the_o sea_n gunner_n publish_v by_o john_n seller_n a_o cascable_a deck_n b_o base_a ring_n c_o touchole_n d_o the_o chamber_n e_o reinfourd_n ring_n f_o trunions_n g_o cornish_n ring_n h_o trunion_n ring_n the_o sea-gunner_n show_v the_o practical_a part_n of_o gunnery_n as_o it_o be_v use_v at_o sea._n and_o as_o a_o introduction_n thereto_o there_o be_v exhibit_v two_o compendium_n one_o of_o vulgar_a the_o other_o of_o decimal_a arithmetic_n with_o necessary_a table_n relate_v to_o that_o art_n to_o which_o be_v add_v a_o appendix_n show_v the_o use_n of_o a_o proportional_a scale_n for_o the_o ready_a work_n of_o any_o question_n in_o gunnery_n and_o the_o use_v of_o the_o sea-gunner_n rule_n of_o a_o excellent_a contrivance_n contain_v a_o epitome_n of_o the_o art_n of_o gunnery_n in_o itself_o compose_v by_o 〈◊〉_d sâ—â—â—â—r_n senior_fw-la london_n print_v by_o h._n clerk_n for_o the_o author_n and_o be_v to_o sell_v by_o he_o at_o the_o hermitage_n in_o wapping_n 1691._o the_o preface_n to_o the_o reader_n courteous_a reader_n have_v observe_v for_o several_a year_n that_o there_o have_v be_v a_o great_a want_n of_o a_o piece_n of_o sea-gunnery_a that_o have_v be_v principal_o adapt_v for_o sea-service_n in_o a_o treatise_n by_o itself_o for_o those_o book_n that_o be_v extant_a be_v chief_o intend_v for_o land_n service_n and_o at_o this_o time_n most_o of_o they_o be_v out_o of_o print_n i_o judge_v this_o a_o fit_a opportunity_n for_o the_o publish_n this_o small_a treatise_n hope_v it_o will_v be_v grateful_o accept_v by_o our_o sea-gunner_n and_o in_o regard_n that_o those_o who_o will_v be_v student_n in_o that_o art_n ought_v in_o some_o competent_a measure_n to_o be_v acquaint_v with_o arithmetic_n for_o the_o sake_n of_o such_o i_o have_v exhibit_v two_o compendium_n thereof_o one_o in_o vulgar_a and_o the_o other_o in_o decimal_a arithmetic_n as_o a_o necessary_a preparation_n for_o the_o work_a those_o question_n that_o be_v incident_a to_o that_o art_n and_o for_o the_o ease_n of_o such_o as_o be_v not_o full_o acquaint_v therewith_o i_o have_v furnish_v they_o with_o a_o proportional_a scale_n whereby_o they_o may_v perform_v all_o the_o operation_n that_o be_v useful_a in_o gunnery_n as_o also_o to_o extract_v the_o square_a and_o cube-root_n and_o how_o to_o perform_v the_o same_o by_o logarithm_n and_o by_o gunter's-scale_n to_o which_o i_o have_v add_v several_a necessary_a table_n useful_a in_o gunnery_n with_o proper_a question_n and_o their_o answer_n and_o useful_a observation_n and_o instruction_n and_o for_o the_o better_a accomplish_v the_o design_n of_o this_o book_n i_o have_v consult_v with_o the_o best_a approve_a author_n that_o have_v write_v on_o this_o subject_a also_o at_o the_o end_n of_o this_o treatise_n i_o have_v present_v you_o with_o a_o small_a tract_n as_o a_o appendix_n particular_o of_o the_o use_n of_o the_o proportional_a scale_n and_o of_o the_o use_n of_o a_o rule_n of_o a_o new_a contrivance_n fit_a for_o the_o pocket_n that_o have_v upon_o it_o a_o epitome_n of_o the_o practical_a part_n of_o gunnery_n in_o itself_o which_o i_o call_v the_o sea-gunner_n rule_n all_o which_o i_o submit_v to_o the_o favourable_a construction_n of_o the_o judicious_a and_o rest_v your_o friend_n to_o serve_v you_o john_n seller_n a_o table_n of_o the_o principal_a matter_n contain_v in_o this_o book_n chap._n i._o of_o vulgar_a arithmetic_n page_z 1_o of_o notation_n of_o number_n ibid._n numeration_n and_o addition_n 3_o subtraction_n 5_o multiplication_n 7_o division_n 15_o the_o rule_n of_o three_o direct_v 19_o the_o rule_n of_o three_o reverse_n 22_o double_a rule_n of_o three_o 23_o chap._n ii_o of_o decimal_a arithmetic_n 28_o notation_n of_o fraction_n ibid._n addition_n of_o decimal_n 33_o subtraction_n of_o decimal_n 34_o multiplication_n of_o decimal_n 35_o division_n of_o decimal_n 37_o a_o decimal_n table_n of_o penny_n and_o farthing_n 46_o a_o table_n of_o decimal_n of_o one_o pound_n sterling_n 48_o a_o table_n of_o the_o decimal_n of_o a_o english_a foot_n to_o every_o inch_n and_o eight_o part_n of_o a_o inch_n 49_o chap._n iii_o the_o extraction_n of_o the_o square-root_n by_o arithmetic_n 52_o to_o extract_v the_o sqaure-root_n by_o logarithm_n and_o gunter's-scale_n 56_o to_o extract_v the_o cube-root_n by_o arithmetic_n 57_o to_o prepare_v a_o