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of_o 60_o degree_n to_o 400_o in_o the_o line_n of_o number_n or_o 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 60_o the_o same_o extent_n will_v reach_v from_o 462_o to_o 400_o which_o be_v the_o length_n of_o the_o base_a basilius_n case_n v._o the_o perpendicular_a and_o angle_v at_o the_o base_a be_v give_v to_o find_v the_o hypothenusall_n if_o the_o perpendicular_a ca_n be_v give_v 231_o and_o the_o angle_n at_o the_o base_a cba_n 30_o degree_n the_o hypothenusall_n bc_n may_v be_v find_v thus_o for_o as_o the_o sine_fw-la of_o the_o angle_n cba_n 30_o degree_n 9,698970_o be_v to_o the_o logarithm_n of_o the_o perpendicular_a ca_n 231_o 12,363612_o so_o be_v the_o sine_fw-la of_o the_o angle_n cab_n 90_o degree_n 10,000000_o to_o the_o logarithm_n of_o the_o hypothenusall_n bc_n 2,664642_o ¶_o here_o because_o the_o angle_n cab_n be_v a_o right_a angle_n or_o 90_o degree_n and_o come_v in_o the_o three_o place_n i_o therefore_o only_o 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the_o angle_n qr_v 45_o degree_n 10_o minute_n and_o the_o side_n q_n 303_o foot_n and_o it_o be_v require_v to_o find_v the_o side_n qr_n ¶_o here_o note_n that_o in_o oblique_a angle_a plain_a triangle_n as_o well_o as_o in_o right_o angle_v the_o side_n be_v in_o proportion_n one_o to_o the_o other_o as_o the_o sin_n of_o the_o angle_n opposite_a to_o those_o side_n therefore_o as_o the_o sine_fw-la of_o the_o angle_n qr_v 45_o deg_fw-la 10_o min._n 9,850745_o be_v to_o the_o logarithm_n of_o the_o side_n q_n 303_o foot_n 2,481443_o so_o be_v the_o sine_fw-la of_o the_o angle_n qsr_n 24_o degree_n 20_o min._n 9,614944_o the_o sum_n of_o the_o second_o and_o three_o term_n 12,096387_o the_o first_o term_v substract_v 9,850745_o to_o the_o logarithme_n of_o the_o side_n qr_n 2,245642_o the_o near_a absolute_a number_n answer_v to_o this_o logarithm_n be_v 176_o and_o so_o many_o foot_n be_v the_o side_n qr_n by_o the_o line_n of_o sines_n and_o number_n the_o line_n of_o sines_n and_o number_n will_v resolve_v these_o triangle_n by_o the_o same_o manner_n of_o work_n as_o in_o the_o other_o before_o for_o if_o you_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 45_o degree_n 10_o min._n to_o 303_o the_o same_o extent_n will_v reach_v from_o the_o sine_fw-la of_o 24_o degree_n 20_o minute_n to_o 176_o and_o so_o much_o be_v the_o side_n qr_n or_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 45_o degree_n 10_o min._n to_o 24_o degree_n 20_o minute_n the_o same_o extent_n will_v reach_v from_o 303_o to_o 176_o the_o length_n of_o the_o inquire_v side_n in_o like_a manner_n if_o the_o angle_n rq_n 110_o degree_n 30_o minute_n and_o the_o angle_n qr_v 45_o degree_n 10_o min._n and_o the_o side_n q_n 303_o foot_n have_v be_v give_v and_o the_o side_n r_v require_v the_o manner_n of_o work_n have_v be_v the_o same_o for_o as_o the_o sine_fw-la of_o the_o angle_n qr_v 45_o degree_n 10_o min._n 9,850745_o be_v to_o the_o logarithm_n of_o the_o side_n q_n 303_o foot_n 2,481443_o so_o be_v the_o sine_fw-la of_o rq_n 110_o deg_fw-la 30_o min._n or_o 69_o de_fw-la 30_o m._n 9,971588_o the_o sum_n of_o the_o second_o and_o three_o term_n 12,453031_o the_o first_o term_v substract_v 9,850745_o to_o the_o logarithm_n of_o the_o side_n r_n 2,602286_o the_o absolute_a number_n answer_v to_o this_o logarithm_n be_v 400_o and_o so_o much_o be_v the_o side_n rs._n ¶_o in_o this_o case_n because_o the_o angle_n rq_n be_v more_o than_o 90_o degree_n you_o must_v therefore_o take_v the_o compliment_n thereof_o to_o 180_o degree_n so_o 110_o degree_n 30_o minute_n be_v take_v from_o 180_o degree_n there_o remain_v 69_o degree_n 30_o min._n who_o sine_fw-la be_v the_o same_o with_o 110_o deg_fw-la 30_o min._n and_o be_v use_v in_o stead_n thereof_o will_v effect_v the_o same_o thing_n by_o the_o line_n of_o sines_n and_o number_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 