Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	day_n	hour_n	minute_n	1,570	5	11.3251	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A52264	Institutio mathematica, or, A mathematical institution shewing the construction and use of the naturall and artificiall sines, tangents, and secants in decimal numbers, and also of the table of logarithms in the general solution of any triangle, whether plain or spherical, with their more particular application in astronomie, dialling, and navigation / by John Newton.	Newton, John, 1622-1678.	1654 (1654)	Wing N1061; ESTC R20441	128,229	417

artificial_a sin_n and_o tangent_n as_o also_o of_o the_o logarithm_n in_o astronomy_n dial_v and_o navigation_n by_o john_n newton_n london_n print_v anno_fw-la domini_fw-la 1654._o a_o mathematical_a institution_n the_o second_o part._n chap._n i._n of_o the_o table_n of_o the_o sun_n motion_n and_o of_o the_o equation_n of_o time_n for_o the_o difference_n of_o meridian_n whereas_o it_o be_v requisite_a that_o the_o reader_n shall_v be_v acquaint_v with_o the_o sphere_n before_o he_o enter_v upon_o the_o practice_n of_o spherical_a trigonometri_n the_o which_o be_v full_o explain_v in_o blundevile_v exercise_n or_o ch●lmades_n translation_n of_o hues_n on_o the_o globe_n to_o who_o i_o refer_v those_o that_o be_v not_o yet_o acquaint_v therewith_o that_o which_o i_o here_o intend_v be_v to_o show_v the_o use_n of_o trigonometrie_n in_o the_o actual_a resolution_n of_o so_o i_o know_v triangle_n of_o the_o sphere_n and_o because_o the_o sun_n place_n or_o distance_n from_o the_o next_o equinoctial_a point_n be_v usual_o one_o of_o the_o three_o term_n give_v in_o astronomical_a question_n i_o will_v first_o show_v how_o to_o compute_v that_o by_o table_n calculate_v in_o decimal_a number_n according_a to_o the_o hypothesis_n of_o bullialdus_n and_o for_o the_o meridian_n of_o london_n who_o longitude_n reckon_v from_o the_o canary_n or_o fortunate_a island_n be_v 21_o deg_n and_o the_o latitude_n north_n 51_o deg_n 57_o part_n min._n or_o centesm_n of_o a_o degree_n nor_o be_v these_o table_n so_o confine_v to_o this_o meridian_n but_o that_o they_o may_v be_v reduce_v to_o any_o other_o if_o the_o place_n be_v east_n of_o london_n add_v to_o the_o time_n give_v but_o if_o it_o be_v west_n make_v substraction_n according_a to_o the_o difference_n of_o longitude_n allow_v 15_o deg_n for_o a_o hour_n and_o 6_o minute_n or_o centesm_n of_o a_o hour_n to_o one_o degree_n so_o will_v the_o sum_n or_o difference_n be_v the_o time_n aequate_v to_o the_o meridian_n of_o london_n and_o for_o the_o more_o speedy_a effect_n of_o the_o say_a reduction_n i_o have_v add_v a_o catalogue_n of_o many_o of_o the_o chief_a town_n and_o city_n in_o diverse_a region_n with_o their_o latitude_n and_o difference_n of_o meridian_n from_o london_n in_o time_n together_o with_o the_o note_n of_o addition_n and_o substraction_n the_o use_n whereof_o be_v thus_o suppose_v the_o time_n of_o the_o sun_n entrance_n into_o taurus_n be_v at_o london_n april_n the_o 10_o 1654._o at_o 11_o of_o the_o clock_n and_o 16_o centesm_n before_o noon_n and_o it_o be_v require_v to_o reduce_v the_o same_o to_o the_o meridian_n of_o vraniburge_n i_o therefore_o seek_v vraniburge_n in_o the_o catalogue_n of_o city_n and_o place_n against_o which_o i_o find_v 83_o with_o the_o letter_n a_o