cube_n number_n for_o extraction_n 59_o to_o extract_v the_o cube-root_n by_o the_o logarithm_n 63_o to_o extract_v the_o cube-root_n by_o gunter's-scale_n 64_o a_o table_n of_o square-root_n 65_o a_o table_n of_o cubick-root_n 66_o to_o make_v the_o table_n of_o square_a and_o cube-root_n 67_o a_o table_n of_o logarithm_n 8_o a_o description_n and_o use_n of_o the_o table_n of_o logarithm_n 82_o multiplication_n by_o logarithm_n 86_o division_n by_o logarithm_n 87_o of_o a_o circle_n 88_o 1._o the_o diameter_n be_v give_v to_o find_v the_o circumference_n by_o the_o logarithm_n ibid._n 2._o the_o circumference_n be_v give_v to_o find_v the_o diameter_n 89_o 3._o the_o diameter_n of_o a_o circle_n be_v give_v to_o find_v the_o area_n or_o superficial_a content_n thereof_o 90_o 4_o the_o circumference_n be_v give_v to_o find_v the_o area_n 91_o chap._n iv_o contain_v geometrical_a rudiment_n useful_a in_o the_o art_n of_o gunnery_n 93_o to_o raise_v a_o perpendicular_a from_o the_o middle_n of_o a_o line_n give_v ibid._n to_o let_v a_o perpendicular_a fall_n from_o a_o point_n assign_v to_o the_o middle_n of_o a_o line_n give_v 94_o to_o raise_v a_o perpendicular_a upon_o the_o end_n of_o a_o line_n give_v 95_o to_o let_v fall_v a_o perpendicular_a from_o a_o point_n assign_v unto_o the_o end_n of_o a_o line_n give_v 96_o to_o draw_v a_o line_n parallel_v to_o a_o line_n give_v 97_o a_o geometrical_a way_n to_o find_v the_o diameter_n of_o a_o bullet_n that_o weigh_v twice_o as_o much_o as_o a_o know_a bullet_n 98_o the_o weight_n of_o a_o shot_n give_v to_o find_v the_o diameter_n geometrical_o 100_o chap._n v._o geometrical_a theorem_n and_o problem_n 102_o arithmetical_a problem_n appertain_v to_o the_o art_n of_o gunnery_n and_o wrought_v by_o decimal_a arithmetic_n by_o logarithm_n and_o gunter_n be_v scale_n prob._n 1._o the_o diameter_n of_o a_o circle_n be_v give_v to_o find_v the_o circumference_n ibid._n prob._n 2._o the_o circumference_n of_o a_o circle_n be_v give_v to_o find_v the_o diameter_n 105_o prob._n 3._o the_o diameter_n of_o a_o circle_n be_v give_v to_o find_v the_o side_n of_o a_o square_n equal_a to_o it_o 107_o prob._n 4._o the_o circumference_n of_o a_o circle_n be_v give_v to_o find_v the_o side_n of_o a_o square_n equal_a in_o content_a to_o that_o circle_n 108_o prob._n 5._o the_o diameter_n of_o any_o spherical_a body_n be_v know_v to_o find_v the_o circumference_n 109_o prob._n 6._o the_o circumference_n of_o any_o spherical_a body_n be_v know_v to_o find_v the_o diameter_n 110_o prob._n 7._o the_o diameter_n and_o circumference_n of_o any_o spherical_a body_n be_v know_v to_o find_v the_o superficial_a content_a 111_o prob._n 8._o the_o axis_n or_o diameter_n of_o a_o globical_a body_n be_v know_v to_o find_v the_o solid_a content_a 112_o prob._n 9_o the_o diameter_n of_o a_o bullet_n be_v give_v with_o the_o weight_n to_o find_v the_o weight_n of_o another_o bullet_n of_o the_o same_o metal_n but_o of_o another_o diameter_n either_o great_a or_o lesser_a 113_o prob._n 10._o have_v the_o weight_n of_o a_o bullet_n of_o one_o kind_n of_o metal_n to_o find_v the_o weight_n of_o a_o bullet_n of_o another_o kind_n of_o metal_n be_v equal_a in_o magnitude_n 116_o prob._n 11._o a_o bullet_n of_o iron_n that_o weight_v seventy_o two_o pound_n what_o will_v a_o bullet_n of_o lead_n weigh_v that_o be_v equal_a to_o it_o in_o bigness_n 118_o prob._n 12._o the_o diameter_n and_o weight_n of_o any_o one_o cylend_a of_o a_o piece_n of_o great_a ordnance_n take_v at_o the_o base_a ring_n be_v know_v to_o find_v the_o weight_n of_o any_o other_o piece_n of_o the_o same_o metal_n and_o shape_n either_o great_a or_o lesser_a its_o diameter_n be_v only_o know_v 119_o prob._n 13._o have_v the_o diameter_n and_o weight_n of_o any_o piece_n of_o great_a ordnance_n to_o find_v the_o weight_n of_o another_o piece_n of_o ordnance_n of_o another_o 〈◊〉_d that_o be_v of_o the_o same_o shaâ—â—_n 121_o â—â—ob_n 14._o to_o find_v the_o supâ—â—fiâ—â—l_a content_a of_o the_o convex_a face_n of_o any_o piece_n of_o ordnance_n and_o also_o the_o solid_a content_n of_o the_o concavity_n thereof_o 123_o prob._n 15._o to_o know_v how_o much_o of_o every_o kind_n of_o metal_n be_v contain_v in_o any_o brass_n piece_n of_o ordnance_n 126_o prob._n 16._o by_o know_v what_o quantity_n of_o powder_n will_v load_v some_o one_o piece_n of_o ordnance_n to_o find_v how_o much_o of_o the_o same_o powder_n will_v
