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end_n draw_v line_n perpendicular_a 3,095 5 14.0786 5 true
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ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A42708 Syntaxis mathematica, or, A construction of the harder problemes of geometry with so much of the conicks as is therefore requisite and other more ordinary and usefull propositions inter-mixed, and tables to several purposes / by Tho. Gibson. Gibson, Thomas, 17th/18th cent. 1665 (1665) Wing G677; ESTC R28671 95,056 272

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require_v shall_v give_v respective_a point_n enough_o in_o each_o hour_n to_o draw_v each_o parallel_n by_o example_n in_o the_o latitude_n 51._o 32′_n the_o sun_n be_v in_o pisces_fw-la the_o beginning_n thereof_o the_o degree_n of_o the_o sun_n height_n above_o the_o horizon_n at_o every_o hour_n be_v as_o follow_v that_o be_v 25._o 37′_n at_o one_o of_o clock_n 21._o 49′_n at_o two_o 15._o 57′_n at_o three_o 8._o 32′_n at_o four_o and_o the_o same_o for_o eight_o nine_o ten_o and_o eleven_o respective_o if_o the_o perpendicular_a stile_n be_v radius_fw-la the_o tangent_n of_o the_o compliment_n of_o 25._o 37′_n 21._o 49′_n 15._o 57′_n 8._o 32′_n be_v apply_v from_o the_o foot_n of_o the_o stile_n to_o the_o respective_a hour_n that_o be_v the_o co-tangent_a of_o 25._o 37′_n from_o the_o foot_n of_o the_o stile_n to_o the_o hour_n of_o 1._o and_o 11._o and_o so_o the_o other_o they_o shall_v give_v point_n in_o every_o hour-line_n one_o by_o which_o a_o line_n be_v draw_v with_o a_o 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consider_v not_o full_o because_o the_o centre_n and_o transverse_a diameter_n of_o the_o ellipsis_n lie_v within_o and_o of_o the_o hyperbola_n without_o the_o section_n and_o if_o h_z or_o any_o point_n within_o a_o section_n be_v give_v and_o require_v through_o it_o to_o draw_v a_o ordinate_a that_o may_v be_v easy_o do_v because_o it_o must_v be_v parallel_n to_o a_o tangent_fw-la at_o the_o vertex_fw-la a._n any_o section_n give_v to_o find_v that_o diameter_n thereof_o which_o shall_v make_v a_o angle_n with_o the_o ordinate_a to_o it_o equal_a to_o a_o angle_n give_v if_o first_o the_o section_n give_v be_v a_o parabola_fw-la find_v any_o diameter_n and_o from_o the_o end_n or_o vertex_fw-la thereof_o draw_v a_o right_a line_n to_o the_o section_n make_v a_o angle_n with_o the_o say_a diameter_n equal_a to_o the_o angle_n give_v to_o which_o if_o a_o parallel_n through_o the_o middle_n of_o the_o other_o right_a line_n be_v draw_v that_o parallel_n be_v the_o diameter_n require_v let_v there_o be_v give_v therefore_o the_o hyperbola_fw-la bac_fw-la and_o the_o angle_n z_o to_o find_v the_o diameter_n egg_n which_o with_o the_o ordinate_a of_o shall_v make_v the_o angle_n ega_fw-la =_o z._n find_v the_o transverse_n axis_fw-la ad_fw-la and_o the_o centre_n e_o and_o upon_o ad_fw-la describe_v by_o the_o 33._o of_o the_o 3._o of_o euclid_n a_o portion_n of_o a_o circle_n dfa_fw-mi capable_a of_o a_o angle_n equal_a to_o z_o then_z draw_z df_n and_o of_o and_o through_o the_o middle_n of_o of_o draw_v egg_n the_o diameter_n require_v the_o work_n be_v altogether_o the_o same_o in_o a_o ellipsis_n only_o the_o lesser_a axis_fw-la be_v to_o be_v use_v midor_n 3.67_o