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

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A38722	The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...; Huict livres des Eléments d'Euclide rendus plus faciles. English	Dechales, Claude-François Milliet, 1621-1678.; Euclid. Elements.; Williams, Reeve, fl. 1682-1703.	1685 (1685)	Wing E3399; ESTC R10241	136,603	430

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be_v equal_a by_o the_o 5_o of_o the_o one_a see_v the_o side_n dam_fw-ge dc_o be_v equal_a it_o shall_v be_v double_a to_o the_o angle_n dac_n in_o like_a manner_n the_o angle_n adc_n be_v double_a to_o the_o angle_n dab_n therefore_o the_o two_o angel_n adb_n adc_fw-la which_o be_v equal_a to_o two_o right_a be_v double_a to_o the_o angle_n bac_n and_o by_o consequence_n the_o angle_n bac_n be_v a_o right_a angle_n second_o the_o angle_n aec_fw-la which_o be_v in_o the_o segment_n aec_fw-la be_v obtuse_a for_o in_o the_o quadrilateral_a abce_n the_o opposite_a angle_n e_z and_o b_o be_v equal_a to_o two_o right_v by_o the_o 22d_o the_o angle_n b_o be_v acute_a therefore_o the_o angle_n e_o shall_v be_v obtuse_a three_o the_o angle_n b_o which_o be_v in_o the_o segment_n abc_n great_a than_o a_o semicircle_n be_v acute_a see_v that_o in_o the_o triangle_n abc_n the_o angle_n bac_n be_v a_o right_a angle_n use_v 31._o use_v 31._o the_o workman_n have_v draw_v from_o this_o proposition_n the_o way_n of_o try_v if_o their_o square_n be_v exact_a for_o have_v draw_v a_o semicircle_n bad_a they_o apply_v the_o point_n a_o of_o their_o square_a bad_a on_o the_o circumference_n of_o the_o circle_n and_o one_o of_o its_o side_n ab_fw-la on_o the_o point_n b_o of_o the_o diameter_n the_o other_o side_n ad_fw-la must_v touch_v the_o other_o point_v d_o which_o be_v the_o other_o end_n of_o the_o diameter_n ptolemy_n make_v use_v of_o this_o proposition_n to_o make_v the_o table_n of_o subtendants_n or_o chord_n of_o which_o he_o have_v occasion_n in_o trigonometry_n 31._o use_v 31._o we_o have_v also_o a_o practical_a way_n to_o erect_v a_o perpendicular_a on_o the_o end_n of_o

b._n by_o consequence_n the_o whole_a angle_n acd_v be_v equal_a to_o the_o two_o angel_n ace_n ecd_n of_o who_o it_o be_v compose_v it_o shall_v then_o be_v equal_a to_o a_o and_o b_o take_v together_o in_o the_o second_o place_n the_o angles_n acd_v acb_n be_v equal_a to_o two_o right_v by_o the_o 13_o and_o i_o have_v demonstrate_v that_o the_o angle_n acd_v be_v equal_a to_o the_o angle_n a_o and_o b_o take_v together_o therefore_o the_o angle_n acd_v be_v equal_a to_o a_o and_o b_o that_o be_v to_o say_v all_o the_o angle_n of_o the_o triangle_n abc_n be_v equal_a to_o two_o right_a or_o 180_o degree_n corollariy_n 1._o the_o three_o angle_n of_o one_o triangle_n be_v equal_a to_o the_o three_o angle_n of_o another_o triangle_n coral_n 2._o if_o two_o angle_n of_o a_o triangle_n be_v equal_a to_o two_o angle_n of_o another_o triangle_n the_o remain_a angle_n in_o the_o one_o shall_v be_v equal_a to_o the_o remain_a angle_n in_o the_o other_o coral_n 3._o if_o a_o triangle_n have_v one_o of_o its_o angle_n right_o the_o other_o two_o shall_v be_v acute_a and_o be_v take_v together_o shall_v be_v equal_a