Selected quad for the lemma: end_n

Word A Word B Word C Word D Occurrence Frequency Band MI MI Band Prominent
end_n draw_v line_n perpendicular_a 3,095 5 14.0786 5 true
View all documents for the selected quad

Text snippets containing the quad

ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A28175 A light to the art of gunnery wherein is laid down the true weight of powder, both for proof and action, of all sorts of great ordnance : also the true ball and allowance for wind, with the most necessary conclusions for the practice of gunnery, either in sea or land-service : likewise the ingredients and making of most necessary fire-works, as also many compositions for the gunner's practice, both at sea and land / by Capt. Thomas Binning ... Binning, Thomas. 1676 (1676) Wing B2934A; ESTC R4473 112,096 190

There are 3 snippets containing the selected quad. | View original text

give_v be_v ab_fw-la have_v place_v the_o one_o foot_n of_o your_o compass_n at_o a_o or_o b_o extend_v the_o other_o at_o pleasure_n above_o the_o half_a of_o the_o line_n by_o estimation_n and_o draw_v two_o arch_n the_o one_o above_o the_o line_n and_o the_o other_o beneath_o then_o keep_v the_o compass_n at_o the_o same_o wideness_n place_v the_o one_o foot_n at_o the_o other_o end_n of_o the_o line_n cut_v the_o former_a arch_n in_o c_o and_o d_o with_o the_o other_o foot_n lay_v a_o ruler_n at_o the_o point_n c_o and_o d_o and_o where_o the_o same_o do_v cut_v the_o line_n ab_fw-la as_o in_o e_o so_o shall_v the_o line_n ab_fw-la be_v cut_v into_o two_o equal_a half_n at_o e_o which_o be_v require_v and_o the_o line_n ce_fw-fr and_z ed_z shall_v be_v both_o perpendicular_a to_o the_o line_n ab_fw-la by_o the_o 19_o the_o definition_n of_o this_o prob._n 2._o a_o angle_n be_v give_v to_o divide_v the_o same_o into_o two_o half_n let_v the_o angle_n be_v abc_n to_o be_v divide_v into_o two_o equal_a half_n place_v the_o one_o foot_n of_o your_o compass_n at_o b_o and_o extend_v the_o other_o at_o your_o pleasure_n to_o d_o and_z e_z describe_v two_o arch_n cut_v each_o other_o in_o the_o point_n f_o so_o shall_v you_o divide_v the_o angle_n abc_n into_o two_o equal_a half_n if_o you_o draw_v a_o line_n from_o b_o to_o f_o with_o two_o angel_n abf_n and_o fbc_n as_o equal_a to_o the_o whole_a angle_n abc_n be_v divide_v into_o two_o half_n prob._n 3._o from_o any_o point_n or_o from_o the_o end_n of_o a_o line_n to_o erect_v or_o raise_v a_o perpendicular_a let_v there_o be_v give_v the_o line_n ab_fw-la and_o the_o point_n in_o the_o same_o give_v c_o from_o which_o a_o perpendicular_a be_v to_o be_v raise_v place_v the_o one_o foot_n of_o your_o compass_n at_o c_o and_o with_o the_o other_o take_v at_o pleasure_n above_o the_o line_n as_o d_o then_o from_o the_o point_n d_o describe_v the_o arch_n of_o a_o circle_n as_o ecf_n cut_v the_o line_n ab_fw-la in_o f_o then_o lay_v your_o ruler_n at_o f_o and_o d_o and_o where_o it_o cut_v the_o arch_n as_o in_o e_o draw_v a_o line_n from_o c_o to_o e_z which_o shall_v be_v perpendicular_a to_o ab_fw-la from_o the_o point_n c._n again_o from_o the_o point_n a_o a_o perpendicular_a be_v raise_v the_o same_o manner_n of_o way_n place_v one_o foot_n of_o your_o compass_n at_o a_o extend_v the_o other_o above_o the_o line_n to_o the_o point_n g_o