annex_v therefore_o i_o conclude_v that_o the_o sun_n do_v that_o day_n at_o uraniburge_n enter_v into_o taurus_n at_o 11_o of_o the_o clock_n and_o 99_o min._n or_o centesm_n before_o noon_n and_o so_o of_o any_o other_o problem_n 1._o to_o calculate_v the_o sun_n true_a place_n the_o form_n of_o these_o our_o table_n of_o the_o sun_n motion_n be_v this_o in_o the_o first_o page_n be_v have_v his_o motion_n in_o julian_n year_n complete_a the_o epochae_n or_o root_n of_o motion_n be_v prefix_v which_o show_v the_o place_n of_o the_o sun_n at_o that_o time_n where_o the_o epocha_n adscribe_v have_v its_o beginning_n the_o table_n in_o the_o follow_a page_n serve_v for_o julian_n year_n month_n day_n hour_n and_o part_n as_o by_o their_o title_n it_o do_v appear_v the_o year_n month_n and_o day_n be_v take_v complete_a the_o hour_n and_o scruple_n current_n after_o these_o table_n follow_v another_o which_o contain_v the_o aequation_n of_o the_o eccentrick_a to_o every_o degree_n of_o a_o semicircle_n by_o which_o you_o may_v thus_o compute_v the_o sun_n place_n first_o write_v out_o the_o epocha_n next_o go_v before_o the_o give_a time_n then_o several_o set_v under_o those_o the_o motion_n belong_v to_o the_o year_n month_n and_o day_n complete_a and_o to_o the_o hour_n and_o scruple_n current_a every_o one_o under_o his_o like_a only_o remember_v that_o in_o the_o bissextile_a year_n after_o the_o end_n of_o february_n the_o day_n must_v be_v increase_v by_o a_o unit_fw-la then_o add_v they_o all_o together_o the_o sum_n shall_v be_v the_o sun_n mean_a motion_n for_o the_o time_n give_v example_n let_v the_o give_v time_n be_v 1654._o may_v 13_o 11_o hour_n 25_o scruple_n before_o noon_n at_o london_n and_o the_o sun_n place_n to_o be_v seek_v the_o number_n be_v thus_o  _fw-fr longit._n ☉_o aphel_n ☉_o the_o epocha_n 1640_o 291.2536_o 96.2297_o year_n compl_a 13_o 359.8508_o 2052_o month_n co_fw-la april_n 118.2775_o 53_o day_n compl_a 12_o 11.8278_o 6_o hour_n 23_o 9444_o  _fw-fr scruple_n 25_o 102_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr sun_n or_o mean_a motion_n 782.1643_o 96.4308_o 2._o subtract_v the_o aphelium_n from_o the_o mean_a longitude_n there_o rest_v the_o mean_a anomaly_n if_o it_o exceed_v not_o 360_o degree_n but_o if_o it_o exceed_v 360_o degr_n 360_o be_v take_v from_o their_o difference_n as_o oft_o as_o it_o can_v the_o rest_n be_v the_o mean_a anomaly_n seek_v example_n the_o ☉_o mean_a longitude_n 782.1643_o the_o aphelium_n substract_v 96.4308_o there_o rest_v 685.7335_o from_o whence_o deduct_v 360._o there_o rest_v the_o mean_a anomaly_n 325.7335_o 3._o with_o the_o mean_a anomaly_n enter_v the_o table_n of_o the_o sun_n eccentrick_a equation_n with_o the_o degree_n descend_v on_o the_o left_a side_n if_o the_o number_n thereof_o be_v less_o than_o 180_o and_o ascend_v on_o the_o right_a side_n if_o it_o exceed_v 180_o and_o in_o a_o straight_a line_n you_o have_v the_o equation_n answer_v thereunto_o use_v the_o part_n proportional_a if_o need_v require_v last_o according_a to_o the_o title_n add_v or_o subtract_v this_o equation_n find_v to_o or_o from_o the_o mean_a longitude_n so_o have_v you_o the_o sun_n true_a place_n example_n the_o sun_n mean_a longitude_n 782.1643_o or_o deduct_v two_o circle_n 720._o the_o sun_n mean_a longitude_n be_v 62.1643_o the_o sun_n mean_a anomaly_n 325.7335_o