sum_n of_o they_o will_v be_v the_o logarithm_n of_o the_o number_n produce_v by_o that_o multiplication_n example_n let_v it_o be_v require_v to_o multiply_v 48_o by_o 5_o first_o set_v down_o the_o two_o number_n to_o be_v multiply_v one_o under_o another_o and_o to_o they_o set_v their_o respective_a logarithm_n as_o in_o the_o margin_n which_o be_v add_v together_o the_o sum_n of_o they_o which_o be_v the_o logarithm_n of_o the_o product_n be_v seek_v in_o the_o table_n the_o absolute_a number_n answer_v thereto_o be_v 240_o the_o product_n of_o those_o two_o number_n multiply_v together_o 48_o 1,68124_o 5_o 0,69897_o 240_o 2,38021_o division_n by_o the_o logarithm_n as_o multiplication_n by_o the_o logarithm_n be_v perform_v by_o addition_n so_o division_n be_v perform_v by_o subtraction_n wherefore_o to_o perform_v division_n you_o must_v subtract_v the_o logarithm_n of_o the_o number_n by_o which_o you_o be_v to_o divide_v from_o the_o logarithm_n of_o the_o number_n which_o be_v to_o be_v divide_v and_o the_o number_n which_o remain_v shall_v be_v the_o logarithm_n of_o the_o quotient_a example_n let_v it_o be_v require_v to_o divide_v 228_o by_o 12_o 228.2,35793_o 12.1,07918_o 19.1,27875_o first_o set_v down_o the_o logarithm_n of_o 228_o and_o under_o it_o set_v the_o logarithm_n of_o 12_o and_o subtract_v the_o lesser_a from_o the_o great_a the_o remainder_n be_v the_o logarithm_n of_o the_o quotient_a which_o be_v seek_v in_o the_o table_n you_o will_v find_v 19_o to_o be_v the_o answer_n of_o the_o question_n be_v the_o quotient_a seek_v and_o so_o many_o time_n be_v 12_o contain_v in_o 228._o of_o a_o circle_n 1._o the_o diameter_n be_v give_v to_o find_v the_o circumference_n by_o the_o logarithm_n the_o proportion_n be_v as_o 7_o to_o 22_o so_o be_v the_o diameter_n to_o the_o circumference_n wherefore_o to_o find_v the_o circumference_n of_o any_o circle_n who_o diameter_n be_v give_v add_v the_o logarithm_n of_o the_o diameter_n give_v to_o the_o logarithm_n of_o 22_o and_o from_o the_o sum_n of_o they_o subtract_v the_o logarithm_n of_o 7_o the_o remainder_n shall_v be_v the_o logatithm_n of_o the_o circumference_n seek_v example_n if_o the_o diameter_n of_o a_o circle_n be_v 113_o what_o be_v the_o circumference_n first_o set_v down_o the_o logarithm_n of_o 22_o which_o be_v 1,34242_o 2,05308_o 3,39550_o 0,84510_o 2,55040_o add_v the_o logarithm_n of_o 113_o which_o be_v from_o which_o subtract_v the_o logarithm_n of_o 7_o which_o be_v which_o be_v seek_v in_o the_o table_n be_v the_o near_a logarithm_n of_o 355_o and_o so_o much_o be_v the_o circumference_n of_o a_o circle_n who_o diameter_n be_v 113._o 2._o the_o circumference_n of_o a_o circle_n be_v give_v to_o find_v the_o diameter_n the_o proportion_n be_v as_o 22_o be_v to_o 7_o so_o be_v the_o circumference_n to_o the_o diameter_n wherefore_o to_o the_o logarithm_n of_o 7_o add_v the_o logarithm_n of_o the_o circumference_n give_v and_o from_o the_o sum_n subtract_v the_o logarithm_n of_o 22_o the_o remainder_n shall_v be_v the_o logarithm_n of_o the_o diameter_n example_n if_o the_o circumference_n of_o a_o circle_n be_v 355_o what_o be_v the_o diameter_n thereof_o first_o set_v down_o the_o logarithm_n of_o 7._o which_o be_v 0,84510_o 2,55023_o 3,39533_o 1,34242_o 2,05291_o and_o to_o it_o add_v the_o logarithm_n of_o 355_o from_o which_o subtract_v the_o logarithm_n of_o 22_o and_o the_o remainder_n be_v the_o near_a logarithm_n of_o 113_o which_o be_v the_o diameter_n require_v 3._o the_o diameter_n of_o a_o circle_n be_v give_v to_o find_v the_o area_n or_o superficial_a content_n thereof_o the_o proportion_n be_v as_o 28_o be_v to_o 22_o so_o be_v the_o square_a of_o the_o diameter_n to_o the_o area_n wherefore_o to_o the_o logarithm_n of_o 22_o add_v the_o logarithm_n of_o the_o diameter_n double_v and_o from_o the_o sum_n subtract_v the_o logarithm_n of_o 28_o the_o remainder_n shall_v be_v the_o logarithm_n of_o the_o area_n require_v example_n if_o the_o diameter_n of_o a_o circle_n be_v 12_o what_o be_v the_o area_n or_o superficial_a content_n thereof_o first_o set_v down_o the_o logarithm_n of_o 22_o which_o be_v 1,34242_o 1,07918_o 1,07918_o 3,50078_o 1,44716_o 2,05362_o and_o to_o that_o the_o logarithm_n of_o 12_o