and_z draw_z from_z g_z the_o arch_n jah_n cut_v the_o line_n at_o h_n then_o lay_v your_o ruler_n by_o g_z and_o h_n and_o where_o it_o cut_v the_o arch_n as_o in_o i_o from_o a_o to_o i_o draw_v a_o line_n so_o shall_v ai_fw-fr be_v a_o perpendicular_a from_o the_o end_n of_o the_o line_n ab_fw-la or_o from_o the_o point_n a._n the_o same_o may_v be_v perform_v by_o the_o line_n of_o chord_n as_o by_o the_o practice_n be_v evident_a prob._n 4._o from_o a_o point_n without_o a_o line_n to_o let_v fall_v a_o perpendicular_a to_o a_o line_n give_v let_v the_o line_n give_v be_v ab_fw-la and_o the_o point_n without_o the_o same_o at_o c_o draw_v a_o line_n to_o ab_fw-la at_o oblique_a angle_n as_o cd_o which_o you_o shall_v cut_v by_o the_o first_o problem_n in_o two_o half_n at_o e_o then_o place_v the_o one_o foot_n of_o your_o compass_n at_o e_o extend_v the_o other_o unto_o c_o drawing_z the_o arch_n cf_n and_o where_o the_o same_o cut_v the_o line_n ab_fw-la as_o in_o f_o from_o c_o to_o f_o draw_v a_o line_n as_o cf_n so_o shall_v cf_n be_v a_o perpendicular_a let_v fall_v from_o the_o point_n c_o upon_o the_o line_n ab_fw-la as_o be_v require_v prob._n 5._o a_o line_n be_v give_v to_o draw_v a_o parallel_n to_o the_o same_o at_o any_o distance_n require_v let_v the_o line_n give_v be_v ab_fw-la and_o the_o distance_n from_o c_o extend_v your_o compass_n to_o the_o length_n of_o c_o and_o place_v one_o foot_n of_o your_o compass_n in_o any_o point_n of_o the_o line_n draw_v two_o arch_n and_o by_o the_o compass_n of_o these_o arch_n draw_v a_o line_n as_o de_fw-fr which_o shall_v be_v parallel_n to_o ab_fw-la require_v as_o ce._n the_o same_o another_o way_n let_v the_o line_n give_v be_v ab_fw-la and_o the_o point_n at_o c_o from_o the_o point_n c_o draw_v a_o line_n at_o pleasure_n unto_o ab_fw-la which_o shall_v touch_v in_o f_o which_o you_o shall_v cut_v by_o the_o first_o problem_n into_o two_o half_n at_o e_o take_v then_o a_o point_n betwixt_o f_o and_o a_o at_o pleasure_n which_o be_v here_o g_o draw_v then_o a_o right_a line_n from_o the_o point_n g_o through_z e_z and_o set_v the_o distance_n de_fw-fr equal_a to_o the_o line_n eglantine_n draw_v then_o a_o line_n through_o the_o point_v cd_o so_o shall_v the_o same_o line_n be_v parallel_n to_o the_o line_n ab_fw-la prob._n 6._o two_o line_n be_v give_v to_o find_v a_o three_o proportional_a line_n let_v a_o and_o b_o be_v two_o line_n give_v and_o let_v it_o be_v require_v to_o find_v a_o three_o line_n in_o proportion_n to_o they_o first_o make_v any_o angle_n as_o cdh_n then_o set_v the_o line_n a_o from_z d_o to_z e_z and_o the_o line_n b_o from_o d_o to_o f_o and_o also_o from_o d_o to_z g_z then_o draw_v of_o that_o do_v by_o the_o point_n g_o draw_v a_o parallel_n to_o of_o as_o gh_v so_o shall_v dh_n be_v the_o three_o proportional_a line_n require_v a._n 4._o b._n 6._o de._n 4._o df._n 6._o dg_n 6._o dh_n 9_o prob._n 7._o three_o line_n be_v give_v to_o find_v a_o four_o in_o proportion_n that_o be_v to_o perform_v the_o rule_n of_o three_o in_o line_n let_v ab_fw-la and_z c_o be_v three_o line_n give_v and_o it_o be_v require_v to_o find_v a_o four_o proportional_a line_n first_o as_o in_o the_o last_o problem_n make_v any_o angle_n at_o pleasure_n as_o the_o angle_n efg_v then_o take_v with_o