to_o one_o right_a angle_n coral_n 4._o from_o one_o and_o the_o same_o point_n of_o a_o line_n there_o can_v be_v draw_v but_o one_o perpendicular_a because_o a_o triangle_n can_v have_v two_o right_a angle_n coral_n 5._o the_o perpendicular_a be_v the_o short_a line_n which_o can_v be_v draw_v from_o a_o point_n to_o a_o line_n coral_n 6._o in_o a_o right_a angle_a triangle_n the_o great_a angle_n be_v a_o right_a angle_n and_o the_o long_a side_n be_v opposite_a thereto_o coral_n 7._o each_o angle_n of_o a_o equilateral_a triangle_n contain_v 60_o degree_n that_o be_v to_o say_v the_o three_o of_o 180._o use_v 32._o use_v 32._o this_o proposition_n be_v of_o use_n to_o we_o in_o astronomy_n to_o determine_v the_o parallax_n let_v the_o point_n a_o represent_v the_o centre_n of_o the_o earth_n and_o from_o the_o point_n b_o on_o the_o superficies_n of_o the_o earth_n let_v there_o be_v take_v by_o observation_n the_o angle_n dbc_n that_o be_v to_o say_v the_o apparent_a distance_n of_o a_o planet_n or_o comet_n from_o the_o zenith_n d._n i_o say_v if_o the_o earth_n be_v transparent_a this_o planet_n or_o comet_n view_v from_o the_o centre_n of_o the_o earth_n a_o will_v appear_v distant_a from_o the_o zenith_n d_o equal_a to_o the_o quantity_n of_o the_o angle_n god_n which_o be_v less_o than_o the_o angle_n cbd_v for_o the_o angle_n cbd_v be_v exterior_a in_o respect_n of_o the_o triangle_n abc_n be_v equal_a by_o the_o 32d._o to_o the_o opposite_a angle_n a_o and_o c._n whence_o the_o angle_n c_o shall_v be_v equal_a to_o the_o excess_n of_o the_o angle_n cbd_v above_o the_o angle_n a._n from_o whence_o i_o conclude_v that_o if_o i_o know_v by_o a_o astronomical_a table_n how_o far_o that_o planet_n or_o comet_n ought_v to_o appear_v distant_a from_o the_o zenith_n to_o one_o which_o shall_v be_v in_o the_o centre_n of_o the_o earth_n and_o if_o i_o observe_v at_o the_o same_o time_n the_o difference_n of_o those_o two_o angle_n shall_v be_v the_o parallax_n viz._n the_o angle_n bca_n proposition_n xxxiii_o theorem_fw-la if_o two_o equal_a and_o parallel_v line_n be_v join_v together_o with_o two_o other_o right_a line_n then_o be_v those_o line_n also_o equal_a and_o parallel_v let_v the_o line_n ab_fw-la gd_v be_v parallel_n and_o equal_a and_o let_v they_o be_v join_v with_o agnostus_n bd._n i_o say_v that_o the_o line_n agnostus_n bd_o be_v equal_a and_o parallel_v draw_v the_o diagonal_a bg_n demonstration_n see_v the_o line_n ab_fw-la gd_v be_v parallel_n the_o alternate_a angle_n abg_n bgd_v shall_v be_v equal_a by_o the_o 29_o and_o the_o side_n gb_n be_v common_a to_o both_o the_o triangle_n abg_n bgd_v and_o the_o side_n ab_fw-la gd_v equal_a with_o the_o angles_n abg_n bgd_v equal_a as_o before_o the_o base_n of_o those_o triangle_n agnostus_n bd_o shall_v be_v equal_a by_o the_o 4._o as_o also_o the_o angle_n dbg_n bga_n which_o because_o alternate_a the_o line_n agnostus_n bd_o shall_v be_v parallel_n by_o the_o 27_o use_v 33._o use_v 33._o this_o proposition_n be_v put_v in_o practice_n to_o measure_v as_o well_o the_o perpendicular_a height_n agnostus_n of_o mountain_n as_o their_o horizontal_n line_n c_o g_o which_o be_v hide_v in_o their_o thickness_n to_o effect_v which_o we_o