the_o give_v diameter_n set_v down_o twice_o add_v all_o three_o together_o from_o which_o subtract_v the_o logarithm_n of_o 28_o the_o remainder_n be_v the_o near_a logarithm_n to_o the_o number_n 113_o and_o some_o small_a matter_n more_o be_v the_o area_n of_o that_o circle_n 4._o the_o circumference_n of_o a_o circle_n be_v give_v to_o find_v the_o area_n the_o proportion_n be_v as_o 88_o be_v to_o 7_o so_o be_v the_o square_a of_o the_o circumference_n to_o the_o area_n wherefore_o to_o the_o logarithm_n of_o 7_o add_v the_o logarithm_n of_o the_o circumference_n twice_o and_o from_o the_o sum_n subtract_v the_o logarithm_n of_o 88_o the_o remainder_n shall_v be_v the_o logarithm_n of_o the_o area_n require_v example_n if_o the_o circumference_n of_o a_o circle_n be_v 38_o what_o be_v the_o area_n thereof_o first_o set_v down_o the_o logarithm_n of_o 7_o which_o be_v 0,84510_o to_o which_o add_v the_o logarithm_n of_o 1,57978_o the_o circumference_n twice_o 1,57978_o the_o sum_n 4,00466_o subtract_v the_o logarithm_n of_o 88_o 1,94448_o the_o remainder_n be_v the_o near_a logarithm_n of_o 115_o the_o area_n seek_v 2,06018_o chap._n iv_o contain_v geometrical_a rudiment_n useful_a in_o the_o art_n of_o gunnery_n how_o to_o raise_v a_o perpendicular_a from_o the_o middle_n of_o a_o line_n give_v let_v the_o line_n give_v be_v a._n b._n and_o let_v c_o be_v a_o point_n therein_o give_v from_o which_o it_o be_v require_v to_o raise_v a_o perpendicular_a first_o therefore_o open_a the_o compass_n to_o any_o convenient_a distance_n and_o set_v one_o foot_n in_o the_o point_n c_o with_o the_o other_o set_v off_o on_o either_o side_n thereof_o the_o equal_a distance_n c_o a_n and_z c_z b_o then_o open_v the_o compass_n to_o any_o convenient_a wide_a distance_n set_v one_o foot_n in_o the_o point_n a_o with_o the_o other_o strike_v the_o occult_a arch_n at_o f_o theâ—_n with_o the_o same_o distance_n set_v one_o foot_n in_o the_o point_n b_o and_o with_o the_o other_o draw_v the_o arch_n f_o cross_v e_o in_o the_o point_n d_o from_o whence_o draw_v the_o line_n dc_o which_o line_n be_v a_o perpendicular_a unto_o the_o give_v line_n a_o b_o as_o be_v require_v to_o let_v a_o perpendicular_a fall_n from_o a_o point_n assign_v to_o the_o middle_n of_o a_o line_n give_v let_v the_o line_n give_v whereupon_o you_o will_v have_v a_o perpendicular_a let_v fall_v be_v the_o line_n b_o c_o d_o and_o the_o point_v a_o to_o be_v the_o point_n assign_v from_o whence_o you_o will_v have_v the_o perpendicular_a let_v fall_v from_o the_o give_v line_n b_o c_o d_o first_o set_v one_o foot_n of_o your_o compass_n in_o the_o point_n a_o and_o open_v they_o to_o any_o convenient_a distance_n so_o that_o it_o be_v more_o than_o the_o line_n a_o c_o describe_v one_o arch_n of_o a_o circle_n with_o the_o other_o foot_n so_o that_o it_o may_v cut_v the_o line_n b_o c_o d_o twice_z that_o be_v at_z e_z and_o at_o f_o then_o find_v the_o middle_n between_o these_o which_o will_v be_v the_o point_n c_o from_o which_o point_n draw_v the_o line_n at_o c_o which_o be_v the_o perpendicular_a which_o be_v to_o be_v let_v fall_n to_o raise_v a_o perpendicular_a upon_o the_o end_n of_o a_o line_n give_v suppose_v the_o line_n whereupon_o you_o will_v have_v the_o perpendicular_a raise_v be_v the_o line_n a_o b_o first_o open_a your_o compass_n to_o a_o convenient_a distance_n and_o set_v one_o foot_n in_o the_o point_n b_o and_o let_v the_o other_o foot_n fall_v any_o where_n above_o the_o line_n as_o at_o the_o point_n d_o and_o in_o that_o point_n let_v one_o foot_n of_o your_o compass_n remain_v turn_v the_o other_o about_o until_o it_o touch_v the_o line_n a_o b_o in_o the_o point_n e_o then_o turn_v the_o same_o foot_n of_o the_o compass_n towards_o c_o and_o draw_v a_o occult_a arch_n and_o lay_v the_o edge_n of_o a_o ruler_n to_o those_o two_o point_n e_o and_o d_o and_o where_o the_o same_o edge_n of_o the_o ruler_n do_v cut_v the_o arch_n c_o from_o that_o point_n draw_v the_o line_n c_o b_o which_o shall_v be_v a_o perpendicular_a at_o the_o end_n of_o the_o line_n a_o b._n to_o let_v fall_v a_o perpendicular_a from_o a_o point_n assign_v unto_o the_o end_n of_o a_o line_n give_v let_v the_o line_n a_o b_o be_v give_v unto_o which_o it_o be_v require_v to_o let_v a_o perpendicular_a fall_n from_o the_o assign_a point_n d_o unto_o the_o end_n a._n first_o from_o the_o assign_a point_n d_o draw_v a_o line_n unto_o any_o point_n of_o