your_o compass_n the_o line_n a_o and_o set_v it_o from_o f_o to_o h_n take_v also_o the_o line_n b_o and_o set_v it_o from_o f_o to_o i_o and_o draw_v the_o line_n he_o that_o do_v take_v the_o three_o line_n c_o and_o set_v it_o upon_o the_o line_n fg_v viz._n always_o upon_o the_o same_o line_n where_o the_o first_o line_n a_o be_v place_v from_o f_o to_o k_o then_o by_o k_o draw_v a_o parallel_n to_o he_o as_o kl_n to_o cut_v fe_o in_o l_o so_o shall_v fl_fw-mi be_v the_o four_o proportional_a line_n require_v fh_o 12._o fi._n 14._o fk_n 18._o fl._n 21._o prob._n 8._o to_o divide_v a_o line_n give_v into_o two_o part_n in_o proportion_n one_o to_o the_o other_o according_a to_o two_o line_n give_v let_v ab_fw-la be_v a_o line_n give_v to_o be_v divide_v into_o two_o such_o part_n that_o the_o lesser_a may_v be_v in_o proportion_n to_o the_o great_a as_o the_o line_n c_o to_o the_o line_n d._n from_o the_o end_n a_o draw_v the_o line_n ae_n make_v the_o angle_n bae_n then_o set_v the_o line_n c_o from_o a_o to_o f_o and_o the_o line_n d_o from_o f_o to_o e_z and_o draw_v the_o line_n ebb_n last_o by_o the_o point_n f_o draw_v a_o parallel_n to_o ebb_v as_o fg_v to_o cut_v ab_fw-la in_o g_z so_o shall_v ab_fw-la be_v divide_v in_o g_o as_o c_z to_z d_o which_o be_v require_v af._n 20._o fe_o 30._o ag._n 16._o gb_n 24._o prob._n 9_o to_o cut_v off_o from_o a_o line_n give_v any_o part_n or_o part_n require_v let_v in_o the_o last_o diagram_n ab_fw-la be_v a_o line_n give_v and_o let_v it_o be_v require_v to_o cut_v off_o from_o it_o â…–_n part_n first_o from_o the_o end_n a_o draw_v the_o line_n ae_n make_v any_o angle_n as_o bae_n then_o set_v on_o any_o five_o equal_a part_n from_o a_o to_o e_o and_o also_o two_o of_o the_o same_o part_n from_o a_o to_o f_o that_o do_v draw_v the_o line_n ebb_n then_o by_o f_o draw_v a_o parallel_n thereunto_o to_o cut_v ab_fw-la in_o g_z so_o shall_v agnostus_n be_v the_o â…–_n part_n of_o ab_fw-la which_o be_v require_v prob._n 10._o to_o divide_v a_o line_n in_o such_o sort_n as_o another_o line_n be_v before_o divide_v this_o problem_n differ_v not_o much_o from_o the_o two_o last_o problem_n draw_v a_o right_a line_n at_o pleasure_n as_o the_o line_n ob_fw-la which_o line_n you_o shall_v divide_v into_o 100_o equal_a part_n as_o exact_o as_o you_o can_v then_o extend_v your_o compass_n to_o the_o distance_n ob_fw-la and_o describe_v a_o arch_n in_o which_o arch_n you_o be_v to_o include_v the_o desire_a line_n a_o then_o draw_v the_o line_n oc_fw-fr and_o you_o have_v the_o angle_n cob_n which_o we_o shall_v hold_v for_o a_o ground-rule_n because_o the_o leg_n ob_fw-la
and_z oc_fw-la contain_v each_o of_o they_o 100_o or_o 1000_o part_n which_o be_v in_o the_o line_n a._n for_o example_n i_o will_v have_v 35_o part_n of_o the_o line_n a_o who_o whole_a length_n make_v 100_o part_n as_o the_o line_n ob_fw-la do_v i_o number_n on_o the_o line_n ob_fw-la 35_o part_n from_o o_o to_o f_o with_o this_o distance_n i_o draw_v the_o arch_a fg_n then_o be_v the_o near_a distance_n from_o f_o to_o g_z a_o right_a line_n which_o be_v the_o 35_o part_n of_o the_o line_n a._n likewise_o if_o it_o be_v desire_v to_o have_v 15_o part_n of_o the_o line_n a_o then_o draw_v from_o o_o the_o arch_n ed_z which_o distance_n be_v the_o desire_v 