chap._n 3._o of_o spherical_a triangle_n pag._n 13._o chap._n 4._o of_o the_o projection_n of_o the_o sphere_n in_o plano_fw-la pag._n 20._o chap._n 5._o of_o planometry_n and_o the_o centre_n of_o gravity_n pag._n 23._o ch._n 6._o of_o solid_a geometry_n p._n 29._o chap._n 7._o of_o the_o scale_n of_o ponderosity_n alias_o the_o stillyard_n p._n 43_o chap._n 8._o of_o the_o 4_o compendium_n for_o quadratique_a equation_n pag._n 45._o chap._n 9_o of_o recreative_a problem_n pag._n 47._o dary_n miscellany_n chap._n i._o of_o the_o inscription_n and_o circumscription_n of_o a_o circle_n 1._o forasmuch_o as_o the_o ratio_fw-la of_o a_o arch_a line_n to_o a_o right_a line_n be_v yet_o unknown_a it_o be_v absolute_o necessary_a that_o right_a line_n be_v apply_v to_o a_o circle_n for_o the_o calculation_n of_o triangle_n wherein_o arch_a line_n come_v in_o competition_n 2._o right_o line_n apply_v to_o a_o circle_n be_v chord_n sin_n tangent_n secant_v and_o verse_v sin_n 3._o the_o chord_n of_o a_o arch_n be_v a_o right_a line_n extend_v from_o one_o end_n of_o that_o arch_n to_o the_o other_o end_n thereof_o the_o sine_fw-la be_v a_o right_a line_n draw_v from_o one_o end_n of_o that_o arch_n perpendicular_o upon_o the_o diameter_n draw_v from_o the_o other_o end_n of_o that_o arch_n the_o tangent_fw-la be_v a_o right_a line_n touch_v one_o end_n of_o that_o arch_n extend_v till_o it_o concur_v with_o the_o secant_fw-la the_o secant_fw-la be_v a_o right_a line_n extend_v from_o the_o centre_n of_o the_o circle_n till_o it_o concur_v with_o the_o tangent_fw-la the_o verse_v sine_fw-la be_v a_o right_a line_n be_v a_o segment_n of_o the_o diameter_n draw_v from_o one_o end_n of_o that_o arch_n till_o it_o be_v cut_v by_o a_o perpendicular_a i._n e._n the_o sine_fw-la from_o the_o other_o end_n of_o that_o arch._n 4._o it_o be_v to_o be_v note_v by_o this_o definition_n in_o prop._n 3._o that_o the_o chord_n of_o a_o arch_n be_v common_a to_o two_o arch_n one_o of_o they_o be_v the_o compliment_n of_o the_o other_o to_o a_o whole_a circle_n and_o likewise_o the_o verse_v sine_fw-la be_v common_a to_o two_o arch_n one_o of_o they_o be_v the_o compliment_n of_o the_o other_o to_o a_o whole_a circle_n but_o the_o sine_fw-la of_o a_o arch_n be_v common_a to_o two_o arch_n one_o of_o they_o be_v the_o compliment_n of_o the_o other_o to_o a_o semicircle_n 5._o as_o the_o sum_n of_o two_o sines_n be_v to_o their_o difference_n so_o be_v the_o tangent_fw-la of_o the_o ½_n sum_n of_o those_o arch_n to_o the_o tangent_fw-la of_o their_o ½_n difference_n 6._o as_o the_o sum_n of_o two_o tangent_n be_v to_o their_o difference_n so_o be_v the_o sine_fw-la of_o the_o sum_n of_o those_o arch_n to_o the_o sine_fw-la of_o their_o difference_n 7._o as_o the_o sine_fw-la of_o the_o sum_n of_o two_o arch_n be_v to_o the_o sum_n of_o their_o sin_n so_o be_v the_o difference_n of_o those_o sin_n to_o the_o sine_fw-la of_o their_o difference_n 8._o if_o you_o put_v r_o =_o the_o radiu_o s_o of_o a_o circle_n a_o =_o a_o arch_n propose_v c_o =_o the_o chord_n of_o