Selected quad for the lemma: ground_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
ground_n	distance_n	file_n	rank_n	1,730	5	11.1963	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A29761	The description and use of the trianguler quadrant being a particular and general instrument, useful at land or sea, both for observation and operation : more universally useful, portable and convenient, than any other yet discovered, with its uses in arithmetick, geometry, superficial and solid, astronomy, dyalling, three wayes, gaging, navigation, in a method not before used / by John Brown, philomath.	Brown, John, philomath.	1671 (1671)	Wing B5041; ESTC R15524	175,812	507

from_o quitrent_n and_o tax_n and_o the_o like_a the_o rent_n be_v usual_o various_a according_a to_o the_o place_n and_o time_n where_o and_o wherein_o the_o purchase_n shall_v happen_v to_o be_v then_o to_o find_v the_o quantity_n of_o the_o whole_a purchase_n say_v as_o 1_o to_o 20_o 18_o 15_o 14_o 12_o 10_o or_o 8_o the_o number_n of_o year_n purchase_v for_o fee-simple_n or_o copyhold_n land_n or_o house_n fee-simple_n or_o copyhold_n for_o lease_n of_o 60_o 50_o 40_o or_o 30_o year_n or_o 21_o year_n so_o be_v the_o yearly_a rent_n to_o the_o whole_a value_n example_n a_o parcel_n of_o land_n worth_a 10_o l._n per_fw-la annum_fw-la fee-simple_n value_v at_o 20_o year_n purchase_n will_v amount_v to_o 200_o l._n for_o the_o extent_n from_o 1_o to_o 20_o will_v reach_v the_o same_o way_n from_o 10_o to_o 200_o the_o whole_a price_n of_o 20_o year_n purchase_n at_o 10_o l._n per_fw-la annum_fw-la chap._n fourteen_o the_o use_n of_o the_o line_n of_o number_n in_o military_a question_n problem_n i._o any_o number_n of_o soldier_n be_v propound_v to_o order_v they_o into_o a_o square_a battle_n of_o man_n that_o be_v as_o many_o in_o rank_a as_o in_o file_n find_v the_o square-root_n of_o the_o number_n of_o soldier_n and_o that_o shall_v be_v the_o number_n of_o man_n in_o rank_n and_o file_n require_v as_o suppose_v it_o be_v require_v to_o order_v 1770_o man_n in_o the_o order_n abovesaid_a you_o shall_v by_o the_o 8_o probl._n of_o the_o 6_o chapt._n find_v that_o the_o square-root_n of_o 1770_o be_v 42_o and_o 6_o over_o which_o here_o be_v not_o considerable_a problem_n ii_o any_o number_n of_o soldier_n propound_v to_o order_v they_o into_o a_o double_a battle_n of_o man_n that_o be_v to_o say_v twice_o as_o many_o in_o rank_a as_o file_v find_v the_o square-root_n of_o half_a the_o number_n of_o man_n and_o that_o be_v the_o number_n of_o man_n in_o file_n and_o the_o double_a the_o number_n in_o rank_n as_o for_o example_n if_o 2603_o be_v so_o to_o be_v place_v the_o half_a of_o 2603_o be_v 1301_o who_o square-root_n by_o the_o 8_o of_o 6_o be_v 36_o the_o number_n of_o man_n in_o file_n and_o 72_o the_o double_a thereof_o be_v the_o number_n in_o rank_n for_o if_o you_o shall_v multiply_v 72_o by_o 36_o the_o product_n be_v 2592_o almost_o the_o number_n of_o man_n propound_v problem_n iii_o any_o number_n of_o soldier_n be_v propound_v to_o order_v they_o into_o a_o quadruple_a battle_n of_o man_n viz._n 4_o time_n as_o many_o in_o rank_a as_o file_v find_v the_o square-root_n of_o a_o four_o part_n of_o the_o number_n of_o man_n and_o that_o shall_v be_v the_o number_n in_o file_n and_o 4_o time_n so_o many_o the_o number_n in_o rank_n so_o the_o four_o part_n of_o 2603_o be_v 650_o who_o square-root_n be_v 25_o ½_o and_o 4_o time_n 25_o be_v 100_o the_o number_n in_o rank_n problem_n iu._n any_o number_n of_o soldier_n be_v give_v together_o with_o