in_o this_o table_n the_o equation_n answer_v to_o 325_o degree_n be_v 1.1525_o the_o equation_n answer_v to_o 326_o degree_n be_v 1.1236_o and_o their_o difference_n 289._o now_o than_o if_o one_o degree_n or_o 10000_o give_v 289_o what_o shall_v 7335_o give_v the_o product_n of_o the_o second_o and_o three_o term_n be_v 2119815_o and_o this_o divide_a by_o 10000_o the_o first_o term_v give_v the_o quotient_a or_o term_v require_v will_v be_v 212_o fere_n which_o be_v deduct_v from_o 1.1525_o the_o equation_n answer_v to_o 325_o degr_n because_o the_o equation_n decrease_v their_o difference_n 1.1313_o be_v the_o true_a equation_n of_o this_o mean_a anomaly_n which_o be_v add_v to_o the_o sun_n mean_a longitude_n their_o aggregate_v be_v the_o sun_n place_n require_v example_n the_o sun_n mean_a longitude_n 62.1643_o equation_n correct_v add_v 1.1313_o the_o sun_n true_a place_n or_o longitude_n 63.2956_o that_o be_v 2_o sign_n 3_o degree_n 29_o minute_n 56_o part_n the_o sun_n equation_n in_o this_o example_n correct_v by_o multiplication_n and_o division_n may_v more_o ready_o be_v perform_v by_o addition_n and_o substraction_n with_o the_o help_n of_o the_o table_n of_o logarithme_n for_o as_o one_o degree_n or_o 10000_o 4.000000_o  _fw-fr  _fw-fr be_v to_o 289_o 2.460898_o so_o be_v 7335_o 3.865400_o  _fw-fr  _fw-fr to_o 212_o fere_n 2.326298_o the_o sun_n mean_v motion_n epochae_fw-la longitud_n ☉_o aphelium_n ☉_o  _fw-fr °_o ′_o ″_o °_o ′_o ″_o per._n jul._n 242_o 99_o 61_o 355_o 85_o 44_o m●●di_fw-la 248_o 71_o 08_o 007_o 92_o 42_o christi_fw-la 278_o 98_o 69_o 010_o 31_o 36_o an._n do._n 1600_o 290_o 95_o 44_o 095_o 58_o 78_o an._n do._n 1620_o 291_o 10_o 41_o 095_o 90_o 39_o an._n do._n 1640_o 291_o 25_o 36_o 096_o 21_o 97_o an._n do._n 1660_o 291_o 40_o 33_o 096_o 53_o 56_o 1_o 356_o 76_o 11_o 0_o 01_o 58_o 2_o 359_o 52_o 22_o 0_o 18_o 17_o 3_o 359_o 28_o 30_o 0_o 04_o 74_o b_o 4_o 000_o 03_o 00_o 0_o 06_o 30_o 5_o 359_o 79_o 11_o 0_o 07_o 89_o 6_o 359_o 55_o 19_o 0_o 09_o 47_o 7_o 359_o 31_o 30_o 0_o 11_o 05_o b_o 8_o 000_o 05_o 97_o 0_o 12_o 64_o 9_o 359_o 82_o 08_o 0_o 14_o 22_o 10_o 359_o 58_o 19_o 0_o 15_o 78_o 11_o 359_o 34_o 30_o 0_o 17_o 36_o b_o 12_o 000_o 08_o 97_o 0_o 18_o 94_o 13_o 359_o 85_o 08_o 0_o 20_o 52_o 14_o 359_o 00_o 19_o 0_o 22_o 11_o 15_o 359_o 37_o 30_o 0_o 23_o 69_o b_o 16_o 000_o 11_o 97_o 0_o 25_o 25_o 17_o 359_o 88_o 08_o 0_o 26_o 83_o 18_o 359_o 64_o 19_o 0_o 28_o 41_o 19_o 359_o 40_o 28_o 0_o 30_o 00_o b_o 20_o 000_o 14_o 97_o 0_o 31_o

fraction_n be_v in_o the_o calculation_n very_o tedious_a beside_o here_o no_o fraction_n almost_o be_v exquisite_o true_a therefore_o the_o radius_fw-la for_o the_o make_n of_o rhese_fw-mi table_n be_v to_o be_v take_v so_o much_o the_o more_o that_o there_o may_v be_v no_o error_n in_o so_o many_o of_o the_o figure_n towards_o the_o left_a hand_n as_o you_o will_v have_v place_v in_o the_o table_n and_o as_o for_o the_o number_n superfluous_a they_o be_v to_o be_v cut_v off_o from_o the_o right_a hand_n towards_o the_o left_a after_o the_o end_n of_o the_o supputation_n thus_o to_o find_v the_o number_n answer_v to_o each_o degree_n and_o minute_n of_o the_o quadrant_n to_o the_o radius_fw-la of_o 10000000_o or_o ten_o million_o i_o add_v eight_o cipher_n more_o and_o then_o my_o radius_fw-la do_v consist_v of_o sixteen_o place_n this_o do_v you_o must_v