that_o arch_n s_o =_o the_o sine_fw-la of_o that_o arch_n t_o =_o the_o tangent_fw-la of_o that_o arch_n and_o z_o =_o the_o secant_fw-la of_o that_o arch._n then_o 9_o if_o twice_o three_o arch_n equi-different_a be_v propose_v then_o as_o the_o sine_fw-la of_o one_o of_o the_o mean_n be_v to_o the_o sum_n of_o the_o sin_n of_o its_o extreme_n so_o be_v the_o sine_fw-la of_o the_o other_o mean_v to_o the_o sum_n of_o the_o sin_n of_o its_o extreme_n 10._o and_o hence_o if_o a_o rank_n of_o arch_n be_v equi-different_a as_o the_o sine_fw-la of_o any_o arch_n in_o that_o rank_n be_v to_o the_o sum_n of_o the_o sin_n of_o any_o two_o arch_n equal_o remote_a from_o it_o on_o each_o side_n so_o be_v the_o sine_fw-la of_o any_o other_o arch_n in_o the_o say_a rank_n to_o the_o sum_n of_o the_o sin_n of_o two_o arch_n next_o to_o it_o on_o each_o side_n have_v the_o same_o common_a distance_n 11._o three_o arch_n equi-different_a be_v propose_v if_o you_o put_v z_o =_o the_o sine_fw-la of_o the_o great_a extreme_a y_fw-fr =_o the_o sine_fw-la of_o the_o lesser_a extreme_a m_n =_o the_o sine_fw-la of_o the_o mean_a m_o =_o the_o cousin_a thereof_o d_o =_o the_o sine_fw-la of_o the_o common_a difference_n d_o =_o the_o cousin_a thereof_o and_o r_o =_o the_o radius_fw-la 12._o from_o the_o last_o before_o go_v it_o be_v evident_a that_o if_o two_o three_o i._o e_o either_o the_o former_a or_o the_o latter_a 60_o deg_fw-la or_o the_o former_a 30_o deg_fw-la and_o the_o latter_a 30_o deg_fw-la of_o the_o quadrant_n be_v complete_v with_o sines_n the_o remain_a three_o part_n of_o the_o quadrant_n maybe_o complete_v by_o addition_n or_o subduction_n only_o 13._o if_o in_o a_o circle_n two_o right_a line_n be_v inscribe_v cut_v each_o other_o the_o rectangle_v of_o the_o segment_n of_o each_o line_n be_v equal_a and_o the_o angle_n at_o the_o point_n of_o intersection_n be_v measure_v by_o the_o half-sum_n of_o its_o intercept_a arch_n 14._o if_o to_o a_o circle_n two_o right_a line_n be_v adscribe_v from_o a_o point_n without_o the_o rectangle_v of_o each_o line_n from_o the_o point_n assign_v to_o the_o convex_a and_o concave_a be_v equal_a and_o the_o angle_n at_o the_o assign_a point_n be_v measure_v by_o the_o half_a difference_n of_o its_o intercept_a arch_n 15._o if_o in_o a_o circle_n or_o a_o elipsis_n three_o right_a line_n shall_v be_v inscribe_v one_o of_o they_o cut_v the_o other_o two_o then_o the_o rectangle_v of_o the_o segment_n of_o each_o line_n so_o cut_v be_v direct_v proportional_a to_o the_o rectangle_v of_o the_o respective_a segment_n of_o of_o the_o cutter_n 16._o if_o a_o plain_a triangle_n be_v inscribe_v in_o a_o circle_n the_o angle_n be_v one_o half_a of_o what_o their_o opposite_a side_n do_v subtend_v 17._o therefore_o the_o angle_n of_o a_o plain_a triangle_n be_v equal_a to_o a_o semicircle_n 18._o and_o hence_o if_o a_o rectangled_a triangle_n be_v inscribe_v in_o a_o circle_n the_o hypothenuse_n thereof_o be_v the_o diameter_n of_o the_o circle_n 19_o as_o the_o diameter_n of_o a_o circle_n be_v to_o the_o