wrought_v right_o if_o not_o then_o resolve_v upon_o the_o length_n or_o breadth_n of_o the_o oval_a take_v more_o or_o less_o for_o the_o breadth_n or_o length_n according_o herein_o also_o be_v see_v the_o excellency_n of_o the_o line_n of_o number_n in_o many_o operation_n problem_n xiii_o the_o length_n and_o breadth_n of_o any_o oblong_a superficies_n give_v in_o foot_n to_o find_v the_o content_a in_o yard_n as_o 9_o foot_n the_o number_n of_o foot_n in_o one_o yard_n to_o the_o length_n in_o foot_n and_o part_n so_o be_v the_o breadth_n in_o foot_n and_o part_n to_o the_o content_a in_o yard_n example_n at_o 13_o foot_n 6_o inch_n long_o and_o 7_o foot_n 6_o inch_n broad_a the_o extent_n of_o the_o compass_n from_o 9_o to_o 13_o ½_o the_o length_n shall_v reach_v the_o same_o way_n from_o 7_o ½_o the_o breadth_n to_o 11_o yard_n and_o a_o quarter_n the_o content_a note_v that_o if_o yoâ—_n measure_n by_o foot_n and_o hundred_o part_n you_o shall_v find_v this_o way_n exceed_v ready_a the_o answer_n be_v give_v in_o yard_n and_o hundred_o part_n of_o a_o yard_n but_o if_o you_o have_v a_o yard_n divide_v into_o a_o 100_o part_n to_o measure_v withal_o then_o the_o rule_n be_v thus_o as_o 1_o to_o the_o length_n oâ—_n breadth_n so_o be_v the_o breadth_n or_o length_n to_o the_o content_a in_o yard_n example_n at_o 3_o yard_n 72_o part_n broad_a and_o 5_o yard_n 82_o part_n long_o the_o extent_n of_o the_o compass_n on_o the_o line_n of_o number_n from_o 1_o to_o 3-72_a shall_v reach_v the_o same_o way_n from_o 5-82_a to_o 21_o yard_n 65_o part_n the_o content_a in_o square_a yard_n and_o 100_o part_n by_o the_o line_n of_o line_n as_o 5-82_a to_o =_o 1_o at_o 10_o the_o end_n so_o be_v =_o 3-72_a to_o 21-65_a yard_n or_o in_o the_o example_n before_o as_o 13-6_a count_v 6_o ½_o for_o 13_o be_v to_o =_o 9_o so_o be_v =_o 7_o ½_o to_o 11_o ¼_n as_o you_o count_v at_o first_o problem_n fourteen_o the_o length_n and_o breadth_n of_o any_o wall_n be_v give_v in_o foot_n and_o 100_o part_n to_o find_v how_o many_o rod_n of_o walling_n there_o shall_v be_v at_o a_o brick_n and_o a_o half_a thick_a first_o you_o must_v note_n that_o 272_o foot_n and_o a_o quarter_n make_v one_o rod_n or_o so_o many_o foot_n be_v in_o a_o rod_n second_o that_o let_v the_o wall_n be_v half_a a_o brick_n one_o brick_n two_o brick_n two_o and_o a_o half_a or_o three_o brick_n thick_a it_o be_v to_o be_v reduce_v to_o brick_n and_o a_o half_a thick_a as_o a_o standard_n thickness_n three_o note_n that_o this_o reduce_n to_o a_o brick_n and_o half_a thick_a may_v be_v at_o the_o measure_n or_o after_o the_o casting-up_a as_o you_o please_v as_o in_o the_o example_n follow_v will_v plain_o appear_v as_o thus_o for_o instance_n a_o front_n or_o side-wall_n of_o a_o house_n be_v to_o be_v measure_v wherein_o the_o celler-story_n wall_n be_v 2_o brick_n and_o a_o half_a thick_a the_o shop_n and_o first_o chamber-story_n be_v two_o brick_n thick_n the_o other_o story_n 1_o brick_n and_o a_o half_a thick_a and_o the_o gable-end_n 1_o brick_n thick_a the_o near_a way_n to_o measure_v this_o wall_n i_o conceive_v be_v thus_o 1._o the_o cellar-story_n be_v 10_o foot_n high_a but_o be_v 2_o brick_n and_o a_o half_a thick_a i_o make_v it_o 16_o foot_n 8_o inch_n high_a by_o add_v two_o three_o of_o 10_o foot_n to_o the_o 10_o foot_n high_a which_o be_v 6_o foot_n 8_o inch_n in_o all_o 16_o foot_n 8_o inch_n 2._o the_o other_o two_o story_n
but_o 43_o inch_n and_o 1_o â—_o about_o who_o four_o part_n be_v under_o 11_o so_o that_o here_o i_o may_v very_o well_o abate_v 1_o inch_n from_o the_o four_o part_n of_o the_o line_n so_o consequent_o if_o the_o rind_n be_v thin_a and_o the_o tree_n less_o a_o less_o allowance_n will_v serve_v and_o if_o the_o rind_n be_v thick_a and_o the_o tree_n large_a there_o ought_v to_o be_v more_o as_o by_o cut_v the_o rind_n away_o and_o then_o take_v the_o true_a diameter_n you_o may_v plain_o see_v this_o measure_n by_o the_o four_o part_n of_o the_o circumference_n for_o the_o side_n of_o the_o square_n and_o allowance_n for_o the_o bark_n be_v allow_v for_o as_o before_z i_o say_v will_v prove_v to_o be_v just_a one_o 5_o part_n over-measure_n especial_o consider_v this_o that_o when_o it_o be_v hew_v and_o large_a wane_v leave_v than_o the_o tree_n be_v mark_v for_o more_o measure_n sometime_o by_o 10_o foot_n in_o 60_o than_o there_o be_v before_o it_o be_v hew_v the_o reason_n be_v because_o when_o the_o tree_n be_v round_o and_o unhewn_a the_o gird_v it_o and_o count_v the_o four_o part_n for_o the_o side_n of_o the_o square_n be_v but_o very_o little_o more_o than_o the_o inscribe_v square_a and_o then_o be_v hew_v and_o that_o scarce_o to_o a_o eight_o square_n and_o measure_v with_o a_o pair_n of_o calliper_n to_o the_o extremity_n of_o that_o do_v not_o then_o allow_v the_o square_n equal_a to_o the_o circle_n for_o the_o side_n of_o the_o square_n as_o by_o the_o work_n by_o those_o several_a square_n will_v very_o plain_o appear_v which_o be_v foretell_v and_o warn_v of_o let_v those_o who_o it_o concern_v look_v to_o it_o but_o this_o be_v premise_v and_o the_o party_n agree_v the_o difference_n be_v as_o 4_o to_z 5_o the_o best_a way_n to_o measure_v round_a timber_n i_o conceive_v be_v by_o the_o diameter_n take_v with_o a_o pair_n of_o calliper_n and_o the_o length_n which_o for_o the_o just_a and_o true_a measure_n be_v large_o handle_v already_o but_o if_o this_o allowance_n be_v agree_v on_o than_o the_o proportion_n for_o it_o be_v thus_o as_o 1-526_a to_o the_o diameter_n so_o be_v the_o length_n to_o a_o four_o and_o so_o be_v that_o four_o to_o the_o content_a in_o foot_n example_n the_o extent_n from_o 1-526_a to_o 15-26_a shall_v reach_v be_v twice_o repeat_v from_o 10_o foot_n the_o length_n to_o 10_o foot_n the_o content_a require_v be_v all_o at_o one_o point_n or_o another_o example_n the_o extent_n from_o 1-526_a to_o 20_o inch_n the_o diameter_n be_v twice_o repeat_v the_o same_o way_n from_o 10_o foot_n the_o length_n shall_v reach_v to_o 17_o foot_n ¼_n the_o content_a or_o if_o you_o have_v the_o circumference_n and_o length_n then_o the_o extent_n from_o 48_o to_o the_o inch_n about_o be_v turn_v twice_o the_o same_o way_n from_o the_o length_n in_o foot_n shall_v reach_v to_o the_o content_a require_v the_o extent_n from_o 48_o to_o 62_o the_o inch_n about_o be_v turn_v twice_o from_o 10_o the_o same_o way_n shall_v reach_v to_o 17_o foot_n ¼_n the_o content_a in_o that_o measure_n thus_o you_o have_v full_a and_o complete_a direction_n for_o the_o measure_n of_o any_o round_a timber_n by_o the_o line_n of_o number_n by_o have_v the_o diameter_n and_o length_n give_v after_o any_o usual_a manner_n there_o remain_v only_o one_o general_a and_o natural_a way_n by_o find_v the_o base_a of_o the_o middle_n or_o one_o end_n by_o the_o 7_o the_o problem_n of_o superficial_a measure_n and_o then_o to_o multiply_v that_o base_a by_o the_o length_n will_v give_v the_o true_a content_a in_o foot_n or_o inch_n thus_o have_v find_v the_o base_a of_o the_o cillender_n by_o the_o seven_o or_o 10_o problem_n of_o superficial-measure_n then_o if_o you_o multiply_v that_o base_a be_v find_v in_o square_a inch_n by_o the_o length_n in_o inch_n you_o shall_v have_v the_o whole_a content_a in_o cube_n inches_n example_n suppose_v a_o cillender_n have_v 10_o inch_n for_o its_o diameter_n then_o by_o the_o seven_o or_o 10_o abovesaid_a you_o shall_v find_v the_o base_a to_o be_v 78-54_a then_o if_o you_o multiply_v 78-54_a by_o 80_o the_o suppose_a length_n in_o inch_n you_o shall_v find_v 2356-20_a cube_n inches_n which_o divide_v by_o 1728_o the_o inch_n in_o a_o cube_n foot_n show_v how_o many_o foot_n there_o be_v etc._n etc._n and_o as_o to_o the_o number_n of_o figure_n and_o the_o fraction_n cut_v off_o you_o have_v ample_a direction_n in_o the_o first_o problem_n and_o the_o three_o section_n of_o the_o sixâ—_o chapter_n problem_n vii_o how_o to_o measure_v a_o pyramid_n or_o taper_n timber_n or_o the_o section_n of_o