any_o hyperbola_n be_v give_v to_o find_v the_o asymptoti_n because_o ah_o touch_v the_o section_n it_o be_v equidistant_a to_o the_o ordinates_n per_fw-la coral_n 2_o ad_fw-la 17_o primi_fw-la but_o to_o the_o rectangle_n or_o parallelogram_n make_fw-mi that_o be_v to_o the_o figure_n comprehend_v of_o the_o two_o side_n ma_fw-fr and_o ag_n be_v make_v equal_a the_o square_n or_o rhombus_fw-la of_o ah_o and_o a_o be_v half_a of_o ah_o therefore_o the_o square_a or_o rhombus_fw-la of_o a_o be_v equal_a to_o a_o four_o part_n of_o the_o square_a or_o rhombus_fw-la of_o ah_o that_o be_v to_o the_o quadrant_a of_o the_o figure_n make_fw-mi and_o therefore_o by_o the_o 38._o of_o the_o first_o and_o coral_n to_o it_o by_o conversion_n it_o may_v be_v show_v that_o the_o right_a line_n en_fw-fr draw_v from_o the_o centre_n and_o produce_v how_o far_o soever_o shall_v never_o meet_v with_o the_o section_n bac_fw-mi and_o by_o the_o same_o reason_n and_o because_o a_o =_o ao_o eo_fw-la draw_v from_o the_o centre_n shall_v do_v the_o like_a etc._n etc._n from_o hence_o it_o appear_v that_o the_o asymptotes_n be_v line_n draw_v from_o the_o centre_n of_o the_o section_n and_o produce_v so_o as_o that_o incline_n towards_o the_o section_n still_o more_o shall_v never_o be_v 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17_o in_o 3._o 18_o in_o 2._o and_o 19_o in_o 1._o prop._n 62._o lib._n 2._o and_o they_o differ_v just_a as_o the_o square_a number_n immediate_o succeed_v to_o unity_n viz._n 1_o 4_o 9_o 16_o 25_o 36_o 49_o 64_o 81_o 100_o &c_n &c_n by_o the_o quantity_n of_o the_o odd_a number_n intercept_v as_o 1_o 3_o 5_o 7_o 9_o 11_o 13_o 15_o 17_o etc._n etc._n which_o be_v the_o same_o proportion_n by_o which_o the_o degree_n of_o velocity_n of_o the_o fall_n of_o any_o thing_n towards_o the_o centre_n of_o the_o earth_n be_v increase_v as_o galileo_n have_v sufficient_o prove_v in_o his_o dialogue_n and_o therefore_o the_o course_n of_o every_o projectile_a or_o thing_n shoot_v from_o gun_n or_o bow_n may_v easy_o be_v prove_v to_o be_v a_o parabolical_a line_n and_o the_o make_v a_o rectilone_n figure_n equal_a to_o a_o parabola_fw-la may_v be_v facilitated_a from_o hence_o if_o it_o be_v not_o needless_a the_o thing_n be_v already_o often_o do_v moreover_o it_o be_v to_o be_v note_v that_o 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the_o 26._o of_o the_o 2._o of_o midorgius_n for_o a_o ellipsis_n numerical_o the_o burn_a point_n and_o vertices_fw-la be_v give_v as_o they_o be_v before_o the_o ellipsis_n also_o may_v be_v describe_v by_o number_n as_o follow_v let_v the_o one_o burn_a point_n be_v at_o a_o the_o other_z at_z b_o and_o let_v the_o diameter_n be_v z_o the_o distance_n betwixt_o a_o and_o b_o let_v that_o be_v x_o equal_a to_o 100_o and_o let_v it_o be_v x″_n 16′_n 100″_n therefore_o also_o z_o =_o 232_o wherefore_o make_v the_o centre_n b_o at_o several_a space_n more_o than_o 16_o and_o less_o than_o 116_o of_o such_o part_n as_o z_o be_v 132_o as_o 110_o 97_o 81_o etc._n etc._n describe_v arch_n again_o make_v the_o centre_n a_o with_o distance_n 22_o 35_o 51_o and_o other_o still_o the_o correspondent_a compliment_n of_o the_o former_a distance_n to_o 132_o draw_v other_o arch_n which_o shall_v cut_v the_o former_a in_o point_n whereby_o the_o ellipsis_n desire_v