chord_n of_o a_o arch_n so_o be_v that_o chord_n to_o the_o verse_v sine_fw-la of_o that_o arch._n 20._o and_o hence_o if_o from_o the_o right_a angle_n of_o a_o rectangled_a triangle_n a_o perpendicular_a be_v let_v fall_n upon_o the_o hypothenuse_n the_o hypothenuse_n be_v thereby_o cut_v according_a to_o the_o ratio_fw-la of_o the_o square_n of_o the_o side_n 21._o if_o in_o a_o circle_n any_o plain_a triangle_n be_v inscribe_v and_o a_o perpendicular_a be_v let_v fall_n upon_o one_o of_o the_o side_n from_o the_o opposite_a angular_a point_n then_o as_o that_o perpendicular_a be_v to_o one_o of_o the_o adjacent_a side_n so_o be_v the_o other_o adjacent_a side_n to_o the_o diameter_n of_o the_o circumscribr_a circle_n 22._o if_o a_o circle_n be_v inscribe_v within_o a_o plain_a triangle_n then_o as_o the_o perimeter_n be_v to_o the_o perpendicular_a so_o be_v the_o base_a on_o which_o it_o fall_v to_o the_o radius_fw-la of_o the_o inscribe_v circle_n 23._o if_o a_o quadrilateral_a figure_n be_v inscribe_v in_o a_o circle_n and_o interfect_a with_o diagonals_n the_o rectangle_n of_o the_o diagonals_n be_v equal_a to_o the_o two_o rectangle_v of_o the_o opposite_a side_n 24._o if_o a_o circle_n be_v both_o inscribe_v and_o circumscribe_v by_o two_o like_o ordinate_a polligon_n then_o as_o the_o co-versed_n sine_fw-la of_o the_o side_n of_o the_o inscribe_v be_v to_o the_o diameter_n so_o be_v the_o area_n of_o the_o inscribe_v to_o the_o area_n of_o the_o circumscribe_v 25._o if_o a_o ordinate_a polligon_n be_v both_o inscribe_v and_o circumscribe_v by_o two_o circle_n then_o as_o the_o diameter_n of_o the_o circumscribe_v be_v to_o the_o co-versed_n sine_fw-la of_o the_o side_n of_o the_o polligon_n so_o be_v the_o area_n of_o the_o circumscribe_v to_o the_o area_n of_o the_o inscribe_v 26._o in_o any_o right_n line_v figure_n if_o a_o circle_n be_v inscribe_v then_o as_o the_o peripheria_fw-la of_o the_o circle_n be_v to_o the_o area_n thereof_o so_o be_v the_o perimeter_n of_o the_o right_n line_v figure_n to_o the_o area_n thereof_o et_fw-la contempt_n 27._o but_o in_o all_o circle_n as_o the_o peripheria_fw-la be_v to_o the_o area_n so_o be_v 2_o to_o the_o radius_fw-la 28._o therefore_o in_o any_o right_n line_v figure_n if_o a_o circle_n be_v inscribe_v as_o 2._o be_v to_o the_o radius_fw-la so_o be_v the_o perimeter_n of_o the_o right_n line_v figure_n to_o the_o area_n thereof_o chap._n ii_o of_o plain_a triangle_n 1._o a_o triangle_n be_v a_o figure_n comprehend_v of_o three_o side_n and_o
whereas_o my_o privilege_n be_v grant_v in_o scotland_n on_o the_o first_o of_o december_n 1698._o yet_o his_o come_v so_o far_o short_a of_o i_o that_o i_o very_o believe_v have_v he_o see_v or_o get_v a_o perfect_a account_n of_o i_o before_o he_o propose_v his_o own_o he_o will_v have_v spare_v the_o pain_n of_o publication_n i_o must_v confess_v that_o for_o any_o thing_n i_o yet_o know_v his_o table_n or_o circle_n for_o multiplication_n and_o division_n which_o indeed_o be_v very_o ingenious_a and_o have_v cost_v he_o much_o thought_n