a_o cone_n 1._o first_o get_v the_o perpendicule_a length_n of_o the_o pyramid_n or_o cone_n thus_o multiply_v half_o the_o diameter_n of_o the_o base_a ab_fw-la by_o itself_o then_o measure_v the_o side_n ad_fw-la and_o multiply_v that_o by_o itself_o then_o take_v the_o lesser_a square_n out_o of_o the_o great_a and_o the_o square_a root_n of_o the_o residue_n be_v the_o perpendicule_a altitude_n require_v viz._n db._n example_n suppose_v the_o half_a diameter_n of_o the_o base_a ac_fw-la be_v 10-25_a and_o the_o side_n dam_fw-ge 100_o ab_fw-la 10-25_a and_o 10-25_a multiply_v together_o call_v square_n make_v 105_o 0625_o dc_o 100_o multiply_v by_o 100_o call_v square_n make_v 10000_o then_o the_o lesser_a square_n 105_o 0625_o take_v out_o of_o 10000_o the_o great_a square_n the_o remainder_n be_v 9894_o 9375_o who_o square_a root_n find_v by_o the_o 8_o the_o problem_n of_o the_o six_o chapter_n be_v 99-475_a the_o true_a length_n of_o the_o line_n db_n the_o length_n or_o height_n of_o the_o cone_n then_o if_o you_o multiply_v the_o area_n or_o content_a of_o the_o base_a ac_fw-la 20-5_a which_o by_o the_o seven_o or_o 10_o of_o superficial_a measure_n be_v find_v to_o be_v 160-08_a by_o 33-158_a a_o three_o part_n of_o 99-475_a the_o whole_a height_n make_v 5308_o cut_v off_o the_o fraction_n for_o the_o true_a content_a of_o the_o cone_n who_o length_n be_v 99_o inch_n and_o near_o a_o half_a and_o who_o base_a be_v 20_o inch_n and_o a_o half_a diameter_n 2._o then_o second_o for_o the_o segment_n or_o section_n of_o a_o cone_n the_o shape_n or_o form_n of_o all_o round_a taper_n timber_n the_o true_a way_n be_v thus_o by_o the_o length_n and_o difference_n of_o diameter_n find_v the_o whole_a length_n of_o the_o cone_n which_o for_o all_o manner_n of_o timber_n as_o it_o grow_v this_o way_n be_v near_o enough_o as_o thus_o as_o the_o difference_n of_o the_o diameter_n at_o the_o the_o two_o end_n be_v to_o the_o length_n between_o the_o two_o end_n so_o be_v the_o diameter_n at_o the_o base_a to_o the_o whole_a length_n of_o the_o cone_n example_n the_o difference_n between_o the_o diameter_n ac_fw-la and_o of_o be_v 13-70_a the_o length_n ae_n be_v 66-32_a than_o the_o extent_n on_o the_o line_n of_o number_n from_o 13-70_a the_o difference_n of_o the_o diameter_n to_o 66-32_a the_o length_n between_o shall_v reach_v the_o same_o way_n from_o 20-50_a the_o great_a diameter_n to_o 99_o and_o better_o the_o length_n that_o make_v up_o the_o cone_n at_o that_o angle_n oâ—_n taper_v in_o the_o timber_n then_o if_o by_o the_o last_o rule_n you_o measure_v it_o as_o a_o cone_n of_o that_o length_n and_o also_o measure_v the_o little_a end_n or_o point_n at_o his_o length_n and_o diameter_n and_o then_o last_o this_o little_a cone_n take_v out_o of_o the_o great_a cone_n there_o remain_v the_o true_a content_a of_o the_o taper-piece_n that_o be_v to_o be_v measure_v viz._n 5246-71_a when_o 61-30_a the_o content_a of_o the_o small_a cone_n at_o the_o end_n be_v take_v out_o of_o 5308_o the_o content_a of_o the_o whole_a pyramid_n 3._o if_o this_o way_n seem_v too_o troublesome_a for_o the_o common_a use_n then_o use_v this_o be_v more_o brief_a to_o the_o content_a that_o be_v find_v out_o by_o the_o diameter_n in_o the_o midst_n of_o the_o timber_n and_o the_o length_n add_v the_o content_a of_o a_o piece_n find_v out_o by_o half_a the_o difference_n of_o diameter_n and_o one_o three_o part_n of_o the_o length_n of_o the_o whole_a piece_n and_o the_o sum_n of_o they_o two_o shall_v be_v the_o whole_a content_a require_v 4._o or_o else_o divide_v the_o length_n of_o the_o tree_n into_o 4_o or_o 5_o part_n and_o measure_v the_o middle_n of_o each_o part_n several_o and_o that_o cast_v up_o by_o his_o proper_a length_n shall_v give_v the_o content_a of_o each_o piece_n then_o the_o sum_n of_o the_o content_n of_o all_o the_o piece_n put_v together_o be_v the_o true_a content_a of_o the_o whole_a taper_n piece_n very_o near_o note_v that_o this_o curiosity_n shall_v never_o need_v to_o be_v
the_o near_a distance_n than_o the_o nd_o from_o the_o right_n sine_fw-la of_o the_o angle_n to_o the_o thread_n shall_v be_v the_o tangent_fw-la require_v example_n i_o make_v ab_fw-la radius_fw-la a_o =_o in_o 50_o the_o cousin_a of_o 40_o than_o the_o =_o sine_fw-la of_o 40_o shall_v be_v be._n again_o if_o i_o make_v ad_fw-la equal_a to_o ab_fw-la the_o =_o cousin_a of_o 30_o viz._n 60_o and_o then_o take_v out_o the_o =_o sine_fw-la of_o 30_o and_o lay_v it_o from_o d_o to_o f_o it_o shall_v be_v a_o angle_n of_o 60_o from_o ab_fw-la or_o 30_o from_o d_o to_o f._n but_o by_o the_o sector_n this_o be_v more_o easy_a use_v x._o to_o take_v out_o ready_o any_o tangent_fw-la above_o 45_o by_o the_o tangent_fw-la to_o 45_o on_o the_o sector-side_n sector_n take_v the_o give_v radius_fw-la make_v it_o a_o =_o in_o the_o co-tangent_a of_o the_o tangent_fw-la require_v then_o the_o =_o tangent_fw-la of_o 45_o shall_v be_v the_o tangent_fw-la require_v example_n i_o will_v have_v a_o tangent_fw-la to_o 80_o degree_n take_v the_o give_v radius_fw-la make_v it_o a_o =_o in_o 10_o the_o compliment_n of_o 80_o then_o the_o =_o tangent_fw-la of_o 45_o shall_v be_v the_o tangent_fw-la of_o 80_o require_v but_o if_o your_o radius_fw-la be_v so_o big_a that_o you_o can_v enter_v it_o then_o take_v the_o half_a or_o a_o quarter_n of_o your_o radius_fw-la and_o then_o =_o 45_o will_v be_v the_o half_a or_o the_o quarter_n of_o the_o tangent_fw-la require_v use_v xi_o how_o to_o work_v proportion_n in_o sines_n alone_o by_o the_o natural_a sin_n there_o be_v 4_o variety_n in_o this_o work_n that_o include_v all_o proportion_n viz._n 1._o when_o the_o sine_fw-la of_o 90_o be_v the_o first_o term_n than_o the_o work_n be_v thus_o lay_v the_o thread_n to_o the_o second_o term_n count_v on_o the_o degree_n from_o the_o head_n quadr_n towards_o the_o loose-piece_n and_o count_v the_o three_o term_n on_o the_o line_n of_o sines_n from_o the_o centre_n downward_o and_o take_v the_o near_a distance_n from_o thence_o to_o the_o thread_n and_o that_o distance_n measure_v from_o the_o centre_n downward_o on_o the_o line_n of_o sines_n give_v the_o four_o term_v require_v example_n as_o sine_fw-la 90_o to_o sine_fw-la 23-30_a so_o be_v 30_o to_o 11-31_a take_v the_o latteral_a second_o term_n sector_n make_v it_o a_o =_o sine_fw-la of_o 90_o then_o take_v out_o the_o =_o three_o term_n and_o measure_v it_o from_o the_o centre_n it_o give_v the_o four_o term_v require_v 2._o when_o the_o sine_fw-la of_o 90_o be_v the_o three_o term_n then_o work_v thus_o take_v the_o sine_fw-la of_o the_o second_o term_n from_o the_o centre_n downward_o quadr._n and_o make_v it_o a_o =_o sine_fw-la in_o the_o first_o term_n lay_v the_o thread_n to_o nd_o then_o on_o the_o degree_n the_o thread_n shall_v give_v the_o four_o term_v require_v example_n as_o the_o sine_fw-la of_o 30_o to_o 23-31_a so_o be_v the_o sine_fw-la of_o 90_o to_o sine_fw-la of_o 52-56_a but_o by_o the_o sector_n sector_n take_v the_o sine_fw-la of_o the_o second_o make_v it_o a_o =_o sine_fw-la of_o the_o first_o term_n then_o take_v out_o =_o 90_o and_o measure_v it_o from_o the_o centre_n and_o it_o shall_v give_v the_o four_o term_v require_v example_n as_o before_o 3._o when_o the_o radius_fw-la or_o sine_fw-la of_o 90_o be_v in_o the_o second_o place_n work_v thus_o quadr._n take_v 90_o from_o a_o lesser_a scale_n as_o the_o uppermost_a sine_fw-la above_o the_o centre_n or_o the_o line_n of_o right-ascentions_a or_o the_o azimuth-scale_n or_o the_o like_a and_o make_v it_o a_o =_o in_o the_o sine_fw-la of_o the_o first_o lay_v the_o thread_n to_o nd_o than_o the_o =_o three_o term_n take_v and_o measure_v on_o the_o same_o scale_n that_o 90_o be_v take_v from_o shall_v give_v the_o four_o term_v require_v example_n as_o 90_o on_o the_o line_n of_o rightascention_n be_v to_o =_o 30_o so_o be_v =_o 20_o to_o 43-12_a measure_v on_o the_o same_o line_n that_o 90_o be_v take_v from_o or_o else_o second_o work_v thus_o