15_o part_n of_o the_o line_n a_o as_o the_o learned_a adriani_n meti_fw-la do_v teach_v and_o also_o how_o to_o find_v the_o part_n of_o a_o long_a line_n than_o the_o line_n of_o measure_n itself_o be_v it_o be_v do_v after_o the_o same_o manner_n and_o be_v of_o great_a use_n to_o a_o gunner_n use_v this_o may_v be_v of_o great_a use_n if_o the_o gunner_n be_v to_o find_v the_o first_o round_a ball_n in_o a_o strange_a place_n and_o the_o weight_n unequal_a to_o we_o prob._n 11._o between_o two_o line_n give_v to_o find_v a_o mean_a proportion_n let_v a_o and_o b_o be_v two_o line_n give_v between_o the_o which_o it_o be_v require_v to_o find_v a_o mean_a proportion_n join_v the_o line_n a_o and_o b_o so_o together_o that_o they_o make_v one_o right_a line_n as_o cd_o be_v join_v together_o in_o the_o point_n e_o and_o upon_o the_o line_n cd_o describe_v the_o semicircle_n viz._n cfd_n then_o upon_o the_o point_n e_o where_o the_o line_n a_o and_z b_o be_v join_v together_o meet_v erect_v a_o perpendicular_a to_o cut_v the_o limb_n in_o f_o as_o of_o which_o shall_v be_v a_o mean_a proportion_n between_o the_o line_n a_o and_o b_o require_v the_o same_o another_o way_n again_o in_o the_o same_o diagram_n let_v the_o line_n a_o and_o g_o be_v give_v between_o the_o which_o it_o be_v require_v to_o find_v a_o mean_a proportion_n take_v the_o line_n g_o and_o lay_v it_o down_o from_o c_o to_o d_o and_o draw_v cd_o whereupon_o describe_v the_o semicircle_n cfd_n then_o take_v the_o line_n a_o and_o set_v it_o from_o d_o to_z e_z then_o upon_o the_o point_n e_o erect_v a_o perpendicular_a to_o cut_v the_o limb_n in_o f_o last_o draw_v df_n which_o shall_v be_v a_o mean_a proportion_n between_o de_fw-fr and_o dc_o or_o between_o the_o line_n a_o and_o g_o require_v and_o if_o you_o draw_v cf_n it_o shall_v be_v a_o mean_a between_o b_o and_z g_z that_o be_v between_o ce_fw-fr and_o cd_o prob._n 12._o to_o divide_v a_o line_n give_v by_o extreme_a and_o mean_a proportion_n let_v ab_fw-la be_v a_o line_n give_v to_o be_v divide_v by_o extreme_a and_o mean_a proportion_n increase_v ab_fw-la at_o length_n to_o c_o then_o upon_o the_o point_n a_o erect_v a_o perpendicular_a as_o ad_fw-la of_o the_o length_n of_o ab_fw-la that_o do_v take_v half_a ad_fw-la or_o ab_fw-la and_o set_v it_o from_o a_o to_o e_o then_z with_o the_o distance_z ed_z make_v the_o arch_a dg_n so_o shall_v ab_fw-la be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o g_o and_z ag_z be_v the_o great_a segment_n and_o gb_v the_o lesser_a the_o same_o another_o way_n prob._n 13._o the_o great_a segment_n of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n to_o find_v the_o whole_a line_n let_v ab_fw-la be_v the_o great_a segment_n give_v and_o the_o whole_a line_n be_v require_v increase_v basilius_n to_o c_o then_o upon_o the_o end_n b_o erect_v a_o perpendicular_a of_o half_a the_o length_n of_o ab_fw-la as_o bd_o and_o draw_v the_o subtendant_a side_n ad_fw-la from_o which_o subtract_v db_v rest_v of_o that_o do_v set_v of_o from_o a_o to_o c_o so_o shall_v cb_n be_v the_o whole_a line_n require_v prob._n 14._o to_o describe_v a_o circle_n upon_o any_o three_o point_n give_v not_o be_v in_o a_o right_a line_n let_v abc_n be_v three_o prob._n 14_o point_n give_v set_a one_o foot_n of_o the_o