their_o distance_n one_o from_o another_o in_o rank_n and_o file_n to_o order_v they_o into_o a_o square_a battle_n of_o ground_n as_o suppose_v i_o will_v order_v 3000_o man_n so_o that_o be_v 7_o foot_n asunder_o in_o file_n and_o 3_o foot_n apart_o in_o rank_n the_o ground_n whereon_o they_o stand_v shall_v be_v square_a extend_v the_o compass_n from_o 7_o foot_n the_o distance_n in_o file_n to_o 3_o foot_n the_o distance_n in_o rank_n then_o that_o extent_n apply_v the_o same_o way_n from_o 3000_o the_o number_n of_o soldier_n reach_n to_o 1286_o who_o great_a square-root_n be_v 35-7_a that_o be_v 35_o the_o number_n of_o man_n to_o be_v place_v in_o file_n then_o if_o you_o divide_v 3000_o the_o whole_a number_n by_o 35-7_a the_o quotient_n be_v 84_o the_o number_n in_o rank_n to_o use_v and_o employ_v a_o square_a plat_n of_o ground_n to_o stand_v in_o as_o 7_o to_z 3_o so_o be_v 3000_o to_o 1286_o who_o square-root_n be_v 35-7_a then_o as_o ●57_a to_o 1_o so_o be_v 3000_o to_o 84._o problem_n v._o any_o number_n of_o soldier_n propound_v to_o order_v they_o into_o rank_n and_o file_n according_a to_o the_o ratio_fw-la of_o any_o two_o number_n give_v this_o question_n be_v all_o one_o with_o the_o former_a for_o as_o the_o number_n give_v for_o the_o distance_n in_o file_n be_v to_o that_o for_o the_o distance_n in_o rank_n so_o be_v the_o whole_a number_n of_o soldier_n to_o a_o four_o who_o square-root_n be_v the_o number_n of_o man_n in_o rank_n then._n the_o whole_a number_n divide_v by_o the_o number_n in_o rank_n the_o quotient_n be_v the_o number_n to_o be_v place_v in_o file_n example_n suppose_v 3000_o soldier_n be_v to_o be_v order_v in_o rank_n and_o file_n as_o 5_o be_v to_o 10_o or_o as_o 5_o be_v to_z 9_o that_o be_v to_o say_v that_o the_o man_n in_o rank_n may_v be_v in_o proportion_n to_o they_o in_o file_n as_o 9_o be_v to_o 5._o say_v thus_o as_o the_o extent_n from_o 5_o to_z 9_o so_o be_v 3000_o to_o 5400_o who_o square-root_n be_v 73_o ½_o the_o number_n of_o man_n in_o rank_n then_o as_o 73_o ½_o to_o 1_o so_o be_v 3000_o to_o near_o 41_o the_o number_n in_o file_n problem_n vi_o there_o be_v 8100_o man_n to_o be_v order_v into_o a_o square_a body_n of_o man_n and_o to_o have_v so_o many_o pike_n as_o to_o arm_v the_o main_a square-body_n round_o about_o with_o 6_o rank_n of_o pike_n the_o question_n be_v how_o many_o rank_n must_v be_v in_o the_o whole_a square_a battle_n and_o how_o many_o pike_n and_o musket_n first_o the_o square-root_n of_o 8100_o be_v 90_o the_o number_n of_o man_n in_o and_o number_n of_o rank_n and_o file_n now_o in_o regard_n that_o there_o must_v be_v 6_o rank_n of_o pike_n round_o about_o the_o musquetier_n there_o will_v be_v 12_o rank_n less_o of_o they_o both_o in_o front_n and_o flank_n than_o in_o the_o whole_a body_n therefore_o substract_v of_o 12_o from_o 90_o rest_v 78_o who_o square_a be_v 6084_o the_o number_n of_o musquetier_n which_o take_v from_o 8100_o there_o remain_v 2061_o the_o number_n of_o pike_n problem_n vii_o to_o three_o number_n give_v to_o find_v a_o four_o in_o a_o double_a proportion_n for_o as_o much_o as_o like_o squar●●_n be_v in_o double_a the_o proportion_n of_o their_o answerable_a side_n therefore_o you_o must_v work_v by_o their_o square_n and_o square-root_n but_o by_o the_o line_n of_o number_n in_o this_o manner_n if_o a_o fathom_n of_o rope_n of_o 6_o inch_n compass_v about_o weigh_v 6_o pound_n 2_o ounce_n or_o 6-●25_a 1000_o what_o shall_v a_o fathom_n of_o rope_n of_o 12_o inch_n compass_v weigh_v here_o note_v always_o that_o