as_o 30_o to_o =_o 90_o so_o be_v 20_o to_o =_o 43-12_a by_o carry_v the_o compass_n till_o it_o so_o stay_v as_o that_o the_o foot_n turn_v about_o will_v but_o just_a touch_n the_o thread_n at_o the_o near_a distance_n or_o else_o thus_o three_o by_o transpose_v the_o term_n when_o the_o three_o be_v not_o great_a than_o the_o first_o thus_o as_o the_o first_o to_o the_o three_o so_o be_v the_o second_o term_n to_o the_o four_o where_o the_o radius_fw-la be_v bring_v to_o the_o three_o place_n it_o be_v wrought_v by_o the_o second_o rule_n as_o before_o by_o the_o sector_n take_v a_o small_a sine_fw-la of_o 90_o make_v it_o a_o =_o in_o 30_o then_o the_o =_o sine_fw-la of_o 20_o take_v and_o measure_v on_o the_o small_a sine_fw-la give_v 43-12_a as_o before_o again_o as_o 90_o to_o =_o 30_o so_o be_v =_o 20_o to_o 43-12_a again_o as_o 30_o to_o =_o 90_o so_o be_v =_o 20_o to_o =_o 43-12_a last_o by_o transpose_v as_o 20_o to_o =_o 30_o so_o be_v =_o radius_fw-la to_o 43-12_a as_o before_o 4._o when_o radius_fw-la be_v none_o of_o the_o give_v term_n then_o when_o the_o first_o term_n be_v great_a than_o the_o second_o and_o three_o quadr._n work_v thus_o take_v the_o second_v term_n make_v it_o a_o =_o in_o the_o first_o lay_v the_o thread_n to_o the_o nd_o then_o the_o near_a distance_n from_o the_o three_o term_n to_o the_o thread_n measure_v from_o the_o centre_n downward_o give_v the_o four_o sine_fw-la require_v example_n as_o 20_o to_o 12_o so_o be_v 18_o to_o 10-50_a by_o the_o quadrant_n as_o 12_o to_z =_o 20_o so_o be_v =_o 18_o to_o 10-50_a when_o only_o the_o second_o term_n be_v great_a than_o the_o first_o then_o transpose_v the_o term_n and_o work_v as_o before_o or_o else_o use_v a_o double_a radius_fw-la which_o be_v on_o this_o instrument_n very_o easy_o do_v have_v several_a radiuss_n or_o last_o use_v a_o parallel_n entrance_n or_o answer_v rather_o as_o before_o which_o be_v careful_o wrought_v will_v do_v very_o well_o by_o the_o sector_n the_o same_o manner_n of_o work_n be_v as_o before_o by_o the_o quadrant_n and_o the_o set_v the_o sector_n be_v all_o one_o to_o the_o lay_v the_o thread_n as_o will_v be_v large_o see_v in_o all_o the_o follow_a proposition_n wrought_v both_o by_o the_o artificial_a and_o natural_a line_n of_o number_n sin_n and_o tangent_n as_o follow_v use_v xii_o have_v the_o day_n of_o the_o month_n or_o sun_n place_n give_v to_o find_v his_o declination_n lay_v the_o thread_n on_o the_o day_n of_o the_o month_n in_o the_o calendar_n particular_o and_o in_o the_o line_n of_o degree_n on_o the_o moving-leg_n you_o have_v his_o declination_n either_o northward_o or_o southward_o according_a to_o the_o time_n of_o the_o year_n count_v from_o 600_o towards_o the_o head_n for_o north-declination_n or_o towards_o the_o end_n for_o south-declination_n by_o the_o artificial_a sin_n and_o tangent_n on_o the_o edge_n of_o the_o instrument_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o to_o the_o sine_fw-la of_o 23_o degree_n 31_o minute_n the_o sun_n great_a declination_n the_o same_o extent_n apply_v the_o same_o way_n from_o the_o sine_fw-la of_o the_o sun_n place_n or_o the_o sun_n distance_n from_o the_o next_o equinoctial-point_n shall_v cause_v the_o moving-point_n to_o fall_v on_z the_o sine_z of_o the_o sun_n declination_n this_o be_v the_o general_a way_n of_o work_v example_n the_o extent_n from_o the_o sine_fw-la of_o 90_o to_o the_o sine_fw-la of_o 23-31_a shall_v reach_v from_o the_o sine_fw-la of_o 30_o to_o 11_o deg_n 31_o min._n the_o sun_n declination_n in_o ♉_o taurus_n 30_o degree_n from_o ♈_o aries_n the_o next_o equinoctial-point_n and_o from_o 60_o degree_n the_o sun_n distance_n in_o ♊_o gemini_fw-la 60_o degree_n from_o ♈_o 20_o deg_n 12_o min._n the_o sun_n declination_n then_o this_o be_v the_o manner_n of_o work_v by_o these_o line_n by_o extend_v the_o compass_n from_o the_o first_o to_o the_o second_o term_n i_o shall_v for_o the_o rest_n wave_v this_o large_a repetition_n of_o extend_v the_o compass_n and_o render_v it_o only_o thus_o by_o the_o word_n of_o the_o cannon-general_a in_o all_o book_n as_o sine_fw-la 90_o to_o sine_fw-la 23-31_a so_o be_v the_o sine_fw-la of_o 30_o to_o sine_fw-la 11-31_a lay_v the_o thread_n to_o 23-31_a generally_n on_o the_o degree_n on_o the_o moveable-piece_n count_v from_o the_o head_n towards_o the_o end_n then_o count_v the_o sun_n place_n from_o the_o next_o equinoctial-point_n on_o the_o line_n of_o sines_n from_o the_o centre_n downward_o and_o take_v the_o nd_o from_o thence_o to_o the_o thread_n then_o this_o distance_n be_v measure_v from_o the_o centre_n downward_o
shall_v be_v the_o sine_fw-la of_o the_o sun_n declination_n require_v for_o that_o distance_n from_o the_o next_o equinoctial-point_n by_o the_o 1_o sy_n rule_n abovesaid_a by_o the_o sector_n take_v 23-31_a from_o the_o sin_n make_v it_o a_o =_o in_o the_o sine_fw-la of_o 90_o then_o the_o =_o sine_fw-la of_o the_o sun_n distance_n from_o the_o next_o equinoctial-point_n shall_v be_v the_o sine_fw-la of_o the_o sun_n declination_n example_n as_o before_o rule_n the_o 1_o st_z use_v xiii_o the_o sun_n declination_n be_v give_v to_o find_v his_o true_a place_n or_o distance_n from_o ♈_o or_o ♎_o the_o two_o equinoctial_a point_n lay_v the_o thread_n to_o the_o declination_n count_v in_o the_o degree_n from_o 600_o particular_o and_o in_o the_o line_n of_o the_o sun_n place_n be_v his_o true_a place_n require_v example_n when_o the_o sun_n declination_n be_v 12_o degree_n northward_o the_o day_n increase_v than_o the_o sun_n will_v be_v 31_o deg_n and_o 23_o min._n from_o ♈_o or_o 1_o deg_n 23_o min._n in_o ♉_o his_o true_a place_n require_v as_o sine_fw-la of_o 23-31_a the_o sun_n great_a declination_n t._n to_o sine_fw-la of_o 90_o so_o sine_fw-la of_o 12-00_a the_o sun_n present_a declination_n to_o sine_fw-la of_o sun_n distance_n from_o ♈_o or_o ♎_o 31-23_a which_o by_o consider_v the_o time_n of_o the_o year_n give_v his_o true_a place_n by_o look_v on_o the_o month_n and_o line_n of_o sun_n place_n on_o the_o quadrantal-side_n take_v the_o sine_fw-la of_o the_o present_a declination_n generally_n make_v it_o a_o =_o sine_fw-la in_o the_o great_a declination_n lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n the_o thread_n shall_v give_v the_o sun_n distance_n from_o ♈_o or_o ♎_o require_v example_n as_o before_o make_v sine_fw-la of_o the_o give_v sun_n declination_n sector_n a_o =_o sine_fw-la in_o the_o sun_n great_a declination_n then_o =_o sine_fw-la of_o 90_o measure_v from_o the_o centre_n be_v the_o =_o sine_fw-la of_o the_o sun_n distance_n from_o ♈_o or_o ♎_o require_v or_o count_v 30_o deg_n for_o one_o sign_n and_o the_o centre_n for_o the_o next_o equinoctial-point_n and_o 90_o for_o the_o two_o tropic_n of_o cancer_n and_o capricorn_n ♋_o ♑_o use_v fourteen_o the_o sun_n place_n or_o day_n of_o the_o month_n and_o great_a declination_n give_v to_o find_v his_o right_a ascension_n from_o the_o same_o equinoctial_a lay_v the_o thread_n to_o the_o day_n of_o the_o month_n quadr._n or_o place_n give_v and_o in_o the_o line_n of_o the_o sun_n right_a ascension_n you_o have_v his_o right_a ascension_n in_o degree_n or_o hour_n and_o minute_n count_v 4_o minute_n for_o every_o degree_n example_n on_o the_o 9th_o of_o april_n near_o night_n the_o sun_n be_v then_o enter_v ♉_o the_o sun_n right_a ascension_n will_v be_v 1_o hour_n 52_o min._n or_o 28_o degree_n of_o right_a ascension_n distant_a from_o ♈_o as_o the_o sine_fw-la of_o 90_o t._n to_o the_o sine_fw-la compliment_n of_o the_o sun_n great_a declination_n or_o c.s._n of_o 23-31_a count_v backward_o from_o 90_o which_o will_v be_v at_o the_o sine_fw-la of_o 66-29′_a so_o be_v the_o tangent_fw-la of_o the_o sun_n distance_n from_o the_o next_o equinoctial-point_n to_o the_o tangent_fw-la of_o the_o sun_n right_a ascension_n from_o the_o same_o equinoctial-point_n generally_n take_v the_o cousin_a of_o the_o great_a declination_n from_o the_o centre_n downward_o be_v the_o sine_fw-la