compass_n in_o the_o middle_a point_n at_o b_o and_o open_v your_o compass_n to_o any_o extension_n that_o be_v above_o one_o half_a of_o the_o distance_n between_o b_o and_o the_o far_a of_o the_o other_o two_o point_n and_o with_o that_o distance_n draw_v the_o blind_a arch_n de_fw-fr and_z fh_n with_o the_o same_o extent_n set_v one_o foot_n in_o c_o draw_v the_o arch_a fh_n again_o with_o the_o same_o extent_n set_v one_o foot_n in_o the_o point_n a_o draw_v the_o arch_n de_fw-fr than_o lay_v a_o ruler_n to_o the_o intersection_n of_o these_o arch_n draw_v the_o line_n dg_n and_o hg_n which_o will_v cross_v each_o other_o in_o the_o point_n g_o and_o there_o be_v the_o centre_n of_o the_o circle_n inquire_v where_o set_v one_o foot_n of_o your_o compass_n and_o extend_v the_o other_o to_o any_o of_o the_o three_o point_n you_o describe_v the_o arch_n of_o a_o circle_n which_o shall_v pass_v through_o the_o three_o point_n give_v and_o give_v the_o whole_a circumference_n require_v which_o have_v you_o may_v find_v the_o diameter_n by_o the_o 12_o the_o definition_n of_o chap._n 10._o of_o this_o book_n observation_n ball._n by_o a_o segment_n or_o a_o piece_n of_o a_o ball_n be_v find_v to_o know_v the_o weight_n of_o the_o whole_a ball._n here_o be_v to_o be_v observe_v that_o if_o any_o town_n fort_n or_o place_n be_v besiege_a or_o block_v up_o by_o a_o enemy_n and_o the_o enemy_n shoot_v continual_o there_o be_v divers_a segment_n or_o piece_n of_o their_o ball_n find_v in_o the_o place_n and_o it_o be_v desire_v to_o know_v what_o ordnance_n they_o do_v batter_v with_o the_o gunner_n shall_v take_v the_o segment_n or_o piece_n of_o the_o ball_n and_o lay_v it_o on_o a_o piece_n of_o paper_n and_o set_v there_o down_o by_o the_o circumference_n three_o prick_n or_o point_n as_o abc_n in_o the_o figure_n follow_v and_o thereby_o and_o by_o the_o precede_a problem_n shall_v find_v the_o diameter_n centre_n and_o circumference_n of_o the_o ball_n whereby_o you_o may_v know_v the_o weight_n of_o that_o ball_n as_o be_v describe_v in_o pag._n 54._o prob._n 15._o how_o to_o reduce_v a_o right_a line_n into_o the_o circumference_n of_o a_o circle_n the_o line_n give_v being_n ab_fw-la to_o be_v reduce_v into_o a_o circumference_n the_o which_o line_n ab_fw-la you_o shall_v first_o divide_v into_o three_o equal_a part_n and_o of_o these_o three_o part_n you_o shall_v make_v a_o equilateral_a triangle_n as_o cde_v the_o half_a of_o cd_o be_v f_o and_o the_o half_a of_o the_o side_n de_fw-fr be_v g_o and_o where_o they_o cut_v one_o another_o as_o in_o h_n that_o be_v the_o centre_n of_o the_o triangle_n cde_v further_a divide_v df_n into_o half_n with_o the_o point_n i_o draw_v then_o a_o right_a line_n from_o the_o centre_n h_n to_o i_o which_o be_v hk_n and_o divide_v the_o line_n he_o into_o four_o equal_a part_n of_o which_o you_o shall_v add_v one_o part_n unto_o he_o which_o shall_v be_v hk_n which_o be_v the_o same_o four_o part_n five_o time_n set_v set_v then_o one_o foot_n of_o your_o compass_n in_o the_o point_n h_n and_o the_o other_o in_o the_o point_n k_o and_o draw_v with_o the_o same_o distance_n a_o round_a circle_n who_o circumference_n shall_v be_v equal_a to_o the_o line_n give_v viz._n the_o line_n ab_fw-la which_o be_v require_v prob._n 16._o how_o