be_v his_o own_o as_o also_o his_o table_n for_o the_o reduction_n of_o penny_n to_o shilling_n &_o shilling_n to_o pound_n but_o in_o that_o part_n which_o be_v common_a with_o his_o rove_n and_o my_o rotula_n he_o seem_v to_o have_v get_v some_o hint_n of_o i_o and_o this_o i_o be_o the_o more_o apt_a to_o belive_o because_o about_o the_o time_n that_o i_o be_v busy_a in_o contrive_v the_o rotula_n there_o be_v a_o very_a smart_n gentleman_n a_o near_a friend_n of_o he_o scholar_n with_o i_o at_o stirline_n but_o that_o which_o give_v i_o great_a evidence_n in_o this_o particular_a be_v some_o expression_n in_o his_o own_o book_n which_o make_v i_o fancy_v that_o he_o have_v at_o least_o get_v some_o imperfect_a description_n of_o i_o before_o he_o contrive_v that_o part_n of_o he_o which_o serve_v for_o addition_n and_o substraction_n for_o whereas_o there_o be_v on_o my_o fix_a plate_n 3_o circle_n he_o speak_v of_o three_o and_o yet_o immediate_o he_o take_v away_o two_o of_o he_o &_o turn_v they_o into_o table_n for_o reduce_v of_o penny_n into_o shilling_n and_o shilling_n into_o pound_n &_o these_o not_o exceed_v the_o limit_n of_o 120._o and_o instead_o of_o the_o three_o on_o the_o fix_a he_o give_v we_o nothing_o but_o a_o little_a segment_n about_o a_o five_o part_n divide_v into_o part_n begin_n at_o 0._o and_o end_v at_o 24._o which_o he_o call_v his_o fix_a index_n as_o also_o whereas_o my_o circle_n be_v divide_v into_o 100_o part_n he_o choose_v to_o make_v his_o differ_v from_o i_o 120._o as_o be_v a_o common_a product_n of_o 10._o 12._o and_o 20._o these_o number_n as_o he_o allege_n in_o the_o begin_n of_o his_o one_a chapter_n be_v preferable_a to_o all_o other_o number_n whatsoever_o and_o yet_o near_o the_o close_a of_o the_o same_o chapter_n he_o acknowledge_v that_o it_o will_v be_v better_a to_o divide_v the_o circle_n for_o addition_n and_o substraction_n into_o 100_o part_n or_o some_o power_n of_o 10._o and_o so_o the_o instrument_n will_v become_v universal_a all_o which_o give_v i_o suspicion_n that_o in_o this_o part_n he_o have_v goten_a at_o least_o some_o lame_a account_n of_o i_o moreover_o his_o instrument_n be_v defective_a and_o come_v far_o short_a of_o i_o even_o in_o addition_n for_o in_o his_o the_o practitioner_n be_v oblige_v to_o mind_n or_o mark_v down_o how_o many_o revolution_n his_o movable_a plate_n make_v and_o every_o one_o be_v 120_o he_o have_v 120_o to_o multiply_v by_o the_o number_n of_o revolution_n which_o be_v not_o only_o troublesome_a but_o likeways_o dangerous_a especial_o in_o real_a bussiness_n where_o a_o man_n who_o mind_n be_v bussy_v both_o about_o the_o figure_n of_o his_o column_n and_o the_o point_n of_o his_o movable_a plate_n be_v oblige_v at_o the_o same_o time_n to_o mind_n the_o several_a revolution_n of_o his_o movable_a plate_n of_o which_o for_o every_o one_o he_o forget_v or_o overlook_v he_o lose_v 120_o for_o his_o pain_n whereas_o in_o my_o a_o man_n be_v not_o tie_v to_o any_o such_o intention_n above_o once_o for_o 1000_o which_o be_v more_o than_o