of_o 66-29′_a make_v it_o a_o =_o sine_fw-la of_o 90_o lay_v the_o thread_n to_o nd_o and_o note_v what_o degree_n and_o minuit_fw-la it_o cut_v for_o this_o be_v fix_v to_o this_o proportion_n then_o take_v the_o tangent_fw-la of_o the_o sun_n distance_n from_o the_o next_o equinoctial-point_n from_o the_o centre_n at_o 600_o on_o the_o degree_n towards_o the_o end_n and_o lay_v it_o on_o the_o sin_n from_o the_o centre_n downward_o and_o note_v the_o point_n where_o it_o stay_v for_o the_o nd_o from_o thence_o to_o the_o thread_n shall_v be_v the_o tangent_fw-la of_o the_o sun_n right_a ascension_n require_v note_v that_o if_o the_o sun_n distance_n from_o ♈_o or_o ♎_o be_v above_o 45_o degree_n than_o the_o tangent_n on_o the_o loose-piece_n be_v to_o be_v use_v instead_o of_o the_o tangent_n on_o the_o moveable-leg_n or_o by_o sines_n only_o thus_o or_o take_v sine_fw-la of_o the_o present_a sun_n declination_n make_v it_o a_o =_o in_o the_o sine_fw-la of_o the_o sun_n great_a declination_n and_o lay_v the_o thread_n to_o nd_o then_o take_v =_o cousin_a of_o the_o sun_n great_a declination_n and_o make_v it_o a_o =_o in_o cousin_a of_o the_o sun_n present_a declination_n and_o lay_v the_o thread_n to_o nd_o and_o in_o the_o degree_n it_o cut_v the_o sun_n right_a ascension_n require_v make_v cousin_a of_o 23-31_a sector_n viz._n the_o right_a sine_fw-la of_o 66-29_a a_o =_o sine_fw-la of_o 90_o than_o the_o =_o tangent_fw-la of_o the_o sun_n distance_n from_o ♈_o or_o ♎_o be_v the_o =_o tangent_fw-la of_o the_o sun_n right_a ascension_n from_o the_o same_o point_n of_o ♈_o or_o ♎_o as_o at_o 30_o from_o ♈_o it_o be_v 28_o degree_n or_o 1_o hour_n and_o 52_o minute_n from_o ♈_o near_o use_v xv._n have_v the_o sun_n right_a ascension_n and_o great_a declination_n to_o find_v the_o angle_n of_o the_o ecliptic_a and_o meridian_n as_o sine_fw-la 90_o to_o sine_fw-la 23-31_a sine_fw-la so_o be_v the_o cousin_a of_o the_o sun_n right_a ascension_n to_o the_o cousin_a of_o the_o angle_n of_o the_o ecliptic_a and_o meridian_n lay_v the_o thread_n to_o 23-31_a quadr._n count_v on_o the_o degree_n from_o the_o head_n then_o count_v the_o cousin_a of_o the_o right_a ascension_n from_o the_o centre_n downward_o or_o the_o sine_fw-la from_o 90_o upward_o and_o take_v the_o nd_o from_o thence_o to_o the_o thread_n and_o measure_v it_o from_o the_o centre_n and_o it_o shall_v reach_v to_o the_o cousin_a of_o the_o angle_n require_v example_n the_o right_a ascention_n be_v 30_o degree_n or_o 2_o hour_n the_o angle_n shall_v be_v 69-50_a sector·_n make_v the_o right_a sine_fw-la of_o 23-31_a a_o =_o sine_fw-la of_o 90_o then_o the_o =_o cousin_a of_o 30_o viz._n =_o 60_o shall_v make_v the_o sine_fw-la of_o 69-50_a the_o angle_n of_o the_o ecliptic_a and_o meridian_n use_v xvi_o have_v the_o latitude_n and_o declination_n of_o the_o sun_n or_o star_n to_o find_v the_o sun_n or_o star_n amplitude_n at_o rise_v or_o set_v quadr._n take_v the_o sun_n declination_n from_o the_o particular_a scale_n of_o sines_n and_o lay_v it_o from_o 6_o in_o the_o hour_n or_o azimuth-line_n and_o it_o shall_v give_v the_o amplitude_n from_o south_n as_o it_o be_v figure_v or_o from_o east_n or_o west_n count_v from_o 90_o observe_v to_o turn_v the_o compass_n the_o same_o way_n from_o 90_o or_o 6_o as_o the_o declination_n be_v northward_o or_o southwards_o example_n the_o sun_n declination_n be_v 10_o degree_n northward_o the_o sun_n amplitude_n or_o line_n be_v 106-12_a from_o the_o south_n or_o 16-12_a from_o the_o east-point_n sines_n as_o cousin_a of_o the_o latitude_n to_o s._n 90_o so_o be_v s._n of_o the_o sun_n declination_n to_o s._n of_o the_o amplitude_n general_o take_v the_o sine_fw-la of_o the_o sun_n declination_n make_v it_o a_o =_o in_o the_o cousin_a of_o the_o latitude_n and_o lay_v the_o thread_n to_o the_o near_a distance_n sector_n and_o on_o the_o degree_n the_o thread_n shall_v show_v the_o true_a amplitude_n require_v make_v the_o right_a sine_fw-la of_o the_o sun_n declination_n a_o =_o in_o cousin_a latitude_n then_o =_o 90_o take_v and_o measure_v from_o the_o centre_n give_v the_o amplitude_n or_o line_n use_v xvii_o have_v the_o same_o amplitude_n and_o declination_n to_o find_v the_o latitude_n as_o s._n of_o the_o sun_n amplitude_n sine_fw-la to_o s._n the_o sun_n declination_n so_o be_v s_o 90_o to_o cousin_a latitude_n take_v the_o sine_fw-la of_o the_o sun_n declination_n quadr._n set_v one_o point_n in_o the_o sine_fw-la of_o the_o sun_n amplitude_n lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o show_v the_o compliment_n of_o the_o latitude_n require_v example_n the_o declination_n be_v 20_o degree_n and_o the_o amplitude_n 33-15_a the_o compliment_n of_o the_o latitude_n will_v be_v 38-28_a count_v from_o the_o head_n toward_z the_o end_n make_v the_o right_n sine_fw-la of_o the_o sun_n declination_n sector_n a_o =_o sine_fw-la in_o the_o sun_n amplitude_n then_o the_o =_o sine_fw-la of_o 90_o shall_v be_v the_o cousin_a of_o the_o latitude_n require_v use_v xviii_o have_v the_o latitude_n and_o sun_n declination_n to_o find_v his_o altitude_n at_o east_n or_o west_n common_o call_v the_o vertical-circle_n or_o azimuth_n of_o east_n or_o west_n take_v the_o sun_n declination_n from_o the_o particular_a line_n of_o sines_n q._n set_v one_o point_n in_o 90_o on_o the_o
azimuth-line_n and_o lay_v the_o thread_n to_o the_o nd_o and_o on_o the_o degree_n it_o show_v the_o altitude_n require_v count_v from_o 600_o towards_o the_o end_n as_o s._n latitude_n s._n of_o 90_o s._n so_o s._n of_o sun_n declination_n to_o s._n sun_n height_n at_o east_n or_o west_n quad._n take_v the_o sine_fw-la of_o the_o sun_n declination_n make_v it_o a_o =_o in_o the_o sine_fw-la of_o the_o latitude_n and_o lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o shall_v show_v the_o sun_n altitude_n at_o east_n and_o west_n require_v example_n declination_n 10._o latitude_n 51-32_a the_o altitude_n be_v 12_o degree_n and_o 50_o minute_n as_o s._n of_o the_o sun_n declination_n to_o =_o s._n of_o latitude_n sector_n so_o be_v the_o =_o s._n of_o 90_o to_o s._n of_o vertical_a altitude_n use_v xix_o have_v the_o latitude_n and_o sun_n declination_n to_o find_v the_o time_n when_o the_o sun_n will_v be_v due_a east_n or_o west_n have_v get_v the_o altitude_n by_o the_o last_o rule_n take_v it_o from_o the_o particular_a sine_fw-la q._n then_o lay_v the_o thread_n to_o the_o sun_n declination_n count_v on_o the_o degree_n then_o set_v one_o point_n in_o the_o hour-line_n so_o as_o the_o other_o turn_v about_o shall_v but_o just_a touch_n the_o thread_n and_o the_o compass-point_n shall_v stay_v at_o the_o hour_n and_o minuit_fw-la of_o time_n require_v as_o tangent_fw-la latitude_n to_o sine_fw-la 90_o t._n so_o be_v the_o tangent_fw-la of_o the_o sun_n declination_n to_o cousin_a of_o the_o hour_n or_o as_o sine_fw-la 90_o to_o tangent_fw-la sun_n declination_n so_o be_v co-tangent-latitude_n to_o cousin_a of_o the_o hour_n from_o noon_n example_n latitude_n 51-32_a declination_n 10_o the_o sun_n will_v be_v due_a east_n at_o 6-32_a and_o west_n at_o 5-28_a take_v the_o tangent_fw-la of_o the_o latitude_n on_o the_o loose-piece_n quad._n count_v from_o 60_o towards_o the_o moveable-leg_n or_o else_o from_o 600_o on_o the_o moving-leg_n or_o degree_n according_a as_o the_o latitude_n be_v above_o or_o under_o 45_o degree_n and_o lay_v it_o from_o the_o centre_n downward_o and_o note_v the_o point_n where_o it_o end_v then_o take_v from_o the_o same_o tangent_fw-la the_o tangent_fw-la of_o the_o sun_n declination_n and_o set_v one_o foot_n in_o the_o point_n last_o note_v lie_v the_o thread_n to_o nd_o than_o the_o =_o sine_fw-la of_o 90_o shall_v be_v the_o sine_fw-la of_o the_o hour_n from_o 6._o or_o by_o the_o sin_n only_o work_v thus_o take_v the_o sine_fw-la of_o the_o sun_n declination_n make_v it_o a_o =_o in_o sine_fw-la of_o the_o latitude_n lay_v the_o thread_n to_o nd_o then_o take_v nd_o from_o the_o cousin_a latitude_n to_o the_o