make_v use_v of_o a_o very_a large_a square_a abdella_n put_v one_o end_n thereof_o in_o a_o in_o such_o manner_n that_o the_o other_o side_n thereof_o bd_o may_v be_v perpendicular_a to_o the_o horizon_n than_o we_o measure_v the_o side_n ad_fw-la bd_o than_o we_o do_v the_o like_a again_o at_o the_o point_n b_o and_o measure_n be_v aec_fw-la the_o side_n parallel_v to_o the_o horizon_n that_o be_v to_o say_v ad_fw-la be_v be_v add_v together_o gives_z the_o horizontal_n line_n cg_n and_o the_o perpendicular_a side_n bd_o aec_fw-la be_v add_v give_v the_o perpendicular_a height_n ag._n proposition_n xxxiv_o theorem_fw-la the_o side_n and_o the_o opposite_a angle_n in_o a_o parallelogram_n be_v equal_a and_o the_o diamter_n do_v divide_v the_o same_o into_o two_o equal_a part_n let_v the_o figure_n ab_fw-la cd_o be_v a_o parallelogram_n that_o be_v to_o say_v let_v the_o side_n ab_fw-la cd_o ac_fw-la bd_o be_v parallel_n i_o say_v that_o the_o opposite_a side_n ab_fw-la cd_o ac_fw-la bd_o be_v equal_a as_o also_o the_o angle_n a_o and_o d_o abdella_n acd_v and_o that_o the_o diameter_n bc_n do_v divide_v the_o figure_n into_o two_o equal_a part_n demonstration_n the_o line_n ab_fw-la cd_o be_v suppose_v parallel_n therefore_o the_o alternate_a angel_n abc_n bcd_v shall_v be_v equal_a by_o the_o 29_o likewise_o the_o side_n ac_fw-la bd_o be_v suppose_a parallel_n the_o alternate_a angle_n acb_n cbd_v be_v equal_a both_o which_o triangle_n have_v the_o same_o side_n bc_n and_o the_o angel_n abc_n bcd_v acb_n cbd_v equal_a they_o shall_v be_v equal_a in_o every_o respect_n by_o the_o 26_o therefore_o the_o side_n ab_fw-la cd_o ac_fw-la bd_o and_o the_o angle_n a_o and_o d_o be_v equal_a and_o the_o diameter_n divide_v the_o figure_n into_o two_o equal_a part_n and_o see_v the_o angel_n abc_n bcd_v acd_v cbd_a be_v equal_a add_v together_o abc_n cbd_v bcd_n acd_a we_o conclude_v that_o the_o opposite_a angle_n abdella_n acd_v shall_v be_v equal_a use_v 34._o use_v 34._o svrveyer_n have_v sometime_o occasion_n to_o make_v use_n of_o this_o proposition_n to_o part_v or_o divide_v land_n if_o the_o field_n be_v a_o parallelogram_n it_o may_v be_v divide_v into_o two_o equal_a part_n by_o the_o diameter_n ad._n but_o if_o one_o be_v oblige_v to_o divide_v the_o same_o into_o two_o equal_a part_n from_o the_o point_n e_o divide_v the_o diameter_n into_o two_o equal_a part_n in_o the_o point_n f_o and_o draw_v the_o line_n efg_v that_o line_n shall_v divide_v the_o field_n into_o two_o equal_a part_n for_o the_o triangle_n aef_n gfd_n which_o have_v their_o alternate_a angel_n eaf_n fdg_n aef_n fgd_v and_o the_o side_n of_o fd_n equal_a be_v equal_a by_o the_o 26_o and_o since_o the_o trapezium_fw-la befd_n together_o with_o the_o triangle_n afe_n that_o be_v to_o say_v the_o triangle_n adb_n be_v one_o half_a of_o the_o parallelogram_n by_o the_o 34●h._n the_o same_o trapezium_fw-la efbd_v together_o with_o the_o triangle_n dfg_v shall_v be_v one_o half_a of_o the_o figure_n or_o field_n and_o the_o line_n eglantine_n divide_v the_o same_o into_o two_o equal_a part_n proposition_n xxxv_o the_o parallellogram_n be_v equal_a when_o have_v the_o same_o base_a they_o be_v between_o the_o same_o parallel_n lines_n