to_o reduce_v the_o circumference_n of_o a_o circle_n give_v into_o a_o right_a line_n the_o mean_a proportion_n betwixt_o dl_o and_o agnostus_n the_o four_o part_n of_o the_o circumference_n give_v the_o side_n of_o a_o square_n equal_a to_o the_o circle_n from_o de_fw-fr subtract_v the_o half_a of_o ac_fw-la there_o remain_v dk_n which_o be_v equal_a to_o bf_n and_o be_v the_o side_n of_o a_o decagon_n inscribe_v in_o the_o circle_n prob._n 17._o the_o diameter_n of_o a_o circle_n be_v give_v to_o find_v its_o proportion_n to_o the_o circumference_n or_o the_o reverse_n land_n the_o true_a knowledge_n of_o this_o problem_n be_v of_o great_a use_n for_o gunnder_n both_o by_o sea_n and_o land_n whensoever_o the_o circumference_n of_o a_o circle_n be_v divide_v into_o 10000_o equal_a part_n then_o do_v the_o diameter_n thereof_o contain_v 3183_o of_o the_o say_a part_n say_v adriani_n meti_fw-la but_o archimedes_n show_v we_o the_o easy_a and_o fit_a for_o our_o purpose_n to_o wit_n that_o if_o the_o circumference_n be_v divide_v into_o 22_o equal_a part_n than_o the_o diameter_n do_v contain_v seven_o of_o the_o like_a part_n as_o be_v demonstrate_v in_o this_o figure_n where_o you_o may_v see_v that_o in_o the_o circle_n abc_n the_o circumference_n be_v divide_v into_o
multiply_v the_o figure_n last_o place_v in_o the_o quotient_a first_o by_o itself_o and_o then_o the_o product_n by_o the_o triple_a number_n before_o subscribe_v this_o do_v subscribe_v the_o last_o product_n under_o the_o say_v triple_a number_n to_o wit_n unite_v under_o unite_v ten_o under_o ten_o etc._n etc._n so_o 4_o being_n square_v or_o multiply_v by_o itself_o the_o product_n be_v 16_o which_o be_v multiply_v by_o the_o triple_a number_n 15_o the_o product_n be_v 240_o this_o therefore_o i_o subscribe_v under_o the_o aforesaid_a triple_a number_n 15._o subscribe_v the_o cube_n of_o the_o figure_n last_o place_v in_o the_o quotient_a under_o the_o resolvend_n in_o such_o manner_n that_o the_o first_o place_n of_o this_o cube_n to_o wit_n the_o place_n of_o unite_v may_v stand_v under_o the_o place_n of_o unite_v in_o the_o resolvend_n so_o 64_o being_n the_o cube_n of_o 4_o i_o write_v it_o under_o the_o resolvend_n 32464_o in_o such_o manner_n that_o the_o figure_n 4_o which_o be_v in_o the_o place_n of_o unite_v in_o the_o cube_n 64_o may_v stand_v under_o the_o figure_n 4_o which_o be_v seat_v in_o the_o place_n of_o unite_v of_o the_o resolvend_n draw_v yet_o another_o line_n under_o the_o work_n add_v the_o three_o last_o number_n together_o in_o the_o same_o order_n as_o they_o be_v seat_v and_o subtract_v the_o sum_n of_o they_o from_o the_o resolvend_n place_v the_o remainder_n orderly_o underneath_o so_o the_o sum_n of_o the_o three_o last_o number_n as_o they_o be_v rank_v in_o the_o work_n be_v 32464_o which_o if_o you_o subtract_v out_o of_o the_o resolvend_n 32464_o the_o remainder_n be_v 0._o thus_o the_o whole_a work_n be_v finish_v the_o cube_n root_n of_o 157464_o the_o number_n propound_v be_v find_v to_o be_v 54._o note_v 1_o when_o the_o sum_n of_o the_o three_o last_o number_n before_o mention_v be_v great_a than_o the_o resolvend_n the_o work_n be_v erroneous_a and_o then_o you_o be_v to_o reform_v it_o by_o place_v a_o lesser_a figure_n in_o the_o quotient_a note_v 2._o