next_o find_v out_o the_o right_a sin_n of_o all_o the_o arch_n less_o than_o a_o quadant_n in_o the_o same_o part_n as_o the_o radius_fw-la be_v take_v of_o whatsoever_o bigness_n it_o be_v and_o from_o those_o right_a sin_n the_o tangent_n and_o secant_v must_v be_v find_v out_o 21._o the_o right_a sin_n in_o make_v of_o the_o table_n be_v either_o primary_n or_o secondary_a the_o primarie_a sin_n be_v those_o by_o which_o the_o rest_n be_v find_v and_o thus_o the_o radius_fw-la or_o whole_a sine_fw-la be_v the_o first_o primary_n sine_fw-la the_o which_o how_o great_a or_o little_o soever_o be_v equal_a to_o the_o side_n of_o a_o sixangled_n figure_n inscribe_v in_o a_o circle_n that_o be_v to_o the_o subtense_n of_o 60_o degree_n the_o which_o be_v thus_o demonstrate_v out_o of_o the_o radius_fw-la or_o subtense_n of_o 60_o degree_n the_o sine_fw-la of_o 30_o degree_n be_v easy_o find_v the_o half_a of_o the_o subtense_n be_v the_o measure_n of_o a_o angle_n at_o the_o circumference_n opposite_a thereunto_o by_o the_o 19_o of_o the_o second_o if_o therefore_o your_o radius_fw-la consist_v of_o 16_o place_n be_v 1000.0000.0000.0000_o the_o sine_fw-la of_o 30_o degree_n will_v be_v the_o one_o half_a thereof_o to_o wit_n 500.0000.0000.0000_o 22._o the_o other_o primary_n sin_n be_v the_o sin_n of_o 60_o 45_o 36_o and_o of_o 18_o degree_n be_v the_o half_a of_o the_o subtense_n of_o 120_o 90_o 72_o and_o of_o 36_o degree_n 23._o the_o subtense_n of_o 120_o degree_n be_v the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v find_v the_o rule_n subtract_v the_o square_a of_o the_o subtense_n of_o 60_o degree_n from_o the_o square_n of_o the_o diameter_n the_o square_a root_n of_o what_o remain_v be_v the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circle●_n or_o the_o subtense_n of_o 120_o degree_n the_o reason_n of_o the_o rule_n the_o subtense_n of_o a_o arch_n with_o the_o subtense_n of_o the_o compliment_n thereof_o to_o 180_o with_o the_o diameter_n make_v in_o the_o meeting_n of_o the_o two_o subtense_n a_o right_a angle_a triangle_n as_o the_o subtense_n ab_fw-la 60_o degree_n with_o the_o subtense_n ac_fw-la 120_o degree_n and_o the_o diameter_n cb_n make_v the_o right_a angle_a triangle_n abc_n right_o angle_v at_o a_o by_o the_o 19_o of_o the_o second_o and_o therefore_o the_o side_n include_v the_o right_a angle_n be_v equal_a in_o power_n to_o the_o three_o side_n by_o the_o 〈◊〉_d of_o the_o second_o therefore_o the_o square_a of_o ab_fw-la be_v take_v from_o the_o square_n of_o cb_n there_o remain_v the_o square_a of_o ac_fw-la who_o squar_fw-la root_n be_v the_o subtense_n of_o 〈◊〉_d degree_n or_o the_o side_n of_o a_o equilateral_a triangle_n inscribe_v in_o a_o circle_n example_n let_v the_o diameter_n cb_n be_v 2000.0000_o 0000.0000_o the_o square_a thereof_o be_v 400000._o 00000.00000.00000.00000.00000_o the_o subtense_n of_o ab_fw-la be_v 100000.00000.00000_o the_o square_a thereof_o be_v 100000.00000.00000_o 00000.00000.00000_o which_o be_v substract_v from_o the_o square_n of_o cb_n the_o remainder_n be_v 300000.00000.00000.00000.00000.00000_o who_o square_a root_n 173205.08075.68877_o the_o subtense_n of_o 120_o degree_n consectary_n hence_o it_o follow_v that_o the_o subtense_n of_o a_o arch_n less_o than_o a_o semicircle_n be_v give_v the_o subtense_n of_o the_o compliment_n of_o that_o arch_n to_o