the_o give_v line_n a_o b_o which_o may_v be_v the_o line_n d_o c_o e_o then_o find_v the_o middle_n of_o the_o line_n d_o e_o which_o be_v at_o c_o place_n one_o foot_n of_o your_o compass_n in_o that_o point_n and_o extend_v the_o other_o foot_n unto_o d_o or_o e_o with_o which_o distance_n draw_v the_o semicircle_n d_o a_o e_o which_o shall_v cut_v the_o give_v line_n a_o b_o in_o the_o point_n a_o from_o which_o point_n draw_v the_o line_n d_o a_o which_o be_v the_o perpendicular_a let_v fall_v from_o the_o assign_a point_n d_o on_o the_o end_n of_o the_o give_v line_n a_o b_o as_o be_v require_v to_o draw_v a_o line_n parallel_n to_o a_o line_n give_v let_v a_o b_o be_v a_o line_n give_v whereunto_o it_o be_v require_v to_o draw_v a_o parallel_n first_o set_v one_o foot_n of_o the_o compass_n in_o the_o point_n c_o and_o open_v the_o other_o foot_n at_o pleasure_n draw_v the_o arch_n e_o then_o with_o the_o same_o distance_n set_v one_o foot_n in_o the_o point_n d_o and_o draw_v the_o other_o arch_n f._n last_o lay_v a_o rule_n to_o the_o convexity_n of_o both_o those_o arch_n and_o draw_v the_o line_n g_o h_n which_o shall_v be_v a_o parallel_n to_o a_o b_o as_o be_v require_v a_o geometrical_a problem_n useful_a in_o the_o art_n of_o gunnery_n a_o geometrical_a way_n to_o find_v the_o diameter_n of_o any_o bullet_n that_o weight_v twice_o as_o much_o as_o a_o know_a bullet_n take_v the_o diameter_n of_o the_o lesser_a bullet_n who_o weight_n you_o know_v and_o square_v that_o diameter_n viz._n make_v a_o geometrical_a square_n each_o side_n to_o be_v equal_a to_o the_o diameter_n of_o the_o bullet_n give_v then_o draw_v a_o diagotal_a line_n from_o either_o of_o the_o two_o opposite_a angle_n and_o that_o diagonal_a shall_v be_v the_o diameter_n of_o a_o bullet_n twice_o the_o weight_n of_o the_o other_o then_o divide_v the_o say_v diagonal_a into_o two_o equal_a part_n set_v one_o foot_n of_o the_o compass_n in_o the_o midst_n of_o that_o diagonal_a and_o with_o the_o other_o foot_n describe_v a_o circle_n and_o that_o circumference_n will_v represent_v a_o bullet_n twice_o as_o much_o weight_n as_o the_o other_o the_o sight_n of_o the_o annex_v figure_n be_v a_o sufficient_a explanation_n a_o b_o be_v the_o diameter_n of_o the_o lesser_a bullet_n a_o c_o the_o diameter_n of_o the_o great_a perform_v by_o arithmetic_n suppose_v the_o diameter_n of_o the_o lesser_a bullet_n be_v five_o inch_n the_o square_a thereof_o be_v twenty_o five_o the_o double_a of_o it_o be_v fifty_o the_o root_n thereof_o be_v 7_o 1_o 7_o and_o so_o much_o be_v the_o diameter_n of_o the_o great_a bullet_n the_o weight_n of_o any_o shot_n give_v to_o find_v the_o diameter_n geometrical_o suppose_v a_o shot_n be_v one_o two_o or_o three_o pound_n weight_n of_o metal_n or_o stone_n assign_v if_o one_o pound_n divide_v the_o diameter_n into_o four_o part_n and_o five_o such_o part_n will_v make_v the_o diameter_n of_o a_o shot_n of_o the_o say_a metal_n or_o stone_n that_o shall_v weigh_v just_a two_o pound_n divide_v the_o diameter_n of_o a_o shot_n weigh_v just_a two_o pound_n in_o seven_o equal_a part_n and_o eight_o such_o part_n will_v make_v a_o diameter_n of_o a_o shot_n of_o three_o pound_n and_o divide_v the_o diameter_n of_o a_o shot_n of_o three_o pound_n into_o ten_o equal_a part_n and_o eleven_o such_o make_v a_o shot_n of_o four_o pound_n divide_v the_o diameter_n of_o a_o shot_n of_o four_o pound_n into_o thirteen_o part_n fourteen_o such_o part_n will_v make_v a_o diameter_n for_o a_o shot_n of_o five_o pound_n and_o so_o divide_v each_o next_o diamter_n into_o three_o equal_a part_n more_o the_o next_o lesser_n be_v divide_v into_o and_o it_o will_v with_o one_o part_n add_v from_o a_o diameter_n of_o a_o shot_n that_o will_v weigh_v just_a one_o pound_n more_o so_o you_o may_v proceed_v infinite_o increase_v or_o decrease_v by_o take_v one_o part_n less_o than_o it_o be_v appoint_v to_o be_v divide_v into_o chap._n v._o geometrical_a theorem_n and_o problem_n theorem_fw-la 1._o all_o circle_n be_v equal_a to_o that_o right_o angle_a triangle_n who_o contain_v side_n the_o one_o be_v equal_a to_o the_o semidiameter_n and_o the_o other_o to_o the_o circumference_n thereof_o theorem_fw-la 2._o the_o proportion_n