when_o the_o two_o number_n of_o like_a denomination_n which_o be_v give_v be_v of_o line_n or_o side_n of_o square_n or_o diameter_n of_o circle_n then_o the_o extent_n of_o the_o compass_n upon_o the_o line_n of_o number_n from_o one_o line_n to_o the_o other_o or_o from_o one_o side_n to_o the_o other_o side_n that_o extent_n turn_v twice_o the_o same_o way_n from_o the_o give_v area_n or_o content_a shall_v reach_v to_o the_o other_o require_v so_o here_o the_o extent_n of_o the_o compass_n from_o 6_o to_o 12_o be_v turn_v two_o time_n the_o same_o way_n from_o 6-125_a shall_v reach_v to_o 24-50_a for_o 24_o pound_n and_o a_o half_a the_o weight_n require_v but_o if_o the_o two_o term_n give_v of_o one_o denomination_n be_v of_o square_n or_o superficies_n or_o areas_n then_o the_o half_a distance_n on_o the_o line_n of_o number_n between_o one_o area_n and_o the_o other_o be_v turn_v the_o same_o way_n on_o the_o line_n from_o the_o give_v line_n or_o side_n it_o shall_v reach_v to_o the_o side_n or_o line_n require_v for_o the_o half-distance_n between_o 24-50_a and_o 6-125_a shall_v reach_v from_o 12_o to_o 6_o or_o the_o contrary_a from_o 6_o to_o 12._o a_o example_n whereof_o you_o have_v in_o the_o four_o and_o 5_o problem_n of_o the_o 12_o chapter_n also_o in_o the_o 6_o and_o seven_o problem_n of_o the_o 8_o chapter_n which_o treat_v of_o superficial-measure_n in_o measure_v of_o land_n note_v also_o that_o if_o you_o have_v three_o line_n of_o number_n viz._n a_o great_a a_o mean_a and_o a_o less_o after_o mr._n windgates_n way_n then_o these_o question_n be_v wrought_v without_o double_v or_o halve_a and_o very_o neat_o and_o speedy_o as_o thus_o the_o extent_n on_o the_o mean_a line_n from_o 24-50_a to_o 6-125_a the_o weight_n of_o the_o two_o rope_n shall_v reach_v on_o the_o great_a line_n from_o 12_o to_o 6_o or_o from_o 6_o to_o 12_o the_o inch_n in_o compass_n about_o of_o each_o rope_n problem_n viii_o to_o three_o number_n give_v to_o find_v a_o four_o in_o a_o tripled_a proportion_n for_o as_o much_o as_o like_a

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

218_o foot_n then_o as_o the_o sine_fw-la of_o 48-15_a the_o angle_n at_o b_o be_v the_o compliment_n of_o the_o angle_n at_o c_o be_v to_o 218_o on_o the_o line_n of_o number_n so_o be_v the_o sine_fw-la of_o the_o angle_n at_o c_o 41-45_a to_o 195_o the_o altitude_n of_o ab_fw-la the_o height_n require_v find_v on_o the_o line_n of_o number_n a_o second_o example_n stand_v at_o d._n i._n but_o if_o i_o be_v stand_v at_o d_o 129_o foot_n and_o a_o half_a from_o a_o and_o will_v find_v the_o height_n ab_fw-la the_o compliment_n of_o the_o angle_n at_o d_o that_o be_v to_o say_v the_o angle_z at_z b_o be_v 33-30_a this_o be_v prepare_v then_o say_v as_o the_o sine_fw-la of_o 33-30_a the_o angle_n at_o b_o to_o the_o measured-side_n dam_fw-ge 129_o ½_o count_v on_o the_o number_n so_o be_v 56-30_a the_o sine_fw-la of_o the_o angle_n at_o d_o to_o 195_o the_o altitude_n require_v ab_fw-la and_o 5_o foot_n more_o the_o usual_a height_n of_o the_o eye_n from_o the_o level_n to_o the_o ground_n make_v 200_o the_o whole_a height_n require_v to_o work_v this_o by_o the_o triangule_a quadrant_n say_v thus_o as_o 129_o ½_o take_v from_o any_o scale_n be_v to_o the_o =_o sine_fw-la of_o 33_o deg_n 30_o min._n lay_v the_o thread_n to_o the_o near_a distance_n so_o be_v the_o =_o sine_fw-la of_o 56-30_a the_o angle_n at_o d_o to_z the_o measure_n of_o 195_o on_o the_o scale_n you_o take_v 129_o ½_o from_o the_o like_a manner_n of_o work_n be_v by_o the_o