a_o semicircle_n be_v also_o give_v 24._o the_o subtense_n of_o 90_o degree_n be_v the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v find_v the_o rule_n multiply_v the_o diameter_n in_o itself_o and_o the_o square_a root_n of_o half_a the_o product_n be_v the_o subtense_n of_o 90_o degree_n or_o the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n the_o reason_n of_o this_o rule_n the_o diagonal_a line_n of_o a_o square_n inscribe_v in_o a_o circle_n be_v two_o diameter_n and_o the_o right_a angle_a figure_n make_v of_o the_o diagonal_o be_v equal_a to_o the_o right_a angle_a figure_n make_v of_o the_o opposite_a side_n by_o the_o 20_o the_o of_o the_o second_o now_o because_o the_o diagonal_a line_n ab_fw-la and_o cd_o be_v equal_a it_o be_v all_o one_o whether_o i_o multiply_v ac_fw-la by_o itself_o or_o by_o the_o other_o diagonal_a cd_o the_o product_n will_v be_v still_o the_o same_o then_o because_o the_o side_n ab_fw-la ac_fw-la and_o bc_n do_v make_v a_o right_a angle_a triangle_n right_o angle_v at_o c_o by_o the_o 〈◊〉_d of_o the_o second_o &_o that_o the_o 〈◊〉_d ac_fw-la and_o ●b_n be_v equal_a by_o the_o work_n the_o half_a of_o the_o square_n of_o ab_fw-la must_v needs_o be_v the_o square_n of_o ac_fw-la or_o cb_n by_o the_o 17_o the_o of_o the_o second_o who_o square_a root_n the_o subtense_n of_o cb_n the_o side_n of_o a_o square_a or_o 90_o degree_n example_n let_v the_o diameter_n ab_fw-la be_v 200000.00000_o 00000_o the_o square_a thereof_o be_v 400000.00000_o 00000.00000.00000.00000_o the_o half_a whereof_o be_v 200000.00000.00000.00000.00000_o 00000._o who_o square_a root_n 14142●_n 356●3_n 73095._o be_v the_o subtense_n of_o 90_o degree_n or_o the_o side_n of_o a_o square_n inscribe_v in_o a_o circle_n 25._o the_o subtense_n of_o 36_o degree_n be_v the_o side_n of_o a_o decangle_n and_o may_v thus_o be_v find_v the_o rule_n divide_v the_o radius_fw-la by_o two_o then_o multiply_v the_o radius_fw-la by_o itself_o and_o the_o half_a thereof_o by_o itself_o and_o from_o the_o square_a root_n of_o the_o sum_n of_o these_o two_o product_n subtract_v the_o half_a of_o radius_fw-la what_o remain_v be_v the_o side_n of_o a_o decangle_n or_o the_o subtense_n of_o 36_o degree_n the_o reason_n of_o the_o rule_n for_o example_n let_v the_o radius_fw-la ebb_n be_v 100000.00000.00000_o then_o be_v bh_n or_o the_o half_a thereof_o 500000._o 00000.00000_o the_o square_a of_o ebb_n be_v 100000_o 00000.00000.00000.00000.00000_o and_o the_o square_a of_o bh_n 250000.00000.00000.00000_o 00000.00000.00000_o the_o sum_n of_o these_o two_o square_n viz_o 125000.00000.00000_o 00000_o 00000._o 00000_o be_v the_o square_a of_o he_o or_o hk_n who_o square_a root_n be_v 1118033●_n 887●9895_n from_o which_o deduct_v the_o half_a radius_fw-la bh_n 500000000000000_o and_o there_o remain_v 618033988749895_o the_o right_a line_n kb_n which_o be_v the_o side_n of_o a_o decangle_n or_o the_o subtense_n of_o 36_o degree_n 26_o the_o subtense_n of_o 72_o degree_n be_v the_o side_n of_o a_o pentagon_n inscribe_v in_o a_o circle_n and_o may_v thus_o be_v sound_a the_o rule_n subtract_v the_o side_n of_o a_o decangle_n from_o the_o diameter_n the_o remainder_n multiply_v by_o the_o radius_fw-la shall_v be_v the_o square_n of_o one_o side_n of_o a_o pentagon_n who_o square_a root_n shall_v be_v the_o side_n itself_o or_o subtense_n of_o 72_o