must_v pass_v by_o the_o say_v 26_o of_o the_o second_o and_o it_o be_v plain_a from_o the_o generation_n of_o a_o ellipsis_n show_v in_o the_o instrumental_a way_n before_o in_o this_o book_n for_o the_o string_n which_o describe_v it_o be_v always_o equal_a to_o z_o +_o obligatory_n that_o be_v 232_o and_o so_o be_v 100_o +_o 110_o +_o 22_o and_o 100_o +_o 97_o +_o 35_o etc._n etc._n wherefore_o this_o be_v evident_a and_o thus_o they_o that_o like_o this_o last_o way_n better_o may_v accomplish_v the_o section_n by_o number_n moreover_o put_v the_o diameter_n of_o a_o parabola_fw-la of_o a_o inch_n ferè_fw-la and_o let_v the_o whole_a base_a incline_v to_o the_o diameter_n at_o angle_n 84_o ferè_fw-la be_v c_o =_o 150_o 64_o last_o let_v the_o perpendicular_a from_o the_o vertex_fw-la to_o the_o base_a be_v d_o =_o 64_o 64_o multiply_v 150_o 64_o by_o 64_o 64_o the_o product_n be_v 9600_o 4096_o of_o which_o two_o three_o be_v equal_a to_o the_o supersicy_n of_o the_o parabola_fw-la and_o be_v 6400_o 4099_o of_o these_o part_n the_o middle_a parallel_n which_o be_v before_o 75_o when_o the_o diameter_n be_v suppose_v 100_o be_v 50_o 64_o which_o double_v be_v 100_o 64_o that_o be_v 6400_o 4096_o as_o before_o so_o that_o in_o this_o case_n the_o residue_n of_o the_o rectangle_n or_o parallelogram_n when_o the_o superficial_a content_n of_o the_o parabola_fw-la be_v take_v from_o it_o and_o the_o length_n of_o the_o middle_a parallel_n be_v both_o denominate_v by_o the_o same_o number_n but_o this_o be_v leave_v to_o the_o reader_n to_o try_v by_o a_o figure_n delineate_v by_o himself_o but_o what_o use_n may_v be_v make_v of_o this_o if_o it_o be_v further_o urge_v either_o in_o natural_a or_o artificial_a number_n i_o
remote_a from_o the_o say_a centre_n note_n this_o aequation_n −_o aaa_n +_o 3_o a_o −_o b_o =_o 0_o be_v natural_o without_o the_o second_o term_n aa_o which_o be_v the_o cause_n that_o it_o have_v the_o false_a root_n not_o discern_v by_o twice_o +_o or_o twice_o −_o succeed_a as_o have_v be_v speak_v of_o chap._n 4._o if_o therefore_o one_o will_v have_v it_o so_o he_o must_v fill_v up_o the_o second_o term_n by_o augment_v the_o root_n never_o so_o little_a putting_z e_z −_o obligatory_n =_o a._n the_o demonstration_n of_o this_o problem_n be_v as_o follow_v it_o be_v to_o be_v prove_v that_o kg_n in_o the_o section_n be_v equal_a to_o be_v the_o subtense_n of_o the_o three_o part_n of_o the_o angle_n give_v put_v kg_a =_o y._n then_o because_o of_o the_o section_n ag_fw-mi =_o yy_a from_o the_o centre_n m_o draw_z mk_n and_o man_fw-mi which_o be_v equal_a because_o of_o the_o circle_n and_o draw_v kn_v parallel_n to_o ae_z and_o produce_v i_o to_o it_o in_o n._n then_o it_o be_v kn_v =_o ge_fw-mi =_o 2_o −_o yy_a the_o square_a therefore_o of_o kn_n be_v 4_o −_o 4_o yy_a +_o yyyy_a and_o mn_v be_v equal_a to_o y_fw-fr plus_fw-fr half_a the_o subtense_n bg_n call_v bg_v by_o the_o single_a letter_n b_o as_o before_z then_o mn_v =_o y_fw-es +_o ½_n b_o the_o square_a of_o which_o be_v yy_a +_o by_o +_o ¼_n bb_v add_v to_o it_o the_o former_a square_n of_o kn_n that_o be_v 4_o −_o 4_o yy_a +_o yyyy_a it_o make_v +_o 4_o −_o 4_o yy_a +_o yyyy_a +_o yy_a +_o by_o +_o ¼_n bb_v equal_a to_o the_o square_n of_o the_o hypothenusal_a mk_n again_o the_o square_a of_o ae_z be_v 