45_o degree_n 10_o min._n to_o 303_o the_o same_o extent_n will_v reach_v from_o the_o sine_fw-la of_o 69_o deg_fw-la 30_o min._n to_o 400._o which_o be_v the_o side_n r_v require_v or_o the_o compass_n be_v open_v to_o the_o distance_n between_o the_o sine_fw-la of_o 45_o deg_fw-la 10_o min._n and_o 69_o deg_fw-la 30_o min._n the_o same_o distance_n will_v reach_v from_o 303_o to_o 400_o as_o before_o case_n viii_o two_o side_n and_o a_o angle_n opposite_a to_o one_o of_o they_o be_v give_v to_o find_v the_o angle_n opposite_a to_o the_o other_o in_o the_o same_o triangle_n let_v there_o be_v give_v the_o side_n q_n 303_o and_o qr_n 176_o together_o with_o the_o angle_n qsr_n 24_o degree_n 20_o minute_n and_o let_v it_o be_v require_v to_o find_v the_o angle_n qr_n the_o proportion_n be_v as_o the_o logarithm_n of_o the_o side_n qr_n 176_o 2,245513_o be_v to_o the_o sine_fw-la of_o the_o angle_n qsr_n 24_o deg_fw-la 20_o min._n 9,614944_o so_o be_v the_o logarithm_n of_o the_o side_n q_n 303_o 2,481443_o the_o sum_n of_o the_o second_o and_o three_o number_n 12,096387_o the_o first_o number_n substract_v from_o the_o sum_n 2,245513_o to_o the_o sine_fw-la of_o the_o angle_n qr_n 9,850374_o the_o near_a degree_n answer_v to_o this_o sine_fw-la be_v 45_o degree_n 10_o min._n which_o be_v the_o quantity_n of_o the_o angle_n qr_n require_v by_o the_o line_n of_o sines_n and_o number_n extend_v the_o compass_n from_o 176_o to_o the_o sine_fw-la of_o 24_o degree_n 20_o minute_n the_o same_o extent_n will_v reach_v from_o 303_o to_o 45_o deg_fw-la 10_o min._n the_o angle_n qr_n or_o the_o distance_n between_o 176_o and_o 303_o will_v be_v equal_a to_o the_o distance_n between_o 24_o degree_n 20_o minute_n and_o 45_o deg_fw-la 10_o min._n case_n ix_o have_v two_o side_n and_o the_o angle_n contain_v by_o they_o give_v to_o find_v either_o of_o the_o other_o angle_n this_o case_n will_v seldom_o come_v in_o use_n in_o survey_v because_o the_o thing_n require_v be_v a_o angle_n which_o be_v most_o common_o give_v they_o be_v observe_v by_o 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on_o the_o frame_n of_o the_o table_n which_o supply_v the_o use_n thereof_o three_o when_o i_o mention_v or_o make_v use_n of_o the_o circumferentor_n i_o mean_v the_o index_n with_o the_o box_n and_o needle_n screw_v to_o the_o staff_n ¶_o having_n thus_o give_v you_o a_o sufficient_a description_n of_o the_o several_a instrument_n and_o their_o part_n i_o come_v now_o to_o the_o use_n of_o they_o show_v how_o any_o angle_n in_o the_o field_n may_v be_v measure_v by_o any_o of_o they_o and_o 1._o how_o to_o observe_v a_o angle_n in_o the_o field_n by_o the_o plain_a table_n suppose_v eke_o and_o kg_o to_o be_v two_o hedge_n or_o two_o side_n of_o a_o field_n include_v the_o angle_n ekg_v and_o that_o it_o be_v require_v to_o draw_v upon_o your_o table_n a_o angle_n equal_a thereunto_o first_o place_v your_o instrument_n as_o near_o the_o angular_a point_n k_o as_o conveniency_n will_v permit_v turn_v it_o about_o till_o the_o north_n end_n of_o the_o needle_n hang_v direct_o over_o 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trouble_n 2._o how_o to_o find_v the_o quantity_n of_o a_o angle_n in_o the_o field_n by_o the_o theodolite_a let_v it_o be_v require_v to_o find_v the_o quantity_n of_o the_o angle_n ekg_v by_o the_o theodolite_a place_n your_o instrument_n at_o k_o lay_v the_o index_n on_o the_o diameter_n thereof_o then_o turn_v the_o whole_a instrument_n about_o the_o index_n still_o rest_v on_o the_o diameter_n till_o through_o the_o sight_n you_o espy_v the_o mark_n at_o e_o then_o screw_v the_o instrument_n fast_o there_o turn_v the_o index_n about_o upon_o the_o centre_n till_o through_o the_o sight_n you_o espy_v the_o mark_n at_o g_o then_z note_z what_o degree_n on_o the_o frame_n of_o the_o table_n be_v cut_v by_o the_o index_n which_o you_o will_v find_v to_o be_v 114_o degree_n and_o that_o be_v