degree_n the_o reason_n of_o the_o rule_n in_o the_o follow_a diagram_n let_v ac_fw-la be_v the_o side_n of_o a_o decangle_n equal_a to_o cx_o in_o the_o diameter_n and_o let_v the_o rest_n of_o the_o semicircle_n be_v bisect_v in_o the_o point_n e_o then_o shall_v either_o of_o the_o right_a line_n ae_n or_o ebb_n represent_v the_o side_n of_o a_o equilateral_a pentagon_n for_o ac_fw-la the_o side_n of_o a_o decangle_n subtend_v a_o arch_n of_o 36_o degree_n the_o ten_o part_n of_o a_o circle_n and_o therefore_o aeb_fw-mi the_o remain_a arch_n of_o a_o semicircle_n be_v 144_o degree_n the_o half_a whereof_o ae_n or_o ebb_n be_v 72_o degree_n the_o five_o part_n of_o a_o circle_n or_o side_n of_o a_o equilateral_a pentagon_n the_o square_a whereof_o be_v equal_a to_o the_o oblong_v make_v of_o db_n and_o bx_n demonstration_n draw_v the_o right_a line_n exit_fw-la ed_z and_o aec_fw-la then_o will_v the_o side_n of_o the_o angle_n ace_n and_o ecx_n be_v equal_a because_o cx_o be_v make_v equal_a to_o ac_fw-la and_o aec_fw-la common_a to_o both_o and_o the_o angle_n themselves_o be_v equal_a because_o they_o be_v in_o equal_a segment_n

to_o swell_v equal_o as_o much_o in_o longitude_n as_o in_o latitude_n till_o it_o join_v itself_o unto_o the_o concavity_n of_o the_o cylinder_n so_o as_o hereby_o no_o part_n thereof_o be_v any_o way_n distort_v or_o displace_v out_o of_o his_o true_a and_o natural_a situation_n upon_o his_o meridian_n parallel_n or_o rumbe_n but_o only_o dilate_v and_o enlarge_v the_o meridians_fw-la also_o parallel_n and_o rumbe_n dilate_v and_o enlarge_n themselves_o likewise_o at_o every_o point_n of_o latitude_n in_o the_o same_o proportion_n now_o then_o let_v we_o diligent_o consider_v of_o the_o geometrical_a lineament_n that_o be_v the_o meridians_fw-la rumbe_n and_o parallel_n of_o this_o imaginary_a nautical_a planisphere_n that_o we_o may_v in_o like_a manner_n express_v the_o same_o in_o the_o mariner_n chart_n for_o so_o undoubted_o we_o shall_v have_v therein_o a_o true_a hydrographical_a description_n of_o all_o place_n in_o their_o longitude_n latitude_n and_o direction_n or_o respective_a situation_n each_o from_o other_o according_a to_o the_o point_n of_o the_o compass_n in_o all_o thing_n correspondent_a to_o the_o globe_n without_o either_o sensible_a or_o explicable_a error_n first_o therefore_o in_o this_o planisphere_n because_o the_o parallel_n be_v every_o where_o equal_a each_o to_o other_o for_o every_o one_o of_o they_o be_v equal_a to_o the_o equinoctial_a or_o circumference_n of_o the_o circumscribe_v cylinder_n the_o meridians_fw-la also_o must_v needs_o be_v parallel_n &_o straight_a line_n and_o consequent_o the_o rumbe_n make_v equal_a angle_n with_o every_o meridian_n must_v likewise_o be_v straight_a line_n second_o because_o the_o spherical_a superficies_n whereof_o this_o planisphere_n be_v conceive_v to_o be_v make_v swell_v in_o every_o part_n thereof_o equal_o that_o be_v as_o much_o in_o latitude_n as_o in_o longitude_n till_o it_o apply_v itself_o round_o about_o to_o the_o concavity_n of_o the_o cylinder_n therefore_o at_o every_o point_n of_o latitude_n in_o this_o planisphere_n a_o part_n of_o the_o meridian_n keep_v the_o same_o proportion_n to_o the_o like_a part_n of_o the_o parallel_n that_o the_o like_a part_n of_o the_o meridian_n and_o parallel_n have_v each_o to_o