thread_n then_o set_v one_o foot_n in_o the_o cousin_a of_o the_o sun_n declination_n lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o give_v the_o hour_n from_o noon_n as_o it_o be_v figure_v or_o the_o hour_n from_o 6_o count_v from_o the_o head_n count_v 4_o minute_n for_o every_o degree_n make_v the_o small_a tangent_fw-la of_o the_o latitude_n if_o above_o 45_o sector_n take_v from_o the_o centre_n a_o =_o sine_fw-la of_o 90_o then_o the_o tangent_fw-la of_o the_o sun_n declination_n take_v from_o the_o same_o small_a tangent_fw-la and_o carry_v paralle_o till_o it_o stay_v in_o like_a sin_n shall_v be_v the_o sine_fw-la of_o the_o hour_n from_o 6._o or_o as_o before_o by_o sines_n only_o make_v sine_fw-la declination_n a_o =_o sine_fw-la latitude_n then_o take_v =_o cousin_a latitude_n and_o make_v it_o a_o =_o cousin_a of_o the_o sun_n declination_n then_o take_v =_o 90_o and_o lay_v it_o from_o the_o centre_n it_o give_v the_o sine_fw-la of_o the_o hour_n from_o 6._o use_v xx._n have_v the_o latitude_n and_o sun_n declination_n to_o find_v the_o ascentional_a difference_n or_o the_o sun_n rise_v and_o set_v and_o oblique_a ascension_n lay_v the_o thread_n to_o the_o day_n of_o the_o month_n or_o to_o the_o sun_n declination_n or_o true_a place_n q._n or_o to_o his_o right_a ascension_n for_o the_o thread_n be_v lay_v to_o any_o one_o of_o they_o be_v then_o also_o lay_v to_o all_o the_o rest_n then_o in_o the_o azimuth-line_n it_o cut_v the_o ascentional_a difference_n if_o it_o you_o count_v from_o 90_o or_o the_o sun_n rise_v as_o you_o count_v the_o morning_n hour_n or_o his_o set_v count_v the_o afternoon_n hour_n the_o oblique_a ascension_n be_v find_v out_o for_o the_o six_o northern_a sign_n oblique-ascention_n or_o summer_n half-year_n by_o substract_v the_o sun_n difference_n of_o ascension_n out_o of_o the_o sun_n right_a ascension_n but_o for_o the_o other_o winter-half_a year_n or_o six_o southern_a sign_n it_o be_v find_v by_o add_v the_o sun_n difference_n of_o ascension_n to_o his_o right_a ascension_n this_o sum_n in_o winter_n and_o the_o remainder_n as_o abovesaid_a in_o summer_n shall_v be_v the_o sun_n oblique_a ascension_n require_v tan._n as_o co-tangent_a lat._n to_o sine_fw-la 90_o to_o be_v the_o tangent_fw-la of_o the_o sun_n declination_n to_o the_o sine_fw-la of_o the_o sun_n ascentional_a difference_n quad._n take_v the_o co-tangent_a latitude_n from_o the_o loose_a or_o moveable-piece_n as_o it_o be_v above_o or_o under_o 45_o degree_n make_v it_o a_o =_o in_o sine_fw-la 90_o lay_v the_o thread_n to_o nd_o then_o take_v the_o tangent_fw-la of_o the_o sun_n declination_n from_o the_o same_o tangent_n and_o carry_v it_o =_o till_o it_o stay_v in_o the_o part_n that_o the_o other_o foot_n turn_v about_o will_v but_o just_a touch_n the_o thread_n which_o part_n shall_v be_v the_o sine_fw-la of_o the_o sun_n ascentional_a difference_n require_v or_o â—hus_o by_o sines_n only_o make_v the_o sine_fw-la of_o declination_n a_o =_o cousin_a of_o the_o latitude_n lay_v the_o thread_n to_o nd_o then_o take_v the_o =_o sine_fw-la of_o latitude_n make_v it_o a_o =_o in_o cousin_a of_o the_o declination_n and_o lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o shall_v cut_v the_o sun_n ascentional-difference_n require_v which_o be_v turn_v into_o time_n by_o count_v 4_o minute_n for_o every_o degree_n and_o add_v to_o or_o take_v from_o 6_o gives_z the_o sun_n rise_v in_o summer_n or_o winter_n make_v the_o co-tangent_a latitude_n sector_n a_o =_o sine_fw-la of_o 90_o then_o take_v tangent_fw-la of_o the_o sun_n declination_n and_o carry_v it_o =_o till_o it_o stay_v in_o like_a part_n viz._n the_o sine_fw-la of_o the_o sun_n ascentional_a difference_n require_v example_n otherwise_o as_o sine_fw-la 90_o to_o =_o tangent_fw-la 38-28_a so_o be_v =_o tangent_fw-la of_o 23-31_a the_o sun_n great_a declination_n to_o the_o sine_fw-la of_o the_o sun_n great_a ascentional_a difference_n 33_o deg_n and_o 12_o min._n use_v xxi_o the_o latitude_n and_o sun_n declination_n give_v to_o find_v the_o sun_n meridian_n altitude_n when_o the_o latitude_n and_o declination_n be_v both_o alike_o viz._n both_o north_n or_o both_o south_n then_o subtract_v the_o declination_n out_o of_o the_o latitude_n or_o the_o less_o from_o the_o great_a and_o the_o remainder_n shall_v be_v the_o compliment_n of_o the_o sun_n meridian_n altitude_n but_o when_o they_o be_v unlike_a then_o add_v they_o together_o and_o the_o sum_n shall_v be_v the_o compliment_n of_o the_o meridian_n altitude_n the_o contrary_a work_n serve_v when_o the_o compliment_n of_o the_o latitude_n and_o declination_n be_v give_v to_o find_v the_o meridian_n altitude_n lay_v the_o thread_n to_o the_o declination_n count_v on_o the_o degree_n from_o 600_o the_o right_a way_n towards_o the_o head_n for_o north_n and_o towards_o the_o end_n for_o south_n declination_n then_o take_v the_o near_a distance_n from_o the_o center-prick_a at_o 12_o in_o the_o hour-line_n to_o the_o thread_n this_o distance_n measure_v on_o the_o particular-line_n of_o sines_n shall_v show_v the_o sun_n meridian-altitude_n require_v use_v xxi_o the_o latitude_n and_o hour_n from_o the_o midnight_n meridian_n give_v to_o find_v the_o angle_n of_o the_o sun_n position_n viz._n the_o angle_n between_o the_o hour_n and_o azimuth-line_n in_o the_o centre_n of_o the_o sun_n t._n as_o sine_fw-la 90_o to_o cousin_a of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o hour_n from_o midnight_n to_o the_o sine_fw-la of_o the_o angle_n of_o position_n example_n as_o sine_fw-la 90_o to_o cousin_a latitude_n 38-28_a so_o be_v the_o cousin_a of_o the_o hour_n from_o midnight_n 120_o for_o which_o you_o must_v use_v 60_o to_o 32-34_a the_o angle_n of_o position_n q._n take_v the_o distance_n from_o the_o hour_n to_o the_o 90_o azimuth_n on_o the_o hour-line_n and_o measure_v it_o in_o the_o particular_a sin_n and_o it_o shall_v show_v the_o angle_n of_o position_n require_v this_o hold_v in_o the_o equinoctial_a take_v cousin_a latitude_n quad_n make_v it_o a_o =_o in_o sine_fw-la 90_o then_o take_v
out_o the_o =_o cousin_a of_o the_o hour_n from_o the_o meridian_n and_o it_o shall_v be_v the_o sine_fw-la of_o the_o sun_n position_n make_v cousin_a latitude_n a_o =_o sine_fw-la 90_o sector_n then_o =_o cousin_a of_o the_o hour_n shall_v be_v sine_z of_o the_o sun_n position_n note_v the_o angle_n of_o the_o sun_n position_n may_v be_v vary_v and_o it_o be_v general_o the_o angle_n make_v in_o the_o centre_n of_o the_o sun_n by_o his_o meridian_n or_o hour-circle_n be_v a_o circle_n pass_v through_o the_o pole_n of_o the_o world_n and_o the_o centre_n of_o the_o sun_n and_o any_o other_o principal_a circle_n as_o the_o meridian_n the_o horizon_n or_o any_o azimuth_n the_o anguler-point_n be_v always_o the_o centre_n of_o the_o sun_n use_v xxii_o the_o sun_n declination_n give_v to_o find_v the_o beginning_n and_o end_n of_o twilight_n or_o daybreak_n lay_v the_o thread_n to_o the_o declination_n on_o the_o degree_n but_o count_v the_o contrary_a way_n q._n viz._n south-declination_n towards_o the_o 〈…〉_z north-declination_n towards_o the_o 〈◊〉_d then_o take_v 18_o degree_n from_o the_o particular_a 〈◊〉_d sin_n for_o twilight_n or_o 13_o degree_n for_o daybreak_n or_o clear_a light_n then_o carry_v this_o distance_n of_o 18_o for_o twilight_n or_o 13_o for_o daybreak_n along_o the_o line_n of_o hour_n on_o that_o side_n of_o the_o thread_n next_o the_o end_n till_o the_o other_o foot_n turn_v about_o will_v but_o just_a touch_n the_o thread_n then_o shall_v the_o point_n show_v the_o time_n of_o twilight_n or_o daybreak_n require_v example_n the_o sun_n declination_n be_v 12_o degree_n north_n the_o twilight_n continue_v till_o 9_o hour_n 24_o minute_n or_o it_o begin_v in_o the_o morning_n at_o 38_o minute_n after_o 2_o but_o the_o daybreak_n be_v not_o till_o 22_o minute_n after_o 3_o in_o the_o morning_n or_o 38_o minute_n after_o 8_o at_o night_n and_o last_v no_o long_o to_o work_v this_o for_o any_o