of_o the_o diameter_n of_o a_o circle_n to_o the_o circumference_n be_v as_o 1,000000_o to_o 3,141593_o fere_n or_o as_o archim_n 7_o to_o 22._o theorem_fw-la 3_o the_o proportion_n of_o the_o diameter_n to_o the_o side_n of_o the_o square_n equal_a to_o the_o circle_n be_v as_o 1,000000_o to_o 886227_o fere_n theorem_fw-la 4._o the_o proportion_n of_o the_o diameter_n to_o the_o side_n of_o the_o inscribe_v square_a be_v as_o 1,000000_o to_o 707107_o fere_n theorem_fw-la 5._o the_o proportion_n of_o the_o circumference_n to_o the_o diameter_n be_v as_o 1_o to_o .318310_o fere_n or_o as_o 22_o to_o 7._o theorem_fw-la 6._o the_o proportion_n of_o the_o circumference_n to_o the_o side_n of_o the_o square_n equal_a to_o the_o circle_n be_v as_o 1_o to_o .282095_o theorem_fw-la 7._o the_o proportion_n of_o the_o circumference_n to_o the_o side_n of_o the_o inscribe_v square_a be_v as_o 1_o to_o .225078_o arithmetical_a problem_n appertain_v to_o the_o art_n of_o gunnery_n and_o wrought_v by_o decimal_a arithmetic_n by_o the_o logarithm_n and_o gunter_n be_v scale_n prob._n 1._o the_o diameter_n of_o a_o circle_n be_v give_v to_o find_v the_o circumference_n the_o analogy_n as_o 1_o be_v to_o the_o diameter_n so_o be_v 3.142_o to_o the_o circumference_n or_o as_o 7_o to_o 22_o so_o be_v the_o diameter_n to_o the_o circumference_n if_o the_o diameter_n of_o a_o circle_n be_v 15_o inch_n what_o be_v the_o circumference_n by_o gunter_n be_v scale_n by_o the_o logarithm_n as_o the_o log._n of_o 15_o the_o diameter_n 1,17609_o be_v to_o the_o logarithm_n of_o 3,142_o 0,49720_o so_o be_v the_o logarithm_n of_o 0,00000_o to_o the_o logar_n of_o the_o answer_n 47,1367329_o extend_v the_o compass_n upon_o the_o line_n of_o number_n from_o 1_o to_o the_o diameter_n the_o same_o extent_n will_v reach_v from_o 3.142_o to_o 47.13_o the_o circumference_n prob._n 2._o the_o circumference_n of_o a_o circle_n be_v give_v to_o find_v the_o diameter_n the_o analogy_n as_o 3,142_o be_v to_o 1_o so_o be_v the_o circumference_n 47_o 13_o to_o the_o diameter_n 15_o inch_n if_o the_o circumference_n of_o a_o circle_n be_v 47_o inch_n and_o 13_o part_n of_o a_o 100_o suppose_v every_o inch_n to_o be_v divide_v into_o 100_o part_n what_o be_v the_o diameter_n or_o as_o 22_o to_o 7_o so_o be_v the_o circumference_n to_o the_o diameter_n by_o the_o logarithm_n as_o the_o logarithm_n of_o 3,142_o 0,49720_o be_v to_o the_o logarithm_n of_o 1_o 0,00000_o so_o be_v the_o logarithm_n of_o 47.13_o 1,67329_o to_o the_o logar_n of_o the_o answer_n  _fw-fr 1,67329_o  _fw-fr 15_o 1,17609_o by_o gunter_n scale_n extend_v the_o compass_n upon_o the_o line_n of_o number_n from_o 47.13_o the_o circumference_n the_o same_o extent_n the_o same_o way_n shall_v reach_v from_o 3,142_o to_o the_o diameter_n 15._o prob._n 3._o the_o diameter_n of_o a_o circle_n be_v give_v to_o find_v the_o side_n of_o a_o square_n equal_a to_o it_o if_o the_o diameter_n of_o a_o circle_n be_v 15_o inch_n what_o shall_v be_v the_o side_n of_o a_o square_n equal_a to_o it_o the_o analogy_n as_o 1_o be_v to_o 15_o so_o this_o number_n 8862_o to_o 13.29_o the_o side_n of_o a_o square_n equal_a in_o content_a to_o that_o circle_n by_o the_o logarithm_n as_o the_o logarithm_n 1_o 0,00000_o be_v to_o the_o logarithm_n 15_o 1,17609_o so_o be_v the_o logarithm_n of_o 8862_o 0,94753_o to_o the_o answer_n 13,29_o 2,12362_o by_o gunter_n be_v scale_n extend_v the_o compass_n from_o 1_o to_o 8862_o the_o same_o extent_n shall_v reach_v from_o 15_o to_o 13.29_o prob._n 4._o the_o circumference_n of_o a_o circle_n be_v give_v to_o find_v the_o side_n of_o a_o square_a equal_a in_o content_a to_o that_o circle_n if_o the_o circumference_n of_o a_o circle_n be_v 47_o 13_o the_o side_n of_o a_o square_n equal_a to_o it_o be_v require_v the_o analogy_n as_o 1_o be_v to_o 47.13_o so_o be_v this_o number_n 2821_o to_o 13.29_o the_o side_n of_o the_o square_n require_v by_o the_o logarithm_n as_o the_o logarithm_n of_o 1_o 0,0000_o be_v to_o the_o logarithm_n of_o 4713_o 0,67329_o â—o_o be_v the_o logarithm_n of_o 2821_o 0,45040_o to_o the_o answer_n 13,29_o 1,12369_o by_o gunter_n scale_n extend_v the_o compass_n upon_o the_o line_n of_o number_n from_o 1_o to_o 2812_o the_o same_o extent_n shall_v reach_v the_o same_o way_n from_o 47.13_o to_o 13.29_o the_o side_n of_o the_o square_n require_v prob._n 5._o the_o diameter_n of_o any_o