for_o every_o one_o of_o the_o particular_a cube_n distinguish_v by_o the_o point_n except_o the_o first_o cube_n on_o the_o left_a hand_n a_o resolvend_n be_v to_o be_v set_v apart_o by_o bring_v down_o to_o the_o remainder_n the_o next_o cube_n and_o as_o often_o as_o a_o resolvend_n be_v set_v apart_o so_o often_o be_v a_o new_a divisor_n to_o be_v find_v by_o add_v the_o triple_a of_o all_o the_o root_n in_o the_o quotient_a consist_v of_o what_o number_n of_o place_n soever_o to_o the_o triple_a of_o the_o square_n of_o such_o root_n after_o they_o be_v orderly_o place_v according_a as_o be_v abovementioned_a note_v 3_o the_o work_n of_o the_o table_n of_o simple_a cube_n in_o folio_n 9_o for_o find_v the_o first_o figure_n of_o the_o root_n as_o before_o declare_v be_v but_o once_o use_v in_o the_o extraction_n of_o the_o root_n of_o any_o number_n whatsoever_o but_o the_o work_n of_o all_o the_o follow_a rule_n be_v to_o be_v use_v for_o the_o find_n of_o every_o place_n in_o the_o root_n except_o the_o first_o the_o practice_n of_o these_o three_o note_n will_v be_v see_v when_o we_o describe_v how_o to_o extract_v the_o cube_n root_n of_o number_n not_o cubical_a chap._n ix_o another_o example_n wrought_v by_o the_o geniture_n suppose_v a_o number_n give_v to_o be_v 16387064_o of_o which_o the_o cube_n root_n be_v require_v first_o you_o must_v cut_v the_o cube_n give_v into_o ternary_n from_o the_o right_a hand_n to_o the_o left_a as_o be_v declare_v in_o chap._n 8._o then_o find_v the_o root_n of_o the_o first_o cube_n from_o the_o left_a hand_n 16._o whereof_o the_o great_a root_n be_v 2_o for_o 2_o be_v multiply_v cubical_o give_v 8_o the_o which_o 8_o be_v deduct_v from_o 16_o the_o first_o cube_n of_o the_o number_n propound_v there_o remain_v 8_o than_o set_v the_o root_n find_v 2_o with_o the_o square_a thereof_o above_o it_o and_o by_o the_o same_o the_o geniture_n and_o then_o find_v a_o second_o figure_n for_o the_o root_n of_o the_o second_o cube_n and_o you_o shall_v have_v 5_o which_o you_o shall_v set_v down_o with_o its_o square_a and_o cube_n under_o it_o right_a against_o the_o geniture_n towards_o the_o right_a hand_n then_o multiply_v each_o one_o by_o another_o and_o add_v the_o product_n together_o there_o come_v 7625_o which_o be_v subtract_v from_o 8387_o there_o do_v remain_v 762_o in_o the_o same_o manner_n find_v a_o root_n for_o the_o number_n remain_v to_o be_v extract_v and_o it_o shall_v be_v the_o root_n of_o your_o three_o cube_n and_o the_o example_n will_v stand_v thus_o and_o see_v there_o remain_v nothing_o it_o be_v manifest_a that_o the_o number_n propound_v 16387064_o be_v a_o cube_n number_n and_o the_o root_n thereof_o be_v 254._o by_o the_o 4_o the_o proposition_n of_o the_o second_o and_o 20_o the_o definition_n of_o the_o seven_o book_n of_o euclid_n when_o we_o come_v to_o calculate_v the_o table_n of_o cube_n by_o which_o you_o may_v make_v a_o inch_n rule_n for_o height_n or_o line_n of_o diameter_n the_o way_n shall_v be_v describe_v how_o to_o extract_v the_o cube_n root_n of_o number_n not_o cubical_a or_o as_o they_o be_v term_v irrational_a number_n from_o which_o no_o true_a root_n can_v be_v obtain_v yet_o many_o time_n the_o error_n will_v not_o be_v ●_o 100000_o part_n of_o a_o unite_n chap._n x._o