let_v the_o parallelograms_n abec_n abdf_n have_v the_o same_o base_a ab_fw-la and_o be_v between_o the_o same_o parallel_n ab_fw-la cd_o i_o say_v they_o be_v equal_a demonstration_n the_o sides_n ab_fw-la ce_fw-fr be_v equal_a by_o the_o 34_o as_o also_o ab_fw-la fd_n wherefore_o ce_fw-fr fd_n be_v equal_a and_o add_v of_o to_o each_o the_o line_n cf_n ed_z shall_v be_v equal_a the_o triangle_n cfa_n ebb_v have_v the_o side_n ca_n ebb_n cf_n ed_z equal_a with_o the_o angles_n deb_n fca_n by_o the_o 29_o the_o one_o be_v exterior_a and_o the_o other_o interiour_n on_o the_o same_o side_n whence_o by_o the_o 4_o the_o triangle_n ace_n bed_n be_v equal_a and_o take_v away_o from_o both_o that_o which_o be_v common_a to_o both_o that_o be_v to_o say_v the_o little_a triangle_n egf_n the_o trapezium_fw-la fgbd_v shall_v be_v equal_a to_o the_o trapezium_fw-la cage_n and_o add_v to_o both_o the_o little_a triangle_n agb_n the_o parallellogram_n abec_n abdf_n shall_v be_v equal_a demonstration_n by_o the_o method_n of_o indivisibles_n this_o method_n be_v new_o invent_v by_o cavalerius_fw-la which_o be_v approve_v

a_o line_n which_o be_v found_v on_o this_o proposition_n for_o example_n to_o erect_v a_o perpendicular_a from_o the_o point_n a_o of_o the_o line_n ab_fw-la i_o put_v the_o foot_n of_o the_o compass_n on_o the_o point_n c_o take_v at_o discretion_n and_o extend_v the_o other_o to_o a_o i_o describe_v a_o circle_n which_o may_v cut_v the_o line_n ab_fw-la in_o the_o point_n b._n i_o draw_v the_o line_n bcd_n it_o be_v evident_a that_o the_o line_n dam_fw-ge shall_v be_v perpendicular_a to_o the_o line_n ab_fw-la see_v the_o angle_n bad_a be_v in_o the_o semicircle_n proposition_n xxxii_o theorem_fw-la the_o line_n which_o cut_v the_o circle_n at_o the_o point_n of_o touch_v make_v with_o the_o touch_n line_n the_o angle_n equal_a to_o those_o of_o the_o alternate_a segment_n let_v the_o line_n bd_o cut_v the_o circle_n in_o the_o point_n b_o which_o be_v the_o point_n where_o the_o line_n ab_fw-la do_v touch_v the_o same_o i_o say_v that_o the_o angle_n cbd_v which_o the_o line_n bd_o comprehend_v with_o the_o touch_n line_n abc_n be_v equal_a to_o the_o angle_n e_o which_o be_v that_o of_o the_o alternate_a segment_n bed_n and_o that_o the_o angle_n abdella_n be_v equal_a to_o the_o angle_n f_o of_o the_o segment_n bfd_n in_o the_o first_o place_n if_o the_o line_n pass_v through_o the_o centre_n as_o do_v the_o line_n be_v it_o will_v make_v with_o the_o touch_n line_n ab_fw-la two_o right_a angle_n by_o the_o 17_o and_o the_o angle_n of_o the_o semicircle_n will_v be_v also_o right_o by_o the_o precede_a so_o the_o proposition_n will_v be_v true_a if_o the_o line_n pass_v not_o through_o the_o centre_n as_o do_v the_o line_n bd_o draw_v the_o line_n be_v through_o the_o centre_n and_o join_v the_o line_n de_fw-fr demonstration_n the_o line_n be_v make_v two_o right_a angle_n with_o the_o touch_n line_n and_o all_o the_o angle_n of_o the_o triangle_n bde_n be_v equal_a to_o two_o right_v by_o the_o 32d_o of_o the_o one_a so_o take_v away_o the_o right_a angle_n abe_n and_o d_o which_o be_v in_o a_o semicircle_n and_o take_v again_o away_o the_o angle_n ebb_v which_o be_v common_a to_o both_o the_o