under_o every_o one_o in_o their_o due_a place_n example_n  _fw-fr l._n a_o man_n owe_v to_o a_o merchant_n 9758_o and_o he_o have_v pay_v of_o that_o debt_n 3514_o there_o remain_v due_a 6244_o case_n ii_o when_o some_o of_o the_o inferior_a number_n be_v great_a than_o the_o superior_a number_n rule_n set_a your_o number_n in_o order_n as_o before_o draw_v a_o line_n under_o they_o and_o begin_v at_o the_o right-hand_a and_o according_a to_o the_o number_n respective_a value_n borrow_v one_o of_o the_o next_o to_o the_o lefthand_n above_o out_o of_o which_o subtract_v what_o remain_v add_v to_o the_o superior_a and_o set_v their_o sum_n under_o the_o line_n than_o what_o you_o borrow_v pay_v to_o the_o next_o number_n on_o the_o lefthand_n below_o and_o so_o proceed_v throughout_o the_o work_n according_a to_o this_o or_o the_o former_a rule_n  _fw-fr l._n s._n d._n as_o from_o 529_o 13_o 4_o take_v 347_o 16_o 7_o rest_n 171_o 16_o 9_o proof_n add_v the_o two_o inferior_n their_o sum_n be_v equal_a to_o the_o superior_a multiplication_n mvltiplication_n serve_v to_o perform_v that_o at_o once_o which_o addition_n do_v at_o many_o time_n and_o to_o multiply_v ready_o it_o be_v necessary_a that_o the_o ensue_a table_n shall_v be_v perfect_o learn_v pythagora_n table_n 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o 2_o 4_o 6_o 8_o 10_o 12_o 14_o 16_o 18_o 3_o 6_o 9_o 12_o 15_o 18_o 21_o 24_o 27_o 4_o 8_o 12_o 16_o 20_o 24_o 28_o 32_o 36_o 5_o 10_o 15_o 20_o 25_o 30_o 35_o 40_o 45_o 6_o 12_o 18_o 24_o 30_o 36_o 42_o 48_o 54_o 7_o 14_o 21_o 28_o 35_o 42_o 49_o 56_o 63_o 8_o 16_o 24_o 32_o 40_o 48_o 56_o 64_o 72_o 9_o 18_o 27_o 36_o 45_o 54_o 63_o 72_o 81_o the_o use_n of_o this_o table_n be_v to_o muliply_v any_o number_n in_o the_o outer_a column_n to_o the_o left_a hand_n by_o any_o figure_n at_o the_o top_n and_o in_o the_o common_a angle_n of_o meeting_n be_v the_o answer_n to_o the_o question_n as_o 7_o time_n 9_o you_o will_v find_v to_o be_v 63._o in_o multiplication_n note_v that_o the_o uppermost_a number_n be_v always_o the_o multplicand_a and_o the_o low_a the_o multiplier_n and_o the_o figure_n which_o remain_v when_o the_o work_n be_v do_v be_v call_v the_o product_n multiplication_n may_v be_v divide_v into_o six_o case_n case_n i._o if_o the_o multiplicand_n have_v dover_n figure_n and_o the_o multiplier_n but_o one_o rule_n draw_v the_o multiplier_n into_o the_o first_o figure_n of_o the_o multiplicand_a and_o subscribe_v the_o unit_v of_o the_o product_n but_o carry_v the_o ten_o to_o the_o next_o place_n then_o draw_v the_o multiplier_n into_o the_o second_o figure_n of_o the_o multplicand_a and_o add_v the_o ten_o you_o carry_v to_o the_o unit_v of_o that_o product_n subscribe_v the_o unit_v of_o their_o sam_n and_o carry_v the_o ten_o to_o the_o three_o place_n according_o proceed_v to_o the_o end_n of_o the_o work_n as_o if_o 5436_o be_v to_o be_v mutiply_v by_o 6_o according_a to_o the_o follow_a example_n 5436_o multiplicand_n 6_o multiplier_n 32616_o product_n case_n ii_o if_o the_o multiplicand_a and_o the_o multiplier_n have_v each_o of_o they_o more_o than_o one_o figure_n rule_n for_o the_o first_o figure_n do_v as_o before_o and_o have_v draw_v the_o second_o figure_n of_o the_o multiplier_n into_o the_o first_o figure_n of_o the_o multiplicand_a 〈◊〉_d the_o unit_v of_o the_o product_n under_o that_o second_o figure_n of_o the_o multiplier_n and_o carry_v the_o ten_o set_v all_o the_o rest_n of_o the_o multiplication_n as_o by_o the_o former_a rule_n and_o this_o direct_v make_v so_o many_o particular_a row_v of_o product_n as_o you_o figure_n in_o your_o multiplier_n at_o last_o add_v they_o together_o for_o a_o total_a product_n example_n 4532_o multiplicand_n 32_o multiplier_n 9064_o particular_a 13596_o product_n 145024_o total_a product_v case_n iii_o if_o cipher_n be_v in_o the_o multiplicand_a and_o multiplier_n or_o either_o of_o they_o rule_n set_a down_o to_o the_o right_a hand_n of_o the_o first_o product_n as_o many_o cipher_n as_o be_v in_o the_o multiplicand_a and_o multiplier_n so_o that_o the_o first_o unit_fw-la of_o the_o product_n of_o the_o first_o multiplier_n may_v stand_v under_o the_o first_o figure_n of_o the_o multiplicand_a and_o work_v the_o rest_n according_a to_o the_o other_o rule_n example_n as_o