any_o column_n do_v ordinary_o contain_v the_o movable_a plate_n at_o every_o revolution_n both_o mark_v and_o give_v notice_n of_o the_o number_n of_o revolution_n but_o beside_o this_o in_o my_o rotula_n the_o same_o circle_n that_o serve_v for_o addition_n and_o substraction_n serve_v likeways_o for_o multiplication_n and_o division_n but_o in_o his_o rove_n he_o have_v one_o for_o addition_n and_o substraction_n and_o ten_o or_o eleven_o for_o multiplication_n and_o division_n and_o yet_o though_o the_o circle_n be_v twice_o as_o large_a and_o though_o they_o contain_v near_o twice_o as_o many_o figure_n as_o they_o do_v they_o will_v be_v no_o more_o than_o what_o be_v necessary_a to_o do_v what_o i_o be_o able_a to_o perform_v by_o i_o last_o whereas_o his_o table_n be_v confine_v only_o to_o shilling_n and_o penny_n and_o these_o of_o limit_a number_n not_o exceed_v 120._o there_o be_v on_o the_o waste_n on_o the_o middle_n of_o my_o movable_a plate_n table_n for_o the_o reduction_n of_o shilling_n penny_n farthing_n weight_n and_o measure_n be_v the_o number_n never_o so_o large_a beside_o the_o decimall_n table_n for_o money_n weight_n measure_n and_o the_o most_o ordinary_a common_a fraction_n by_o the_o help_n of_o which_o six_o last_v sort_n of_o table_n the_o multiplication_n and_o division_n of_o complex_n number_n do_v become_v just_a as_o easy_a as_o that_o of_o integer_n without_o all_o that_o tediousness_n which_o mr_n glover_n propose_v in_o his_o book_n to_o conclude_v what_o i_o have_v say_v here_o be_v no_o more_o than_o be_v necessary_a for_o the_o vindication_n of_o my_o own_o invention_n and_o to_o satisfy_v those_o who_o already_o be_v or_o hereafter_o may_v be_v misinform_v either_o by_o the_o story_n of_o delamain_n or_o mr._n glover_n rove_n arithmetic_n who_o for_o what_o be_v peculliar_o he_o deserve_v a_o good_a degree_n of_o commendation_n and_o encouragement_n chap._n i._o concern_v the_o rotula_n and_o the_o rectification_n thereof_o albeit_o in_o book_n of_o this_o nature_n it_o be_v usual_a to_o prefix_v a_o scheme_n of_o the_o machine_n of_o which_o they_o treat_v yet_o i_o have_v think_v fit_a in_o this_o to_o omit_v that_o because_o such_o as_o have_v a_o rotula_n need_v not_o a_o scheme_n and_o such_o as_o want_v one_o have_v no_o use_n for_o a_o book_n i_o shall_v therefore_o as_o brief_o as_o i_o can_v describe_v the_o rotula_n and_o then_o show_v you_o how_o to_o use_v it_o the_o rotula_n consist_v of_o two_o principal_a part_n to_o wit_n a_o circular_a plain_n move_v upon_o a_o center-pin_n this_o we_o call_v the_o movable_a plate_n and_o a_o ring_n who_o circle_n be_v describe_v from_o the_o same_o centre_n this_o we_o call_v the_o fix_a plate_n because_o it_o be_v fix_v to_o the_o box_n to_o secure_v it_o from_o move_v about_o the_o centre_n as_o the_o other_o do_v the_o fix_a plate_n be_v divide_v into_o three_o part_n by_o two_o circle_n the_o innermost_a of_o which_o be_v double_v with_o a_o little_a interstice_n for_o peg-holes_n near_o the_o circumference_n of_o the_o movable_a there_o