other_o in_o the_o globe_n without_o any_o explicable_a error_n and_o because_o like_a part_n of_o whole_n keep_v the_o same_o proportion_n that_o their_o whole_n have_v therefore_o the_o like_a part_n of_o any_o parallel_n and_o meridian_n of_o the_o globe_n have_v the_o same_o proportion_n that_o the_o same_o parallel_n and_o meridian_n have_v for_o example_n sake_n as_o the_o meridian_n be_v double_a to_o the_o parallel_n of_o 60_o degree_n so_o a_o degree_n of_o the_o meridian_n be_v double_a to_o a_o degree_n of_o that_o parallel_n or_o a_o minute_n to_o a_o minute_n and_o what_o proportion_n the_o parallel_n have_v to_o the_o meridian_n the_o same_o proportion_n have_v their_o diameter_n and_o semidiameter_n each_o to_o other_o but_o the_o sine_fw-la of_o the_o compliment_n of_o the_o parallel_n latitude_n or_o distance_n from_o the_o equinoctial_a be_v the_o semidiameter_n of_o the_o parallel_n as_o here_o you_o see_v ae_n the_o sine_fw-la of_o ah_o the_o compliment_n of_o of_o the_o latitude_n or_o distance_n of_o the_o parallel_n abcd_v from_o the_o equinoctial_a be_v the_o semidiameter_n of_o the_o same_o parallel_n and_o as_o the_o semidiameter_n of_o the_o meridian_n or_o whole_a sine_fw-la be_v to_o the_o semidiameter_n of_o the_o parallel_n so_o be_v the_o secant_fw-la or_o hypothenusa_fw-la of_o the_o parallel_n latitude_n or_o of_o the_o parallel_n distance_n from_o the_o equinoctial_a to_o the_o semidiameter_n of_o the_o meridian_n or_o whole_a sine_fw-la as_o fk_v that_o be_v ak_v to_o ae_n that_o be_v gk_n so_o be_v lk_v to_o kf_n therefore_o in_o this_o nautical_a planisphere_n the_o semidiameter_n of_o each_o parallel_n be_v equal_a to_o the_o semidiameter_n of_o the_o equinoctial_a that_o be_v to_o the_o whole_a sine_fw-la the_o part_n of_o the_o meridian_n at_o every_o point_n of_o latitude_n must_v needs_o increase_v with_o the_o same_o proportion_n wherewith_o the_o secant_v of_o the_o ark_n contain_v between_o those_o point_n of_o latitude_n and_o the_o equinoctial_a do_v increase_v now_o than_o we_o have_v a_o easy_a way_n lay_v open_a for_o the_o make_n of_o a_o table_n by_o help_n of_o the_o natural_a canon_n of_o triangle_n whereby_o the_o meridians_fw-la of_o the_o mariner_n chart_n may_v most_o easy_o and_o true_o be_v divide_v into_o part_n in_o due_a proportion_n and_o from_o the_o equinoctial_a towards_o either_o pole_n for_o suppose_v each_o distance_n of_o each_o point_n of_o latitude_n or_o of_o each_o parallel_n from_o other_o to_o contain_v so_o many_o part_n as_o the_o secant_fw-la of_o the_o latitude_n of_o each_o point_n or_o parallel_n contain_v by_o perpetual_a addition_n of_o the_o secant_v answerable_a to_o the_o latitude_n of_o each_o point_n or_o parallel_v unto_o the_o sum_n compound_v of_o all_o the_o former_a secant_v begin_v with_o the_o secant_fw-la of_o the_o first_o parallel_v latitude_n and_o thereto_o add_v the_o secant_fw-la of_o the_o second_o parallel_n latitude_n and_o to_o the_o sum_n of_o both_o these_o adjoin_v the_o secant_fw-la of_o the_o three_o parallel_n latitude_n and_o so_o forth_o in_o all_o the_o rest_n we_o may_v make_v a_o table_n which_o shall_v true_o show_v the_o section_n and_o point_n of_o latitude_n in_o the_o meridian_n of_o the_o nautical_a planisphere_n by_o which_o section_n the_o parallel_n must_v be_v draw_v as_o in_o the_o table_n of_o meridional_a