4_o and_o the_o square_a of_o i_o be_v ¼_n bb_v which_o two_o square_n be_v equal_a also_o to_o the_o square_n of_o mk_n because_o mk_v =_o ma._n therefore_o 4_o −_o 4_o yy_a +_o yyyy_a +_o yy_a +_o by_o +_o ¼_n bb_a =_o =_o 4_o +_o ¼_n bb_v that_o be_v −_o 3_o yy_a +_o yyyy_a +_o by_o =_o 0._o that_o be_v by_o add_v on_o each_o part_n 3_o yy_a and_o subtract_v yyyy_a +_o 3_o yy_a −_o yyyy_a =_o by_o or_o last_o divide_v all_o by_o y_fw-mi +_o 3_o y_z −_o yyy_a =_o b._n but_o this_o aequation_n be_v alike_o graduate_v and_o like_a affect_v as_o the_o first_o aequation_n +_o 3_o a_o −_o aaa_o =_o b._n wherefore_o you_o =_o a._n but_o a_o =_o be_v and_o y_fw-mi =_o kg._n and_o therefore_o kg_a =_o be_v which_o be_v to_o be_v prove_v in_o like_a sort_n it_o may_v be_v prove_v that_o fd_a be_v a_o true_a root_n of_o the_o aequation_n 3_o a_o −_o aaa_o =_o b_o in_o the_o first_o figure_n and_o the_o subtense_n of_o the_o three_o part_n of_o the_o compliment_n of_o the_o angle_n give_v bag_n to_o a_o circle_n and_o by_o such_o work_a one_o may_v find_v it_o evident_a that_o when_o a_o circle_n cut_v a_o parabola_fw-la in_o point_n how_o many_o soever_o the_o vertex_fw-la except_v perpendicular_n let_v fall_v from_o all_o those_o point_n to_o the_o axis_fw-la be_v all_o the_o several_a root_n of_o one_o and_o the_o same_o aequation_n nor_o have_v that_o aequation_n any_o more_o root_n then_o those_o perpendicular_o to_o the_o axis_fw-la note_n 1._o in_o the_o aequation_n +_o aaa_o −_o bca_fw-mi =_o bbd_v the_o construction_n differ_v somewhat_o from_o the_o former_a for_o b_o be_v repute_a unity_n if_o c_o as_o here_o be_v sign_v with_o −_o the_o axis_fw-la of_o the_o parabola_fw-la must_v be_v produce_v from_o the_o point_n c_o in_o the_o axis_fw-la within_o the_o section_n distant_a from_o a_o by_o ½_n beyond_o the_o vertex_fw-la till_o the_o continuation_n be_v equal_a to_o ½_n c_o and_o at_o the_o end_n thereof_o raise_v a_o perpendicular_a equal_a to_o ½_n d_o at_o the_o end_n of_o that_o be_v the_o centre_n of_o the_o circle_n desire_v and_o according_a to_o this_o method_n may_v any_o aequation_n not_o above_o biquadratical_a be_v resolve_v after_o by_o take_v away_o the_o second_o term_n if_o there_o be_v any_o by_o the_o second_o rule_n of_o chapter_n the_o four_o it_o be_v reduce_v to_o such_o a_o form_n as_o this_o aaa_o *_o bca_fw-la =_o bbd_v if_o the_o quantity_n unknown_a have_v but_o three_o dimension_n or_o if_o it_o have_v four_o then_o thus_o aaaa_o *_o bcaa_o *_o bbda_fw-es =_o =_o bbbf_n or_o else_o taking_z b_o for_o unity_n than_o thus_o aaa_o *_o caa_o =_o d_o and_o thus_o aaaa_o *_o caa_n *_o da_fw-mi =_o f_o the_o sign_n +_o and_o −_o be_v hear_v omit_v for_o they_o must_v be_v supply_v as_o the_o nature_n of_o the_o aequation_n require_v note_n 2._o note_v that_o in_o this_o breviate_v the_o line_n b_o be_v that_o which_o be_v ba_o in_o the_o example_n of_o trisection_n and_o that_o which_o be_v r_o or_o unity_n in_o the_o example_n of_o two_o mean_n also_o the_o line_n c_o be_v that_o which_o in_o the_o former_a example_n of_o trisection_n be_v 2_o ce_fw-fr or_o 3._o and_o if_o this_o quantity_n be_v nothing_o than_o the_o perpendicular_a equal_a to_o half_o d_o be_v to_o be_v erect_v at_o the_o end_n of_o half_a b_o or_o ½_n set_v off_o from_o the_o vertex_fw-la upon_o