other_o place_n where_o the_o latitude_n do_v vary_v do_v thus_o find_v the_o hour_n that_o answer_v to_o 18_o degree_n of_o altitude_n in_o as_o much_o declination_n the_o contrary_a way_n and_o that_o shall_v be_v the_o time_n of_o twilight_n or_o at_o 13_o degree_n for_o daybreak_n according_a to_o the_o rule_n in_o the_o 26_o use_n where_o the_o way_n how_o be_v large_o handle_v to_o the_o 33_o use_n both_o way_n general_o use_v xxiii_o to_o find_v for_o what_o latitude_n your_o instrument_n be_v particular_o make_v for_o take_v the_o near_a distance_n from_o the_o centre_n on_o the_o head-leg_n quadr._n to_o the_o azimuth-line_n on_o the_o moveable-leg_n this_o distance_n measure_v on_o the_o particular_a scale_n of_o sines_n shall_v show_v the_o latitude_n require_v or_o the_o extent_n from_o 0_o to_o 90_o on_o the_o azimuth-line_n shall_v show_v the_o compliment_n of_o the_o latitude_n be_v measure_v as_o before_o use_v xxiv_o have_v the_o meridian_n altitude_n give_v to_o find_v the_o time_n of_o sun_n rising_n or_o set_v true_a place_n or_o declination_n take_v the_o sun_n meridian_n altitude_n from_o the_o particular_a scale_n quadrant_n and_o set_v on_o point_n in_o ☉_o on_o the_o azimuth-line_n lay_v the_o thread_n to_o the_o nd_o and_o on_o the_o hour-line_n it_o show_v the_o time_n of_o rise_v or_o set_v and_o on_o the_o degree_n the_o declination_n and_o the_o rest_n in_o their_o respective_a line_n example_n the_o meridian_n altitude_n be_v 50_o the_o sun_n rise_v at_o 5_o and_o set_v at_o 7._o use_v xxv_o the_o latitude_n and_o declination_n give_v to_o find_v the_o sun_n height_n at_o 6._o lay_v the_o thread_n to_o the_o day_n of_o the_o month_n or_o declination_n quadrant_n then_o take_v the_o nd_o from_o the_o hour-point_n of_o 06_o and_o 6_o to_o the_o thread_n and_o that_o distance_n measure_v on_o the_o particular_a scale_n of_o sines_n shall_v be_v the_o sun_n altitude_n at_o 6_o in_o summer_n time_n or_o his_o depression_n under_o the_o horizon_n in_o the_o winter_n time_n t._n as_o sine_fw-la of_o 90_o to_o sine_fw-la of_o the_o sun_n declination_n so_o be_v sine_fw-la latitude_n to_o sine_fw-la of_o the_o sun_n altitude_n at_o 6._o quad._n count_n the_o sun_n declination_n on_o the_o degree_n from_o 90_o towards_o the_o end_n and_o there_o lie_v the_o thread_n then_o the_o least_o distance_n from_o the_o sine_fw-la of_o the_o latitude_n to_o the_o thread_n measure_v from_o the_o centre_n downward_o shall_v be_v the_o sine_fw-la of_o the_o sun_n altitude_n at_o 6._o make_v the_o sine_fw-la of_o the_o declination_n a_o =_o sine_fw-la of_o 90_o sector_n then_o the_o =_o sine_fw-la of_o the_o latitude_n shall_v be_v the_o sine_fw-la of_o the_o sun_n height_n at_o 6._o example_n latitude_n 51-32_a declination_n 23-31_a the_o height_n at_o 6_o be_v 18_o deg_n 13_o min._n use_v xxvi_o have_v the_o latitude_n the_o sun_n declination_n and_o altitude_n to_o find_v the_o hour_n of_o the_o day_n take_v the_o sun_n altitude_n quadrant_n from_o the_o particular_a scale_n of_o sines_n between_o the_o compass_n then_o lay_v the_o thread_n to_o the_o day_n of_o the_o month_n or_o declination_n then_o carry_v the_o compass_n along_o the_o line_n of_o hour_n between_o the_o thread_n and_o the_o end_n till_o the_o other_o point_n be_v turn_v about_o will_v but_o just_a touch_n the_o thread_n and_o then_o the_o fix_a point_n shall_v show_v the_o true_a hour_n and_o min._n require_v in_o the_o fore_n or_o afternoon_n if_o you_o be_v in_o doubt_n which_o it_o be_v than_o another_o observation_n present_o after_o will_v determine_v it_o example_n may_n 10_o at_o 30_o degree_n of_o altitude_n the_o hour_n will_v be_v 32_o minute_n after_o 7_o in_o the_o morning_n or_o 28_o minute_n after_o 4_o in_o the_o afternoon_n this_o work_n be_v somewhat_o more_o difficult_a than_o the_o former_a i_o shall_v part_v it_o thus_o 1._o first_o to_o find_v the_o hour_n the_o sun_n be_v in_o the_o equinoctial_a take_v the_o sine_fw-la of_o the_o sun_n altitude_n quad._n make_v it_o a_o =_o cousin_a of_o the_o latitude_n lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o shall_v give_v the_o hour_n from_o 12_o as_o it_o be_v figure_v count_v 15_o degree_n for_o a_o hour_n or_o from_o 6_o count_v from_o the_o head_n at_o 90._o example_n latitude_n 51-30_a altitude_n 20_o the_o hour_n be_v 8_o &_o 12′_n in_o the_o forenoon_n or_o 3-48′_a in_o the_o afternoon_n the_o same_o by_o artificial_a sin_n &_o tangent_n as_o cousin_a latitude_n to_o sine_fw-la 90_o so_o be_v the_o sine_fw-la of_o the_o sun_n altitude_n to_o sine_fw-la of_o the_o hour_n from_o 6._o sector_n make_v s._n ☉_o altitude_n a_o =_o s._n in_o ☉_o latitude_n then_o take_v out_o =_o s._n 90_o and_o it_o shall_v be_v the_o sine_fw-la of_o the_o hour_n from_o 6._o 2._o the_o latitude_n declination_n and_o altitude_n give_v to_o find_v the_o hour_n at_o any_o time_n quad_n first_o by_o the_o 25_o use_n find_v the_o sun_n altitude_n or_o depression_n at_o 6_o then_o in_o summertime_n lay_v this_o distance_n from_o the_o centre_n downward_o and_o in_o wintertime_n lay_v it_o upward_o from_o the_o centre_n towards_o the_o end_n of_o the_o head-leg_n and_o note_v that_o point_n for_o that_o day_n or_o degree_n of_o declination_n for_o by_o take_v the_o distance_n from_o thence_o to_o the_o sun_n altitude_n on_o the_o general_n scale_n you_o have_v add_v or_o substract_v the_o altitude_n at_o 6_o to_z or_o from_o the_o present_a altitude_n for_o by_o take_v the_o distance_n from_o that_o note_a point_n over_o or_o under_o the_o centre_n to_o the_o sun_n present_a altitude_n you_o have_v in_o summer_n the_o difference_n between_o the_o sun_n present_a altitude_n and_o his_o altitude_n at_o 6._o and_o in_o winter_n you_o have_v the_o sum_n of_o the_o present_a altitude_n and_o the_o altitude_n at_o 6._o this_o operation_n be_v plain_o hint_v at_o in_o the_o four_o chapter_n and_o 9th_o and_o 10_o section_n which_o be_v understand_v take_v the_o whole_a operation_n in_o short_a term_n thus_o count_v the_o sun_n declination_n from_o 90_o towards_o the_o end_n and_o thereunto_o lie_v the_o thread_n the_o near_a distance_n from_o the_o sine_fw-la of_o the_o latitude_n to_o the_o thread_n be_v the_o sun_n height_n or_o depression_n at_o 6_o in_o winter_n use_v the_o sum_n of_o in_o summer_n the_o difference_n between_o the_o sun_n altitude_n at_o 6_o and_o his_o present_a altitude_n with_o this_o distance_n between_o your_o compass_n set_v one_o point_n in_o the_o cousin_a of_o the_o latitude_n lay_v the_o thread_n to_o nd_o then_o take_v the_o nd_o from_o 90_o to_o the_o thread_n then_o set_v one_o foot_n in_o the_o cousin_a of_o the_o sun_n declination_n and_o lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o give_v the_o hour_n require_v from_o 6_o count_v from_o 90_o
extent_n shall_v reach_v in_o the_o same_o line_n from_o the_o true_a hour_n of_o the_o night_n to_o the_o star_n hour_n from_o the_o meridian_n then_o lay_v the_o thread_n to_o the_o star_n declination_n the_o nd_o from_o the_o star_n hour_n in_o the_o line_n of_o hour_n to_o the_o thread_n measure_v on_o the_o particular_a scale_n of_o altitude_n give_v the_o star_n altitude_n then_o by_o his_o declination_n and_o altitude_n you_o may_v soon_o find_v his_o azimuth_n by_o use_v 27._o and_o if_o the_o instrument_n be_v neat_o fix_v to_o a_o foot_n to_o set_v north_n and_o south_n and_o turn_v to_o any_o azimuth_n and_o altitude_n you_o may_v find_v any_o star_n at_o any_o time_n convenient_a and_o visible_a use_v xlvii_o the_o altitude_n and_o azimuth_n of_o any_o star_n be_v give_v to_o find_v his_o declination_n lay_v the_o thread_n to_o the_o altitude_n on_o the_o degree_n count_v from_o 600_o towards_o the_o end_n then_o set_v one_o point_n on_o the_o star_n azimuth_n count_v in_o the_o azimuth_n line_n and_o take_v the_o nd_o from_o thence_o to_o the_o thread_n which_o distance_n measure_v from_o the_o begin_n of_o the_o particular_a scale_n of_o altitude_n shall_v give_v the_o declination_n if_o the_o compass_n stand_v