case_n iv_o if_o a_o cypher_n or_o cipher_n be_v in_o the_o middle_n of_o the_o multiplicand_a rule_n work_n according_a to_o the_o former_a rule_n till_o you_o come_v to_o the_o cipher_n then_o under_o the_o first_o o_o subscribe_v the_o ten_o you_o carry_v but_o under_o the_o rest_n of_o the_o cipher_n set_v cipher_n except_o under_o the_o last_o where_o subscribe_v the_o unit_v remain_v of_o the_o product_n of_o the_o next_o figure_n of_o the_o multiplier_n draw_v into_o the_o multiplicand_a the_o rest_n be_v according_a to_o the_o other_o rule_n example_n as_o case_n v._o if_o a_o cypher_n or_o cipher_n be_v in_o the_o middle_n of_o the_o multiplier_n rule_n multiply_v as_o before_o be_v teach_v until_o you_o come_v to_o the_o cipher_n in_o the_o multiplier_n which_o subscribe_v in_o order_n before_o the_o particular_a product_n of_o the_o next_o multipler_n draw_v into_o the_o multiplicand_a then_o set_v the_o unit_v of_o its_o product_n under_o that_o multiplier_n and_o observe_v the_o other_o rule_n for_o the_o rest_n as_o case_n vi_o if_o cipher_n be_v both_o in_o the_o middle_n of_o the_o multiplicand_a and_o also_o in_o the_o multiplier_n rule_n when_o you_o come_v to_o the_o cipher_n in_o the_o multiplicand_a then_o under_o the_o first_o cipher_n place_n set_v the_o ten_o you_o carry_v if_o any_o be_v and_o after_o that_o as_o many_o cipher_n as_o be_v in_o the_o multiplier_n no_o figure_n intervening_a then_o multiply_v into_o the_o next_o figure_n of_o the_o multiplicand_a subscribe_v the_o unit_v of_o the_o product_n and_o carry_v the_o ten_o in_o the_o same_o row_n and_o so_o do_v in_o every_o row_n of_o the_o particular_a product_n according_a as_o this_o or_o some_o of_o the_o other_o rule_v require_v example_n you_o may_v abbreviate_v multiplication_n by_o the_o help_n of_o subtraction_n especial_o when_o to_o be_v multiply_v by_o 5_o or_o 9_o as_o case_n i._o to_o multiply_v any_o number_n by_o 5._o rule_n subtract_v half_o the_o number_n and_o to_o it_o add_v a_o cypher_n example_n as_o 45276_o product_n 226380_o be_v to_o be_v multiply_v by_o 5_o halue_fw-mi the_o number_n and_o add_v a_o cypher_n at_o the_o latter_a end_n and_o the_o work_n be_v do_v case_n ii_o to_o multiply_v any_o number_n by_o 9_o rule_n add_v a_o cypher_n to_o the_o number_n give_v to_o be_v multiply_v by_o 9_o and_o subtract_v the_o first_o number_n out_o of_o it_o and_o the_o remainder_n be_v the_o product_n or_o answer_n of_o the_o question_n example_n let_v the_o number_n be_v 6789_o to_o which_o add_v o_o cypher_n and_o the_o number_n be_v thus_o 67890_o out_o of_o which_o subtract_v the_o first_o number_n and_o the_o remainder_n be_v 61101_o the_o product_n or_o answer_n of_o the_o question_n division_n division_n serve_v to_o divide_v any_o number_n into_o as_o many_o part_n as_o you_o please_v and_o consist_v of_o three_o number_n the_o divisor_n the_o dividend_n and_o the_o quotient_a for_o see_v how_o often_o the_o divisor_n be_v contain_v in_o the_o dividend_n so_o many_o figure_n it_o produce_v in_o the_o quotient_a or_o see_v how_o often_o 1_o be_v contain_v in_o the_o divisor_n so_o many_o time_n the_o quotient_n be_v contain_v in_o the_o dividend_n which_o be_v all_o one_o if_o you_o be_v to_o divide_v 888_o pound_n among_o 4_o man_n the_o question_n be_v what_o each_o man_n must_v have_v order_n your_o work_n as_o in_o this_o example_n the_o first_o demand_n be_v how_o many_o time_n 4_o can_v you_o have_v in_o 8_o the_o answer_n be_v 2_o which_o 2_o place_n in_o the_o quotient_a then_o multiply_v the_o 2_o in_o the_o quotient_a by_o 4_o the_o divisor_n and_o that_o make_v 8_o place_z 8_o under_z the_o 8_o on_o the_o least_o figure_n of_o the_o dividend_n and_o draw_v a_o line_n under_o it_o and_o subtract_v 8_o from_o 8_o and_o there_o remain_v o._n then_o take_v down_o the_o next_o 8_o and_o demand_v how_o many_o time_n the_o divisor_n be_v contain_v in_o the_o dividend_n 8_o which_o be_v 2_o time_n set_v that_o 2_o in_o the_o quotient_a and_o multiply_v the_o divisor_n 4_o by_o that_o 2_o which_o make_v 8_o set_v that_o 8_o under_o the_o second_o figure_n of_o the_o dividend_n and_o draw_v a_o line_n as_o before_o and_o subtract_v it_o from_o the_o 8_o in_o the_o dividend_n and_o there_o remain_v 0._o proceed_v in_o the_o same_o manner_n as_o you_o have_v do_v with_o the_o rest_n and_o you_o