angle_n cbd_v shall_v be_v equal_a to_o the_o angle_n bed_n three_o the_o angle_n abdella_n be_v equal_a to_o the_o angle_n f_o because_o in_o the_o quadrilateral_a bfde_v which_o be_v inscribe_v in_o a_o circle_n the_o opposite_a angle_n e_o and_o f_o be_v equal_a to_o two_o right_n by_o the_o 22d_o the_o angles_n abdella_n dbc_n be_v also_o equal_a to_o two_o right_v by_o the_o 13_o of_o the_o one_a and_o the_o angle_n dbc_n and_o e_o be_v equal_a as_o just_a now_o i_o do_v demonstrate_v therefore_o the_o angles_n abdella_n and_o bfd_n be_v equal_a use_v this_o proposition_n be_v necessary_a for_o that_o which_o follow_v proposition_n xxxiii_o problem_n to_o describe_v upon_o a_o line_n a_o segment_n of_o a_o circle_n which_o shall_v contain_v a_o give_v angle_n it_o be_v propose_v to_o describe_v on_o the_o line_n ab_fw-la a_o segment_n of_o a_o circle_n to_o contain_v the_o angle_n c._n make_v the_o angle_n bid_v equal_a to_o the_o angle_n c_o and_o draw_v to_o ad_fw-la the_o perpendicular_a ae_n make_v also_o the_o angle_n abf_n equal_a to_o the_o angle_n baf_n and_o last_o describe_v a_o circle_n on_o the_o point_n f_o as_o centre_n at_o the_o open_a bf_n or_o favorina_n the_o segment_n beaumont_n contain_v a_o angle_n equal_a to_o the_o angle_n c._n demonstration_n the_o angel_n baf_n abf_n be_v equal_a the_o line_n favorina_n fb_n be_v equal_a by_o the_o 6_o and_o the_o circle_n which_o be_v describe_v on_o the_o centre_n f_o pass_v through_o a_o and_o b_o now_o the_o angle_n dae_n be_v right_o the_o line_n da_fw-la touch_v the_o circle_n in_o the_o point_n a_o by_o the_o 16_o therefore_o the_o angle_n which_o the_o segment_n beaumont_n comprehend_v as_o the_o angle_n e_o be_v equal_a to_o the_o angle_n dab_n that_o be_v to_o say_v to_o the_o angle_n c._n but_o if_o the_o angle_n be_v obtuse_a we_o must_v take_v the_o acute_a angle_n which_o be_v its_o compliment_n to_o 180_o degree_n proposition_n xxxiv_o problem_n a_o circle_n be_v give_v to_o cut_v therefrom_o a_o segment_n to_o contain_v a_o assign_a angle_n to_o cut_v from_o the_o circle_n be_v a_o segment_n to_o contain_v the_o angle_n a._n draw_v by_o the_o 17_o the_o touch_n line_n bd_o and_o make_v the_o angle_n dbc_n equal_a to_o the_o angle_n a._n it_o be_v evident_a by_o the_o 32d_o that_o the_o segment_n bec_n contain_v a_o angle_n equal_a to_o dbc_n and_o by_o consequence_n to_o the_o angle_n a._n use_v i_o have_v make_v use_n of_o this_o proposition_n to_o find_v geometrical_o the_o excentricity_n of_o the_o annual_a circle_n of_o the_o sun_n and_o his_o apogeon_n three_o observation_n be_v give_v it_o be_v also_o make_v use_n of_o in_o optic_n two_o unequal_a line_n be_v propose_v to_o find_v a_o point_n where_o they_o shall_v appear_v equal_a or_o under_o equal_a angle_n make_v on_o each_o segment_v which_o may_v contain_v equal_a angle_n proposition_n xxxv_o theorem_fw-la if_o two_o line_n cut_v each_o other_o in_o a_o circle_n the_o rectangle_n comprehend_v under_o the_o part_n of_o the_o one_o be_v equal_a to_o the_o rectangle_n comprehend_v under_o the_o part_n of_o the_o other_o in_o the_o first_o place_n if_o two_o line_n cut_v each_o other_o in_o the_o centre_n of_o the_o circle_n they_o shall_v be_v equal_a and_o divide_v