sector_n as_o thus_o in_o the_o forego_n example_n as_o 218_o take_v from_o the_o line_n of_o line_n to_o the_o =_o sine_fw-la of_o 48_o deg_n 15_o min._n so_o be_v the_o =_o sine_fw-la of_o 41-45_a to_o 195_o on_o the_o line_n of_o line_n latteral_o and_o yet_o further_o so_o be_v the_o =_o sine_fw-la of_o 90_o to_o 291_o the_o line_n cb._n use_v ii_o to_o find_v a_o altitude_n at_o two_o station_n but_o if_o you_o can_v come_v to_o measure_v to_o the_o foot_n of_o the_o object_n i._n than_o you_o must_v observe_v at_o two_o place_n as_o thus_o for_o example_n first_o as_o before_z find_v the_o angle_n at_o d_o or_o rather_o the_o compliment_n thereof_o viz._n 33-30_a then_o go_v further_o backward_o in_o a_o right_a line_n with_o the_o object_n and_o first_o station_n any_o competent_a number_n of_o foot_n as_o suppose_v 88_o ½_o to_o c_o there_o also_o observe_v the_o altitude_n or_o compliment_n viz._n the_o angle_n abc_n 48-15_a then_o find_v the_o difference_n between_o 48-15_a and_o 33-30_a and_o it_o be_v 14-45_a then_o as_o the_o sine_fw-la of_o the_o difference_n last_o find_v viz._n the_o angle_n cbd_v 14-45_a to_o 88_o ½_o on_o the_o line_n of_o number_n so_o be_v the_o sine_fw-la of_o the_o angle_n at_o c_o 41-45_a to_o the_o measure_n of_o the_o side_n db_n 233_o on_o the_o line_n of_o number_n again_o for_o the_o second_o operation_n as_o the_o sine_fw-la of_o 90_o the_o angle_n at_o a_o to_o the_o hypothenusa_n db_n 233_o so_o be_v the_o sine_fw-la of_o 56-30_a the_o angle_n at_o d_o to_o 195_o the_o altitude_n require_v the_o same_o by_o the_o triangule_a quadrant_n or_o sector_n as_o 88_o ½_o the_o measure_a distance_n cd_o to_o the_o =_o sine_fw-la of_o 14-45_a cbd_v so_o be_v =_o sine_fw-la of_o 41-45_a to_o the_o measure_n of_o 233_o the_o opposite-side_n db._n again_o as_o 233_o take_v from_o the_o line_n of_o line_n to_o =_o sine_fw-la of_o 90_o so_o be_v the_o =_o sine_fw-la of_o 56-30_a the_o angle_n at_o d_o to_z 195_o on_o the_o line_n of_o line_n the_o height_n require_v use_v iii_o another_o way_n to_o save_v one_o operation_n from_o i_fw-mi first_o i._n observe_v the_o compliment_n of_o the_o angle_n at_o d_o and_o also_o the_o compliment_n of_o the_o angle_n at_o c_o then_o count_v these_o two_o compliment_n on_o the_o line_n of_o natural_a tangent_n on_o the_o loose-piece_n or_o moving-leg_n and_o take_v the_o distance_n between_o they_o and_o measure_v it_o on_o the_o same_o tangent-line_n from_o the_o begin_n thereof_o and_o note_n what_o tangent_fw-la the_o compass-point_n stay_v at_o and_o count_v that_o for_o the_o first_o term_n in_o degree_n and_o minute_n then_o as_o the_o tangent_fw-la of_o this_o first_o term_n to_o the_o measure_a distance_n cd_o 88_o ½_o on_o the_o line_n of_o number_n so_o be_v the_o tangent_fw-la of_o 45_o to_o the_o altitude_n require_v thus_o in_o our_o example_n the_o distance_n measure_v be_v 88_o ½_o the_o two_o compliment_n 33-30_a and_o 48-15_a the_o distance_n between_o they_o make_v the_o tangent_fw-la of_o 24-34_a to_o be_v use_v as_o a_o first_o term_n then_o as_o the_o tangent_fw-la of_o ●4-34_a the_o first_o term_n last_o find_v to_o 88_o ½_o on_o the_o number_n so_o be_v the_o tangent_fw-la of_o 45_o to_o 195_o ferè_fw-la on_o the_o number_n the_o height_n require_v but_o if_o the_o distance_n from_o d_o or_o c_o to_z a_o the_o foot_n of_o the_o object_n be_v require_v than_o the_o manner_n of_o calculation_n run_v thus_o as_o the_o tangent_fw-la of_o the_o difference_n of_o the_o co-tangents_a first_o find_v 24-34_a be_v to_o the_o distance_n between_o d_o