the_o quantity_n of_o the_o angle_n ekg_v 3._o how_o to_o find_v the_o quantity_n of_o any_o angle_n in_o the_o field_n by_o the_o circumferentor_n if_o it_o be_v require_v to_o find_v the_o 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end_n of_o the_o needle_n have_v cut_v 79_o degree_n and_o then_o direct_v the_o sight_n to_o g_o the_o same_o end_n of_o the_o needle_n have_v cut_v 325_o degree_n now_o if_o from_o 325_o you_o subtract_v 79_o the_o remainder_n be_v 246_o but_o because_o this_o remainder_n 246_o be_v great_a than_o 180_o you_o must_v therefore_o subtract_v 246_o the_o remainder_n from_o 360_o and_o there_o will_v remain_v 114_o the_o true_a quantity_n of_o the_o inquire_v angle_n and_o thus_o you_o must_v always_o do_v when_o the_o remainder_n exceed_v 180_o degree_n ¶_o this_o add_v and_o substract_n for_o the_o find_n of_o angle_n may_v seem_v tedious_a to_o some_o but_o here_o the_o reader_n be_v desire_v to_o take_v notice_n that_o for_o quick_a dispatch_n the_o circumferentor_n be_v as_o good_a a_o instrument_n as_o the_o best_a for_o in_o go_v round_o a_o field_n or_o in_o survey_v of_o a_o whole_a manor_n you_o be_v not_o to_o take_v notice_n of_o the_o quantity_n of_o any_o angle_n but_o only_o to_o observe_v what_o degree_n the_o needle_n cut_v which_o in_o those_o case_n be_v sufficient_a as_o will_v appear_v hereafter_o but_o in_o take_v of_o distance_n by_o the_o circumferentor_n it_o be_v altogether_o necessary_a as_o may_v appear_v by_o the_o 7_o chap._n follow_v and_o for_o that_o reason_n i_o have_v here_o show_v how_o to_o find_v a_o angle_n by_o the_o circumferentor_n and_o also_o that_o you_o may_v thereby_o perceive_v what_o congruity_n and_o harmony_n there_o be_v in_o all_o the_o three_o instrument_n 4._o how_o to_o set_v the_o index_n and_o label_n horizontall_a upon_o the_o staff_n when_o you_o have_v screw_v the_o index_n and_o sight_n to_o the_o staff_n as_o a_o circumferentor_n before_o you_o put_v the_o label_n upon_o the_o brass_n pin_n or_o wire_n you_o must_v hang_v a_o line_n and_o plummet_n upon_o that_o pin_n and_o then_o put_v on_o the_o label_n then_o move_v the_o index_n up_o and_o down_o till_o the_o thread_n and_o plummet_n hang_v direct_o upon_o a_o line_n which_o be_v gage_v from_o under_o the_o pin_n all_o along_o the_o sight_n and_o then_o do_v the_o instrument_n stand_v horizontall_a or_o level_n which_o it_o must_v always_o do_v when_o you_o take_v a_o altitude_n therewith_o 5._o how_o to_o observe_v a_o angle_n of_o altitude_n the_o label_n which_o be_v to_o be_v hang_v on_o one_o of_o the_o sight_n of_o the_o circumferentor_n as_o be_v intimate_v in_o the_o description_n thereof_o and_o the_o tangent_fw-la line_n on_o the_o edge_n of_o the_o index_n be_v only_o for_o the_o find_n of_o angle_n of_o altitude_n and_o be_v therefore_o only_o useful_a in_o take_v of_o height_n and_o in_o survey_v of_o mountainous_a and_o uneven_a ground_n the_o manner_n how_o to_o observe_v a_o angle_n of_o altitude_n by_o this_o label_n and_o the_o tangent_fw-la line_n on_o the_o index_n be_v thus_o suppose_v ca_n to_o be_v a_o tree_n tower_n or_o hill_n who_o height_n be_v require_v your_o instrument_n be_v place_v at_o b_o exact_o level_v direct_v the_o sight_n thereof_o towards_o ca_n and_o there_o fix_v it_o hang_v the_o label_n on_o the_o farthermost_a fight_n upon_o a_o pin_n for_o that_o purpose_n then_o move_v the_o label_n too_o and_o fro_o along_o the_o side_n of_o the_o index_n till_o through_o the_o sight_n at_o the_o end_n of_o the_o label_n and_o by_o the_o pin_n on_o which_o the_o label_n hang_v you_o espy_v the_o very_a top_n of_o the_o object_n to_o be_v measure_v at_o c_o then_z note_z what_o degree_n of_o the_o tangent_fw-la line_n be_v cut_v by_o the_o label_n which_o suppose_v 30_o and_o that_o be_v the_o quantity_n of_o the_o angle_n of_o altitude_n it_o be_v equal_a to_o the_o angle_n cba_n thus_o by_o the_o rule_n in_o this_o chapter_n