principle_n of_o geometry_n there_o be_v divers_a reason_n which_o make_v i_o to_o give_v these_o few_o principle_n of_o geometry_n because_o the_o whole_a work_n of_o this_o book_n be_v either_o to_o be_v do_v by_o arithmetic_n or_o geometry_n and_o beside_o that_o a_o gunner_n can_v obtain_v to_o know_v the_o truth_n of_o a_o diameter_n of_o a_o ball_n except_o he_o can_v geometrical_o extract_v the_o wind_n of_o the_o boar_n of_o the_o piece_n and_o thereby_o find_v the_o ball_n fit_v such_o a_o piece_n and_o in_o general_n will_v be_v most_o useful_a for_o any_o gunner_n definition_n 1._o a_o point_n or_o prick_v be_v that_o which_o be_v the_o least_o of_o all_o material_n and_o it_o be_v the_o begin_n of_o thing_n as_o be_v void_a of_o length_n breadth_n and_o depth_n have_v neither_o part_n nor_o quantity_n expressible_a in_o number_n and_o therefore_o it_o admit_v of_o no_o division_n but_o that_o which_o be_v mental_a only_o this_o point_n or_o prick_n be_v represent_v unto_o you_o by_o the_o letter_n a._n thus_o a._n 2._o a_o line_n be_v a_o magnitude_n extend_v itself_o in_o length_n without_o breath_n or_o thickness_n whether_o it_o be_v a_o straight_a line_n or_o crooked_a and_o in_o respect_n of_o its_o length_n may_v be_v divide_v into_o part_n but_o will_v admit_v of_o no_o other_o division_n but_o in_o length_n only_o as_o be_v set_v forth_o to_o you_o by_o the_o line_n bc_n the_o extremity_n whereof_o be_v point_n as_o b_o and_z c._n 3._o a_o right_a or_o straight_a line_n be_v the_o near_a distance_n that_o can_v be_v betwixt_o two_o point_n as_o be_v the_o former_a line_n bc._n 4._o circular_a or_o crooked_a line_n be_v long_o though_o they_o be_v extend_v no_o further_o than_o right_a line_n as_o be_v the_o line_n de_fw-fr and_o fg._n 5._o rightlined_n parallel_n be_v two_o straight_a line_n so_o draw_v as_o they_o be_v equi-distant_a in_o all_o place_v one_o from_o the_o other_o so_o that_o although_o they_o be_v infinite_o extend_v yet_o can_v they_o never_o meet_v as_o may_v be_v see_v by_o the_o line_n he_o and_o kl_n 6._o a_o circular_a parallel_n be_v a_o circle_n draw_v either_o within_o or_o without_o another_o circle_n upon_o one_o and_o the_o same_o centre_n as_o be_v see_v by_o the_o two_o circle_n viz._n nopq_n and_o rstv_n be_v both_o draw_v upon_o the_o same_o centre_n w_n and_o therefore_o be_v parallel_v one_o to_o another_o 7._o a_o superficies_n be_v the_o second_o kind_n of_o quantity_n and_o to_o it_o be_v attribute_v two_o dimension_n length_n and_o breadth_n but_o not_o thickness_n for_o a_o superficies_n be_v the_o term_n or_o end_n of_o a_o body_n as_o a_o line_n be_v the_o end_n and_o term_v of_o a_o superficies_n as_o wxyz_n be_v a_o superficies_n 8._o the_o extreme_n of_o a_o superficies_n be_v line_n as_o the_o end_n limit_n or_o bound_n of_o a_o line_n be_v point_n confine_v the_o line_n so_o be_v line_n the_o limit_n bound_n and_o end_n enclose_v a_o superficies_n as_o in_o the_o forego_n figure_n you_o may_v see_v the_o superficies_n enclose_v with_o four_o line_n viz._n yw_n wx_n xz_n and_o yz_n which_o be_v the_o extreme_n or_o limit_n thereof_o 9_o a_o plain_a superficies_n be_v that_o which_o lie_v equal_o between_o his_o line_n like_v as_o a_o right_a line_n be_v the_o short_a extension_n or_o draught_n from_o one_o