and_o c_o 88_o ½_o so_o be_v the_o co-tangent_a of_o the_o great_a ark_n 48-15_a to_o the_o great_a distance_n ca_n 218._o or_o so_o be_v the_o co-tangent_a of_o the_o lesser_a ark_n 33-30_a to_o the_o lesser_a distance_n da_fw-la 129_o ½_o but_o if_o the_o hypothenusaes_n be_v require_v than_o reason_n thus_o as_o the_o tangent_fw-la of_o the_o difference_n first_o find_v be_v 24-34_a to_o the_o distance_n between_o the_o station_n d_o and_o c_o 88_o ½_o so_o be_v the_o secant_fw-la of_o the_o angle_n at_o b_o the_o great_a viz._n 48-15_a count_v beyond_o 90_o to_o cb_n 291._o or_o so_o be_v the_o secant_fw-la of_o 33-30_a the_o lesser_a angle_n at_o b_o to_o 233_o the_o lesser_a distance_n db_n the_o hypothenusa_n require_v to_o work_v these_o two_o last_o by_o the_o triangule_a quadrant_n first_o prick_v off_o the_o tangent_n and_o secant_v to_o be_v use_v parallelly_n from_o the_o loose-piece_n on_o the_o great_a general_n scale_n and_o note_v those_o point_n for_o your_o present_a use_n as_o thus_o the_o tangent_fw-la of_o 24-34_a take_v from_o the_o loose-piece_n from_o 60_o count_v as_o 00_o will_v reach_v to_o the_o sine_fw-la of_o 10-40_a on_o the_o general_a scale_n second_o the_o secant_fw-la of_o 33-30_a be_v the_o measure_n from_o the_o tangent_fw-la of_o 33-30_a on_o the_o loose-piece_n count_v from_o 60_o to_o the_o centre_n will_v reach_v on_o the_o general_a scale_n from_o the_o centre_n to_o 28-50_a three_o the_o measure_n from_o the_o tangent_fw-la of_o 48-15_a on_o the_o loose-piece_n to_o the_o centre_n be_v the_o secant_fw-la of_o 48-15_a will_v reach_v from_o the_o centre_n to_o 32-5_a on_o the_o general_a scale_n this_o be_v prepare_v the_o work_n be_v thus_o as_o distance_n between_o the_o two_o station_n to_o =_o tangent_fw-la of_o the_o first_o term_n at_o 10-40_a so_o be_v =_o tangent_fw-la of_o 45_o to_o the_o altitude_n require_v again_o for_o the_o distance_n as_o distance_n between_o the_o two_o station_n to_o the_o =_o tangent_fw-la of_o the_o first_o term_n so_o be_v the_o =_o tangent_fw-la of_o the_o great_a angle_n compliment_n at_o 26-36_a to_o the_o great_a distance_n ca_n 218._o or_o so_o be_v the_o =_o tangent_fw-la of_o the_o lesser_a angle_n compliment_n at_o 15-25_a to_o the_o lesser_a distance_n da_fw-la 129_o ½_o or_o so_o be_v the_o =_o secant_fw-la of_o the_o great_a angle_n compliment_n at_o 32-5●_a to_o the_o great_a hypothenusa_n cb_n 291._o or_o so_o be_v the_o =_o secant_fw-la of_o the_o lesser_a angle_n compliment_n at_o 28-50_a to_o the_o lesser_a hypothenusa_n db_n 233._o use_v iu._n another_o way_n for_o altitude_n by_o the_o line_n of_o shadow_n either_o accessable_a or_o unaccessable_a by_o one_o or_o two_o station_n if_o this_o way_n be_v desire_v it_o may_v be_v put_v on_o this_o as_o well_o as_o any_o other_o quadrant_n then_o the_o use_n be_v thus_o figure_n ii_o suppose_v that_o ab_fw-la be_v the_o height_n of_o a_o tree_n or_o other_o object_n to_o be_v find_v go_v so_o far_o back_o from_o it_o as_o suppose_v to_o c_o till_o look_v up_o by_o the_o two_o pin_n put_v for_o sight_n the_o thread_n fall_v on_o 45_o degree_n on_o the_o quadrant_n or_o on_o 1_o on_o the_o line_n of_o shadow_n then_o i_o say_v that_o the_o height_n ab_fw-la be_v equal_a to_o the_o distance_n ca_n more_o by_o the_o height_n of_o your_o eye_n from_o the_o ground_n but_o if_o you_o go_v further_o back_o still_o to_o d_o till_o the_o thread_n fall_v on_o 2_o on_o the_o line_n of_o shadow_n that_o be_v to_o say_v at_o 26_o deg_n 34_o min._n the_o altitude_n will_v be_v but_o half_a the_o distance_n