equal_o so_o than_o it_o be_v evident_a that_o the_o rectangle_n comprehend_v under_o the_o part_n of_o the_o one_o be_v equal_a to_o the_o rectangle_n comprehend_v under_o the_o part_n of_o the_o other_o second_o let_v one_o of_o the_o line_n pass_v through_o the_o centre_n f_o as_o ac_fw-la and_o divide_v the_o line_n bd_o in_o two_o equal_o in_o the_o point_n e_o i_o say_v that_o the_o rectangle_n comprehend_v under_o ae_n aec_fw-la be_v equal_a to_o the_o rectangle_n comprehend_v under_o be_v ed_z that_o be_v to_o say_v to_o the_o square_n of_o be._n the_o line_n ac_fw-la be_v perpendicular_a to_o bd_o by_o the_o three_o demonstration_n see_v that_o the_o line_n ac_fw-la be_v divide_v equal_o in_o f_o and_o unequal_o in_o f_o the_o rectangle_n comprehend_v under_o ae_n aec_fw-la with_o the_o square_n of_o fe_o be_v equal_a to_o the_o square_n of_o fc_n or_o fb_n by_o the_o 5_o of_o the_o 2d_o now_o the_o angle_n e_o being_n right_n the_o square_a of_o fb_n be_v equal_a to_o the_o square_n of_o be_v of_o therefore_o the_o rectangle_n comprehend_v under_o ae_n aec_fw-la with_o the_o square_n of_o of_o be_v equal_a to_o the_o square_n of_o be_v of_o and_o take_v away_o the_o square_n of_o of_o there_o remain_v that_o the_o square_a of_o be_v be_v equal_a to_o the_o rectangle_n under_o be_v ed._n three_o let_v the_o line_n ab_fw-la pass_v through_o the_o centre_n f_o and_o let_v it_o divide_v the_o line_n cd_o unequal_o in_o the_o point_n e_o draw_v fg_v perpendicular_o to_o cd_o and_o by_o the_o 3d._n the_o line_n cg_n gd_v shall_v be_v equal_a demonstration_n see_v the_o line_n ab_fw-la be_v divide_v equal_o in_o f_o and_o unequal_o in_o e_z the_o rectangle_n comprehend_v under_o ae_n ebb_n with_o the_o square_n of_o of_o be_v equal_a to_o the_o square_n of_o bf_n or_o fc_n by_o the_o 5_o of_o the_o 2d_o in_o the_o place_n of_o of_o put_v the_o square_n of_o fg_n ge_z which_o be_v equal_a thereto_o by_o the_o 47th_o of_o the_o one_a in_o like_a manner_n the_o line_n cd_o be_v equal_o divide_v in_o g_o and_o unequal_o in_o e_z the_o rectangle_n ce_v with_o the_o square_n of_o ge_z shall_v be_v equal_a to_o the_o square_n of_o gc_n add_v the_o square_n of_o gf_n the_o rectangle_n of_o ce_fw-fr ed_z with_o the_o square_n of_o ge_z fg_v shall_v be_v equal_a to_o the_o square_n of_o gc_n gf_n that_o be_v to_o say_v by_o the_o 47th_o of_o the_o one_a to_o the_o square_n of_o cf._n therefore_o the_o rectangle_n aeb_n with_o the_o square_n of_o ge_z gf_n and_o the_o rectangle_n of_o ce_fw-fr ed_z with_o the_o same_o square_n be_v equal_a and_o by_o consequence_n take_v away_o the_o same_o square_n the_o rectangle_n aeb_n be_v equal_a to_o the_o rectangle_n cfd_n four_o let_v the_o line_n cd_o he_o cut_v each_o other_o in_o the_o point_n e_o so_fw-mi that_o neither_o of_o they_o pass_v through_o the_o centre_n i_o say_v that_o the_o rectangle_n ce_v be_v equal_a to_o the_o rectangle_n hei_fw-la for_o draw_v the_o line_n afb_n the_o rectangle_v ce_v hei_fw-la be_v equal_a to_o the_o rectangle_n aeb_fw-mi by_o the_o precede_a case_n therefore_o they_o be_v equal_a use_v one_o might_n by_o this_o proposition_n have_v a_o practical_a way_n to_o find_v the_o four_o proportional_a to_o three_o give_v line_n or_o