the_o axis_n within_o but_o if_o c_o have_v any_o length_n then_o at_o the_o distance_n of_o ½_n c_o from_o that_o end_n upon_o the_o axis_fw-la and_o this_o which_o have_v be_v say_v be_v enough_o for_o all_o cubiques_n prob._n 3._o but_o where_o the_o equation_n be_v aaaa_o −_o caa_n +_o da_fw-mi =_o f_o so_o place_v as_o here_o if_o there_o be_v +_o f_o and_o the_o problem_n be_v to_o find_v the_o value_n of_o the_o root_n a_o then_o produce_v ma_fw-fr towards_o a_o make_v as_o equal_a to_o the_o right_a parameter_n of_o the_o section_n and_o make_v axe_n =_o f_o and_o upon_o the_o diameter_n x_v describe_v the_o circle_n xh_v cut_v by_o a_o perpendicular_a to_o ma_fw-fr namely_o ah_o in_o h_z then_o make_v the_o centre_n m_o and_o the_o space_n mh_o describe_v the_o circle_n desire_v but_o if_o it_o be_v −_o f_o as_o in_o this_o example_n i_o put_v it_o then_o after_o ah_o be_v find_v as_o before_o upon_o the_o diameter_n be_o describe_v a_o circle_n and_o in_o it_o from_o a_o apply_v a_o line_n ai_fw-fr =_o ah_o and_o make_v the_o centre_n m_o and_o the_o space_n mi_fw-mi describe_v the_o circle_n fik_fw-mi which_o be_v the_o circle_n seek_v for_o now_o this_o circle_n fik_fw-mi may_v cut_v or_o touch_v the_o parabola_fw-la in_o 1_o 2_o 3_o or_o 4_o points_z from_o all_o which_o perpendicular_o let_v fall_v to_o the_o axis_fw-la give_v all_o the_o root_n of_o the_o aequation_n as_o well_o the_o true_a as_o false_a one_o namely_o if_o the_o quantity_n d_o be_v mark_v −_o than_o those_o perpendicular_o which_o be_v on_o that_o side_n the_o parabola_fw-la where_o the_o centre_n m_o be_v be_v the_o true_a root_n but_o if_o it_o be_v +_o d_o as_o here_o the_o true_a root_n be_v those_o of_o the_o other_o side_n as_o gk_v and_o no_o and_o those_o of_o the_o centre_n side_n as_o fz_n pq_fw-fr be_v the_o false_a demonstration_n put_v ce_fw-fr =_o c_o 2_o and_o draw_v i_o perpendicular_a to_o ag_v and_o gl_o equal_a and_o parallel_v to_o it_o last_o put_v gk_v =_o a_o then_o ag_v =_o aa_o and_o take_v from_o it_o ae_z that_o be_v ½_n c_o +_o ½_n then_o ge_o =_o aa_o −_o −_o ½_n c_o −_o ½_n who_o square_a be_v aaaa_o −_o caa_n −_o −_o aa_o +_o ¼_n cc_o +_o ½_n c_o +_o ¼_n and_o because_o by_o construction_n gl_o =_o ½_n d_o therefore_o kl_v =_o a_o +_o ½_n d_o and_o the_o square_a of_o it_o be_v aa_o +_o da_fw-mi +_o ¼_n dd_v which_o add_v to_o the_o former_a square_n of_o ge_fw-mi it_o give_v the_o square_a of_o km_n that_o be_v a4_n −_o caa_n +_o ¼_n cc_o +_o da_fw-mi +_o ¼_n dd_fw-mi +_o ½_n c_o +_o ¼_n again_o the_o square_a of_o ae_z be_v ¼_n cc_o +_o ½_n c_o +_o ¼_n to_o which_o add_v the_o square_a of_o i_o that_o be_v ¼_n dd_v the_o whole_a be_v the_o spuare_fw-la of_o ma_fw-fr to_o wit_n ¼_n cc_o +_o ¼_n dd_fw-mi +_o ½_n c_o +_o ¼_n but_o the_o square_n of_o ah_o that_o be_v ai_fw-fr be_v equal_a to_o f_o because_o sa_o =_o 1_o and_o xa_fw-la =_o f_o between_o which_z ah_o or_o ai_fw-fr be_v a_o mean_v therefore_o the_o square_a of_o mi_fw-mi be_v ¼_n cc_o +_o ¼_n dd_fw-mi +_o ½_n c_o +_o ¼_n −_o f_o but_o mi_fw-mi =_o mk_v therefore_o their_o square_n be_v equal_a that_o be_v aaaa_o −_o caa_n +_o ¼_n cc_o +_o ¼_n dd_fw-mi +_o da_fw-mi +_o ½_n c_o +_o ¼_n =_o ¼_n cc_o +_o ¼_n dd_fw-mi +_o ½_n c_o +_o ¼_n −_o f._n that_o be_v aaaa_o −_o caa_n +_o da_fw-mi =_o −_o f._n or_o else_o aaaa_o =_o