be_v another_o double_a circle_n with_o a_o small_a interstice_n also_o betwixt_o they_o for_o peg_n holes_n the_o space_n without_o the_o double_a circle_n on_o the_o movable_a and_o within_o that_o on_o the_o fix_a be_v both_o of_o they_o equal_o divide_v into_o 100_o part_n and_o both_o be_v number_v begin_v at_o 0._o 1._o 2._o 3._o and_o so_o proceed_v in_o a_o natural_a order_n to_o 99_o all_o the_o division_n be_v draw_v straight_o from_o the_o centre_n on_o the_o fix_v many_o of_o these_o division_n be_v protract_v some_o only_a to_o the_o middle_a part_n and_o other_o run_v over_o both_o for_o confine_v the_o several_a single_a coefficents_n of_o the_o respective_a tabular_a number_n to_o which_o they_o be_v prefix_v with_o this_o caution_n when_o the_o coefficient_o be_v the_o same_o they_o be_v set_v down_o in_o the_o uttermost_a part_n and_o when_o any_o number_n admit_v of_o two_o pair_n of_o coefficient_o the_o one_o pair_n be_v set_v in_o the_o midmost_fw-mi and_o the_o other_o in_o the_o out-most_a part._n thus_o against_o 18._o on_o the_o fix_a you_o will_v find_v in_o the_o midmost_fw-mi part_n 2_o ×_o 9_o that_o be_v two_o time_n nine_o or_o nine_o time_n two_o this_o ×_o cross_n signifie_v the_o word_n time_n and_o 3_o ×_o 6_o in_o the_o outmost_a also_o on_o the_o movable_a there_o be_v a_o segment_n of_o a_o circle_n within_o the_o peg-hole_n circle_n begin_v at_o 9_o of_o the_o natural_a number_n and_o end_n at_o 72._o this_o segment_n be_v likewise_o divide_v by_o the_o same_o line_n that_o divide_v the_o outmost_a circle_n of_o natural_a number_n into_o equal_a part_n on_o the_o fix_a plate_n at_o the_o division_n betwixt_o 99_o and_o 0_o there_o be_v a_o little_a bit_n of_o metal_n screw_v or_o rivet_v reach_v likeways_o a_o lit_fw-fr le_fw-fr further_o than_o the_o peg-hole_n circle_n on_o the_o movable_a this_o piece_n of_o metal_n we_o call_v the_o stop_n and_o must_v always_o be_v place_v next_o your_o left_a hand_n with_o the_o number_n 25_o or_o 30._o towards_o your_o
yield_v the_o total_a product_n i_o shall_v subjoin_v another_o example_n and_o so_o end_v with_o multiplication_n in_o this_o last_o example_n the_o two_o cipher_n of_o the_o multiplier_n be_v set_v to_o the_o right_n of_o the_o unite_v of_o the_o multiplicand_a and_o then_o multiply_v by_o 9_o i_o set_v the_o two_o cipher_n behind_o the_o product_n and_o so_o what_o be_v but_o 9_o time_n before_o do_v now_o become_v 900_o time_n the_o multiplicand_v you_o see_v also_o the_o unite_v of_o the_o product_n make_v by_o 7_o set_z under_z 7_o of_o the_o multiplier_n and_o the_o unite_v of_o that_o make_v by_o 3_o under_z 3_o of_o the_o multiplier_n all_o the_o rest_n due_o observe_v rank_a and_o file_n to_o conclude_v this_o chapter_n and_o make_v you_o prompt_v in_o find_v your_o coefficient_o observe_v that_o all_o the_o product_n of_o any_o coefficient_o be_v contain_v within_o ten_o time_n the_o least_o of_o the_o two_o so_o that_o all_o the_o product_n of_o 2_o be_v within_o 20_o and_o 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