part_n place_v at_o the_o end_n of_o this_o discourse_n we_o make_v the_o distance_n of_o each_o parallel_n from_o other_o to_o be_v one_o minute_n or_o centesm_n of_o a_o degree_n and_o we_o suppose_v the_o space_n between_o any_o two_o parallel_n next_o to_o each_o other_o in_o the_o planispere_n to_o contain_v so_o many_o part_n as_o the_o secant_fw-la answerable_a to_o the_o distance_n of_o the_o further_a of_o those_o two_o parallel_n from_o the_o equinoctial_a and_o so_o by_o perpetual_a addition_n of_o the_o secant_v of_o each_o minute_n or_o centesm_n to_o the_o sum_n compound_v of_o all_o the_o former_a secant_v be_v make_v the_o whole_a table_n as_o for_o example_n the_o secant_fw-la of_o one_o centesm_n in_o master_n briggs_n be_v trigonometrica_n britannica_fw-la be_v 100000.00152_o which_o also_o show_v the_o section_n of_o one_o minute_n or_o centesm_n of_o the_o meridian_n from_o the_o equinoctial_a in_o the_o nautical_a planisphere_n whereunto_o add_v the_o secant_v of_o two_o minute_n or_o centesme_n that_o be_v 100000._o 00609_o the_o sum_n be_v 200000.00761_o which_o show_v the_o section_n of_o the_o second_o minute_n of_o the_o meridian_n from_o the_o equinoctial_a in_o the_o planisphere_n to_o this_o sum_n add_v the_o secant_fw-la of_o three_o minute_n which_o be_v 100000.01371_o the_o sum_n will_v be_v 3000●0_n 02132_o which_o show_v the_o section_n of_o the_o three_o minute_n of_o the_o meridian_n from_o the_o equinoctial_a and_o so_o ●orth_a in_o all_o the_o rest_n but_o after_o the_o table_n be_v thus_o finish_v it_o be_v too_o large_a for_o so_o small_a a_o volume_n we_o have_v content_v ourselves_o with_o every_o ten_o number_n and_o have_v also_o cut_v off_o eight_o place_n towards_o the_o right_a hand_n so_o that_o in_o this_o table_n the_o section_n of_o 10_o minute_n be_v 100_o of_o one_o degree_n 1000_o and_o this_o be_v sufficient_a for_o the_o make_v either_o of_o the_o general_a or_o any_o particular_a chart._n i_o call_v that_o a_o general_a chart_n who_o line_n ae_n in_o the_o follow_a figure_n represent_v the_o equinoctial_a as_o here_o it_o do_v the_o parallel_n of_o 50_o degree_n and_o so_o contain_v all_o the_o parallel_n successive_o from_o the_o equinoctial_a towards_o either_o pole_n but_o they_o can_v never_o be_v extend_v very_o near_o the_o pole_n because_o the_o distance_n of_o the_o parallel_n increase_v as_o much_o as_o secant_v do_v but_o notwithstanding_o this_o it_o may_v be_v reme_v general_n because_o a_o more_o general_a chart_n can_v be_v contrive_v in_o plano_fw-la except_o a_o true_a projection_n of_o the_o sphere_n itself_o and_o i_o call_v that_o a_o particular_a chart_n which_o be_v make_v proper_o for_o one_o particular_a navigation_n as_o if_o a_o man_n be_v to_o sail_v between_o the_o latitude_n of_o 50_o and_o 55_o degree_n and_o his_o difference_n of_o longitude_n be_v not_o to_o exceed_v six_o degree_n than_o a_o chart_n make_v as_o this_o figure_n be_v for_o such_o a_o voyage_n may_v be_v call_v particular_a and_o be_v thus_o to_o be_v project_v probl._n 3_o the_o latitude_n of_o two_o place_n be_v know_v to_o find_v the_o meridional_a difference_n of_o the_o same_o latitude_n in_o this_o proposition_n there_o be_v three_o variety_n first_o when_o one_o of_o the_o place_n be_v under_o the_o equinoctial_a and_o the_o other_o without_o and_o in_o this_o case_n the_o