on_o the_o rightside_n of_o the_o thread_n than_o the_o declination_n be_v north_n if_o on_o the_o left_a it_o be_v south_n according_a as_o you_o work_v for_o the_o sun_n azimuth_n in_o a_o particular_a latitude_n use_v xlviii_o the_o altitude_n and_o declination_n of_o any_o star_n with_o the_o right_a ascension_n of_o the_o sun_n and_o the_o true_a hour_n of_o the_o night_n give_v to_o find_v the_o right_a ascension_n of_o that_o star_n first_o by_o the_o 43d_o use_v find_v the_o star_n hour_n viz._n how_o many_o hour_n and_o minute_n it_o want_v of_o come_v to_o or_o be_v past_o the_o meridian_n then_o the_o extent_n of_o the_o compass_n on_o the_o line_n of_o 24_o hour_n on_o the_o head-leg_n from_o the_o star_n hour_n to_o the_o true_a hour_n shall_v reach_v the_o same_o way_n from_o the_o sun_n right_a ascension_n to_o the_o star_n right_a ascension_n on_o the_o line_n of_o twice_o 12_o or_o 24_o hour_n use_v xlix_o to_o find_v when_o any_o fixed-star_n come_v to_o south_n by_o the_o line_n of_o twice_o 12_o or_o 24_o hour_n in_o use_v 42_o section_n 4_o you_o have_v the_o way_n by_o substraction_n with_o its_o caution_n but_o by_o the_o line_n of_o twice_o 12_o or_o 24_o hour_n work_v thus_o count_v the_o sun_n right_a ascension_n on_o that_o line_n and_o take_v the_o distance_n from_o thence_o to_o the_o next_o 12_o backward_o viz._n that_o at_o ♈_o at_o the_o begin_n of_o the_o line_n when_o the_o sun_n right_a ascension_n be_v under_o 12_o hour_n or_o to_o the_o next_o 12_o in_o the_o middle_n of_o the_o rule_n at_o ♎_o when_o the_o sun_n right_a ascension_n be_v above_o 12_o hour_n which_o be_v nothing_o but_o a_o reject_v 12_o for_o more_o conveniency_n then_o the_o same_o extent_n lay_v the_o same_o way_n from_o the_o star_n right_a ascension_n shall_v reach_v to_o the_o star_n come_v to_o south_n or_o the_o extent_n from_o the_o sun_n to_o the_o star_n right_a ascension_n shall_v reach_v the_o same_o way_n from_o 12_o to_o the_o star_n come_v to_o south_n example_n for_o the_o lyons-heart_n august_n 20._o the_o sun_n right_a ascension_n the_o 20_o of_o august_n be_v 10_o hour_n 36_o minute_n the_o right_a ascension_n of_o the_o lions-heart_n be_v 9_o hour_n and_o 50_o min._n therefore_o the_o extent_n from_o 10_o hour_n 35_o min._n to_o the_o beginning_n shall_v reach_v the_o same_o way_n from_o 9_o hour_n 50_o min._n by_o borrow_v 12_o hour_n because_o the_o sun_n right_a ascension_n be_v more_o than_o the_o star_n to_o 11_o hour_n 13_o min._n of_o the_o next_o day_n viz._n at_o a_o quarter_n past_o 11_o or_o at_o 11_o hour_n and_o 13_o min._n the_o same_o day_n where_o you_o may_v observe_v that_o the_o remainder_n be_v above_o 12_o if_o you_o add_v 24_o hour_n the_o time_n of_o sothe_v be_v between_o midnight_n and_o midday_n next_o follow_v use_v l._n to_o find_v what_o two_o day_n in_o the_o year_n be_v of_o equal_a length_n and_o the_o sun_n rise_v and_o set_v lay_v the_o thread_n on_o any_o one_o day_n in_o the_o upper_a line_n of_o month_n and_o day_n and_o at_o the_o same_o time_n the_o thread_n cut_v in_o the_o lower-line_n of_o month_n the_o day_n that_o be_v answerable_a to_o it_o in_o length_n rise_v set_v and_o declination_n and_o other_o requisite_n example_n the_o one_a of_o april_n and_o the_o 21_o of_o august_n be_v day_n of_o equal_a length_n and_o the_o sun_n rise_v and_o set_v be_v the_o same_o on_o both_o those_o day_n only_o in_o the_o upper-line_n the_o day_n be_v increase_v in_o length_n and_o in_o the_o lower-line_n they_o be_v decrease_v use_v li._n to_o find_v how_o many_o degree_n the_o sun_n be_v under_o the_o horizon_n at_o any_o hour_n the_o declination_n and_o hour_n be_v give_v count_n the_o sun_n declination_n on_o the_o degree_n the_o contrary_a way_n viz._n for_o north_n declination_n count_v from_o 600_o towards_o the_o end_n and_o count_v for_o southern_a declination_n towards_o the_o head_n and_o thereunto_o lie_v the_o thread_n then_o take_v the_o near_a distance_n from_o the_o hour_n give_v to_o the_o thread_n this_o distance_n measure_v in_o the_o particular_a scale_n of_o altitude_n shall_v show_v the_o sun_n depression_n under_o the_o horizon_n at_o that_o hour_n example_n january_n the_o 10_o at_o 8_o at_o night_n how_o many_o degree_n be_v the_o sun_n under_o the_o horizon_n on_o that_o day_n and_o hour_n the_o sun_n declination_n be_v about_o 20_o degree_n south_n then_o if_o i_o lay_v the_o thread_n to_o 20_o degree_n of_o declination_n north_n and_o take_v the_o near_a distance_n from_o 8_o to_o the_o thread_n that_o distance_n i_o say_v measure_v in_o the_o particular_a scale_n give_v 34_o degree_n and_o 9_o min._n for_o the_o sun_n depression_n under_o the_o horizon_n of_o 8_o afternoon_n to_o do_v this_o in_o other_o latitude_n you_o be_v to_o find_v the_o sun_n altitude_n at_o 8_o in_o northern_a declination_n by_o use_v 37._o chap._n xvii_o the_o use_n of_o the_o triangule_a quadrant_n in_o find_v of_o height_n and_o distance_n accessable_a or_o inaccessible_a use_v i._o to_o find_v a_o altitude_n at_o one_o station_n first_o the_o triangule_a quadrant_n be_v rectify_v and_o fix_v to_o a_o ball_n and_o socket_n and_o three-legged-staff_n be_v necessary_a in_o these_o operation_n to_o perform_v they_o exact_o especial_o for_o distance_n look_v up_o to_o the_o object_n as_o you_o will_v to_o a_o star_n and_o observe_v what_o degree_n and_o minute_n the_o thread_n cut_v and_o set_v it_o down_o also_o observe_v the_o place_n where_o you_o stand_v at_o the_o time_n of_o observation_n and_o the_o distance_n from_o your_o eye_n to_o the_o ground_n and_o the_o place_n on_o the_o object_n that_o be_v level_a with_o your_o eye_n also_o as_o the_o play_v of_o the_o thread_n and_o plummet_n will_v plain_o show_v also_o you_o must_v have_v the_o measure_n from_o the_o place_n where_o you_o stand_v observe_v to_o the_o point_n exact_o right_a under_o the_o object_n who_o height_n you_o will_v have_v in_o foot_n yard_n perch_n or_o what_o you_o please_v to_o integer_n and_o fraction_n in_o decimal_n if_o it_o may_v be_v also_o note_v that_o in_o all_o right-angle-triangle_n one_o acute_a angle_n be_v always_o the_o compliment_n of_o the_o other_o so_o that_o observe_v or_o find_v one_o by_o observation_n by_o consequence_n you_o have_v the_o other_o by_o take_v that_o from_o 90._o these_o thing_n be_v premise_v the_o operation_n follow_v by_o the_o artificial_a number_n sin_n and_o tangent_n and_o also_o by_o the_o natural_a note_v also_o by_o the_o way_n that_o in_o regard_n the_o compliment_n of_o the_o angle_n observe_v be_v frequent_o use_v if_o you_o count_v the_o degree_n the_o contrary_a way_n that_o be_v to_o say_v from_o the_o head_n you_o shall_v have_v the_o compliment_n require_v as_o have_v be_v oftentimes_o hint_v before_o then_o as_o the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o measure_a side_n be_v to_o the_o measure_a side_n count_v on_o the_o number_n so_o be_v the_o sine_fw-la of_o the_o angle_n find_v to_o the_o altitude_n or_o height_n require_v on_o number_n example_n at_o one_o station_n stand_v at_o c_o i_o look_v up_o to_o b_o the_o object_n who_o height_n be_v require_v and_o i_o find_v the_o thread_n to_o fall_v on_o 41_o degree_n and_o 45_o minute_n but_o if_o you_o count_v from_o the_o head_n it_o be_v 48-15_a the_o compliment_n thereof_o as_o in_o the_o figure_n you_o see_v also_o the_o measure_n from_o c_o to_o a_o be_v find_v to_o be_v
from_o a_o but_o if_o you_o remove_v to_o e_o the_o thread_n fall_v on_o 3_o on_o the_o shadow_n the_o altitude_n will_v be_v but_o one_o three_o part_n of_o the_o distance_n ea_fw-la from_o hence_o you_o may_v observe_v that_o observe_v at_o c_o and_o at_z d_o where_o the_o thread_n fall_v on_o 1_o and_o on_o 2_o the_o distance_z between_z c_z and_o d_o be_v equal_a to_o the_o altitude_n so_o likewise_o at_o d_o and_o at_z e_z and_o so_o by_o consequence_n at_o 1_o ½_o and_o 2_o ½_o and_o 3_o ½_o or_o any_o other_o equal_a part_n this_o be_v a_o excellent_a easy_a way_n the_o like_a will_v be_v if_o you_o observe_v at_o d_o and_o c_o look_v up_o to_o f_o where_o the_o altitude_n of_o be_v twice_o the_o distance_n ac_fw-la use_v v._o another_o way_n by_o the_o line_n of_o shadow_n at_o one_o station_n measure_v any_o distance_n as_o foot_n yard_n ii_o or_o the_o like_a from_o any_o object_n as_o suppose_v from_o a_o to_o d_o be_v 200_o foot_n and_o look_v up_o to_o b_o the_o thread_n cut_v the_o stroke_n by_o 2_o on_o the_o line_n of_o shadow_n then_o by_o the_o line_n of_o number_n say_v as_o 2_o the_o part_n cut_v be_v to_o 1_o so_o be_v 200_o the_o distance_n measure_v to_o 100_o the_o height_n or_o suppose_v i_o measure_v any_o other_o uneven_a number_n and_o the_o thread_n fall_v between_o 00_o on_o the_o loose-piece_n and_o 1_o on_o the_o shadow_n common_o call_v contrary_a shadow_n the_o rule_n be_v always_o thus_o as_o the_o part_n cut_v by_o the_o thread_n be_v to_o 1_o so_o be_v the_o measure_a distance_n to_o the_o height_n require_v be_v less_o than_o the_o measure_a distance_n but_o when_o the_o thread_n fall_v between_o 1_o and_o 90_o at_o the_o head_n call_v right_a shadow_n then_o the_o rule_n go_v thus_o as_o 1_o to_o the_o part_n cut_v by_o the_o thread_n so_o be_v the_o measure_a distance_n to_o the_o height_n be_v always_o more_o than_o the_o measure_a distance_n from_o the_o foot_n of_o the_o object_n to_o the_o station_n use_v vi_o another_o way_n by_o the_o line_n of_o shadow_n and_o the_o sun_n shine_v when_o the_o sun_n shine_v find_v his_o altitude_n and_o also_o as_o the_o thread_n lie_v see_v what_o division_n on_o the_o line_n of_o shadow_n be_v cut_v by_o the_o thread_n and_o then_o straightway_o measure_v the_o shadow_n length_n on_o the_o ground_n and_o if_o the_o sun_n be_v under_o 45_o degree_n high_a the_o shadow_n be_v long_o than_o the_o length_n of_o that_o object_n which_o cause_v the_o shadow_n but_o if_o the_o sun_n be_v above_o 45_o degree_n high_a than_o the_o object_n be_v long_o than_o the_o shadow_n and_o the_o operation_n be_v thus_o by_o the_o line_n of_o number_n only_o with_o a_o pair_n of_o compass_n the_o height_n of_o the_o sun_n be_v under_o 45_o say_v as_o the_o part_n cut_v by_o the_o thread_n on_o the_o shadow_n be_v to_o 1_o so_o be_v the_o shadow_n measure_v to_o the_o height_n require_v the_o height_n of_o the_o sun_n be_v above_o 45_o say_v as_o 1_o to_o the_o part_n cut_v by_o the_o thread_n on_o the_o line_n of_o shadow_n so_o be_v the_o measure_n of_o the_o shadow_n to_o the_o height_n in_o the_o same_o part_n use_v vii_o to_o find_v a_o inaccessible_a altitude_n by_o the_o quadrat_fw-la and_o shadow_n otherwise_o observe_v the_o altitude_n at_o both_o station_n and_o count_v the_o observe_v altitude_n at_o both_o station_n on_o the_o quadrat_fw-la or_o shadow_n according_a as_o it_o happen_v to_o be_v either_o above_o or_o under_o 45_o degree_n and_o take_v the_o lesser_a out_o of_o the_o great_a note_v the_o remainder_n for_o the_o first_o term_n and_o the_o divisor_n to_o divide_v the_o distance_n between_o the_o station_n increase_v with_o cipher_n if_o need_v be_v and_o the_o quotient_n be_v the_o answer_v require_v but_o by_o the_o line_n of_o number_n work_v thus_o the_o extent_n from_o the_o difference_n to_o 1_o shall_v reach_v the_o same_o way_n from_o the_o measure_a distance_n to_o the_o height_n require_v example_n figure_n ii_o let_v abcde_v represent_v the_o object_n and_o three_o station_n let_v the_o line_n ac_fw-la represent_v the_o altitude_n the_o point_n b_o one_o station_n 50_o foot_n from_o a_o d_o another_o station_n 100_o foot_n from_o a_o or_o 50_o from_o b_o and_o e_z another_o station_n 73_o foot_n from_o d_o or_o 173_o foot_n from_o a_o all_o which_o measure_v you_o need_v not_o know_v before_o but_o only_o bd_o and_o de_fw-fr also_o the_o angle_n at_o b_o 63-27_a and_o his_o compliment_n count_v the_o other_o way_n be_v the_o angle_n at_o c_o 26_o degree_n 33_o minute_n the_o angle_n at_o d_o 45_o and_o his_o compliment_n so_o also_o the_o angle_n at_o e_o 30_o and_o his_o compliment_n 60._o now_o mind_n the_o operation_n by_o either_o of_o these_o first_o lay_v the_o thread_n on_o 26-33_a and_o in_o the_o quadrat_fw-la it_o cut_v 50_o lay_v the_o thread_n on_o 45_o and_o in_o the_o shadow_n or_o quadrat_fw-la it_o cut_v 100_o or_o 1_o or_o if_o you_o lay_v the_o thread_n to_o 60_o then_o in_o the_o shadow_n it_o cut_v 173._o the_o difference_n between_o 173_o and_o 100_o be_v 73._o then_o as_o 73_o the_o difference_n in_o tangent_n between_o the_o two_o observation_n be_v to_o the_o distance_n in_o foot_n 73_o so_o be_v radius_fw-la 100_o or_o the_o side_n of_o the_o quadrat_fw-la to_o 100_o the_o hight_v require_v again_o for_o the_o two_o near_a observation_n who_o difference_n of_o tangent_n be_v 50._o as_o 50_o the_o difference_n in_o tangent_n to_o 50_o foot_n the_o measure_a distance_n so_o be_v 100_o the_o side_n of_o the_o quadrat_fw-la to_o 100_o the_o height_n again_o last_o by_o the_o observation_n at_o b_o &_o e_o the_o difference_n of_o tangent_n be_v 123._o as_o 123_o the_o difference_n in_o tangent_n to_o 123_o the_o measure_a distance_n so_o be_v 100_o the_o radius_fw-la or_o side_n of_o the_o quadrat_fw-la to_o 100_o the_o height_n require_v or_o in_o the_o first_o figure_n the_o angle_n at_o the_o top_n be_v 33-30_a and_o 48-15_a and_o the_o measure_a distance_n 88_o foot_n and_o a_o half_a the_o difference_n in_o tangent_n will_v be_v 45-8_a then_o as_o 45-8_a to_o 100_o the_o side_n of_o the_o quadrat_fw-la so_o be_v 88_o ½_o the_o measure_a distance_n to_o 194_o the_o altitude_n require_v this_o way_n be_v general_a for_o any_o station_n though_o both_o of_o right_a shadow_n or_o both_o of_o contrary_n or_o mix_v of_o right_a and_o contrary_a and_o do_v by_o the_o line_n of_o number_n or_o by_o multiplication_n and_o division_n also_o note_v that_o you_o may_v find_v this_o difference_n in_o tangent_n or_o secant_v by_o the_o natural_a tangent_n or_o natural_a secant_v on_o the_o sector_n and_o the_o scale_n of_o equal_a part_n belong_v to_o they_o thus_o take_v the_o distance_n between_o the_o compleplement_n of_o the_o two_o observation_n on_o the_o great_a or_o lesser_a line_n of_o tangent_n as_o be_v most_o convenient_a and_o measure_v this_o distance_n in_o the_o line_n of_o line_n or_o equal_a part_n equal_a to_o that_o radius_fw-la and_o that_o shall_v be_v the_o difference_n in_o tangent_n require_v the_o like_a for_o the_o secant_v also_o by_o the_o artificial_a number_n sin_n and_o tangent_n you_o may_v come_v by_o this_o difference_n in_o tangent_n or_o secant_v very_o well_o thus_o just_o right_a against_o the_o tangent_fw-la of_o the_o co-altitude_n count_v on_o the_o line_n of_o tangent_n in_o the_o line_n of_o number_n be_v one_o number_n and_o against_o the_o tangent_fw-la of_o the_o compliment_n of_o the_o other_o angle_n be_v the_o other_o number_n only_o with_o this_o caution_n that_o if_o the_o tangent_fw-la be_v above_o 45_o then_o take_v the_o distance_n from_o 45_o to_o the_o tangent_fw-la as_o it_o be_v count_v backward_o with_o compass_n and_o set_v the_o same_o the_o increase_a way_n from_o 1_o on_o the_o number_n to_o the_o other_o number_n require_v then_o the_o lesser_a take_v from_o the_o great_a leave_v the_o difference_n in_o tangent_n that_o be_v require_v in_o the_o same_o manner_n the_o sin_n count_v from_o 90_o and_o lay_v the_o contrary_a way_n from_o 1_o increase_a will_v give_v the_o difference_n in_o secant_v to_o measure_v the_o baâ—â—_n and_o hypothenusa_n by_o number_n only_o use_v viii_o another_o pretty_a way_n by_o scale_n and_o compass_n without_o arithmetic_n from_o t._n s._n then_o draw_v the_o line_n that_o the_o thread_n make_v on_o the_o board_n then_o measure_v from_o your_o stand_n to_o the_o foot_n of_o the_o object_n and_o take_v the_o number_n of_o foot_n or_o yard_n from_o any_o scale_n and_o lay_v it_o from_o the_o right_a angle_n on_o the_o other_o line_n and_o raise_v a_o perpendicule_a from_o thence_o to_o the_o plumb-line_n make_v by_o the_o thread_n and_o