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	straight_a	2,138	5	12.9115	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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of_o the_o one_a 2._o the_o triangle_n ebc_n dbf_n shall_v be_v equal_a in_o every_o respect_n have_v the_o side_n bc_n bd_o be_v bf_n equal_a and_o the_o angel_n cbe_n dbf_n opposite_a at_o the_o vertex_fw-la be_v equal_a so_o than_o the_o angle_n be_v bdf_n bec_n bfd_n shall_v be_v equal_a by_o the_o four_o of_o the_o first_o and_o their_o base_n aec_fw-la df_n equal_a 3._o the_o triangle_n gbc_n dbh_n have_v their_o opposite_a angle_n cbg_n dbh_n equal_a as_o also_o the_o angle_n bdh_fw-mi bcg_n and_o the_o side_n bc_n bd_o they_o shall_v then_o have_v by_o the_o 26_o of_o the_o one_a their_o side_n bg_n bh_n cg_n dh_n equal_a 4._o the_o triangle_n ace_n afd_v have_v their_o side_n ac_fw-la ad_fw-la ae_n of_o equal_a and_o the_o base_n aec_fw-la df_n equal_a they_o shall_v have_v by_o the_o 8_o of_o the_o one_a the_o angles_n adf_n ace_n equal_a 5._o the_o triangle_n acg_n adh_n have_v the_o side_n ac_fw-la ad_fw-la cg_n dh_n equal_a with_o the_o angles_n adh_n agc_n thence_o they_o 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the_o square_n of_o the_o other_o two_o sides_n ab_fw-la ac_fw-la draw_v the_o line_n ah_o parallel_n to_o bd_o ce_fw-fr and_o draw_v also_o the_o line_n ad_fw-la ae_n fc_n bg_n i_o prove_v that_o the_o square_a of_o be_v equal_a to_o the_o right_o angle_a figure_n or_o long_a square_a bh_n and_o the_o square_a agnostus_n to_o the_o right_o angle_a figure_n ch_z and_o that_o so_o the_o square_n be_v be_v equal_a to_o the_o two_o square_n of_o ag._n demonstration_n the_o triangle_n fbc_n abdella_n have_v their_o sides_n ab_fw-la bf_n bd_o bc_n equal_a and_o the_o angel_n fbc_n abdella_n be_v equal_a see_v that_o each_o of_o they_o beside_o the_o right_a angle_n include_v the_o angle_n abc_n thence_o by_o the_o 4_o the_o triangle_n abdella_n fbc_n be_v equal_a now_o the_o square_n of_o be_v double_a to_o the_o triangle_n fbc_n by_o the_o 41_o because_o they_o have_v the_o same_o base_a bf_n and_o be_v between_o the_o same_o parallel_n bf_a ac_fw-la likewise_o the_o right_o line_a 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in_o the_o first_o place_n trigonometry_n can_v be_v without_o it_o because_o it_o be_v necessary_a to_o make_v the_o table_n of_o all_o the_o line_n that_o can_v be_v draw_v within_o a_o circle_n that_o be_v to_o say_v of_o chord_n of_o sines_n also_o tangent_n and_o secant_v which_o i_o shall_v here_o show_v by_o one_o example_n let_v it_o be_v suppose_v that_o the_o semi-diameter_n ab_fw-la be_v divide_v into_o 10000_o part_n and_o that_o the_o arch_a bc_n be_v 30_o degree_n see_v the_o chord_n or_o subtendent_fw-la of_o 60_o degree_n be_v equal_a to_o the_o semi-diameter_n ac_fw-la bd_o the_o sine_fw-la of_o 30_o degree_n shall_v be_v equal_a to_o the_o half_a of_o ac_fw-la it_o shall_v therefore_o be_v 5000_o in_o the_o right_o angle_a triangle_n adb_n the_o square_a of_o ab_fw-la be_v equal_a to_o the_o square_n of_o bd_o and_o ad_fw-la make_v then_o the_o square_n of_o ab_fw-la by_o multiply_v 10000_o by_o 10000_o and_o from_o that_o product_n subtract_v the_o square_n of_o bd_o 5000_o there_o remain_v the_o square_a of_o ad_fw-la or_o bf_n the_o sine_fw-la of_o the_o compliment_n and_o extract_v the_o square_a root_n there_o be_v find_v the_o line_n fb_n then_o if_o by_o the_o rule_n of_o three_o you_o say_v as_o ad_fw-la be_v to_o bd_o so_o be_v ac_fw-la to_o ce_fw-fr you_o shall_v have_v the_o tangent_fw-la ce_fw-fr and_o add_v together_o the_o square_n of_o ac_fw-la ce_fw-fr you_o shall_v have_v by_o the_o 47_o the_o square_a of_o ae_n and_o by_o extract_v the_o root_n thereof_o you_o shall_v have_v the_o length_n of_o the_o line_n ae_n the_o secant_fw-la use_v 47._o we_o augment_v figure_n as_o much_o as_o we_o please_v by_o this_o proposition_n example_n to_o double_v the_o square_n abcd_v continue_v the_o side_n cd_o and_o make_v de_fw-fr equal_a to_o ad_fw-la the_o square_a of_o ae_n shall_v be_v the_o double_a of_o the_o square_n of_o abcd_n see_v that_o by_o the_o 47_o it_o be_v equal_a to_o the_o square_n of_o ad_fw-la and_o de._n and_o make_v a_o right_a angle_n aef_n and_o take_v of_o equal_a to_o ab_fw-la the_o square_a of_o of_o shall_v be_v triple_a to_o abcd._n and_o make_v again_o the_o right_a angle_n afg_v and_o fg_v equal_a to_o ab_fw-la the_o square_a of_o agnostus_n shall_v be_v quadruple_a to_o to_o abcd._n what_o i_o here_o say_v of_o a_o square_a be_v to_o be_v understand_v of_o all_o figure_n which_o be_v alike_o that_o be_v to_o say_v of_o the_o same_o species_n proposition_n xlviii_o theorem_fw-la if_o the_o two_o square_n make_v upon_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n make_v on_o the_o other_o side_n than_o the_o angle_n comprehend_v under_o the_o two_o other_o side_n of_o the_o triangle_n be_v a_o right_a angle_n if_o the_o square_a of_o the_o side_n np_n be_v equal_a to_o the_o square_n of_o the_o sides_n nl_n lp_n take_v together_o the_o angle_n nlp_n shall_v be_v a_o right_a angle_n draw_v lr_n perpendicular_a to_o nl_n and_o equal_a to_o lp_v then_o draw_v the_o line_n nr_n demonstration_n in_o the_o right_o angle_a triangle_n nlr_n the_o square_a of_o nr_n be_v equal_a to_o the_o square_n of_o nl_n and_o of_o lr_n or_o lp_v by_o the_o 47_o now_o the_o square_n of_o np_n be_v equal_a to_o the_o same_o square_n of_o nl_n lp_v therefore_o the_o square_n of_o nr_n be_v equal_a to_o that_o of_o np_n and_o by_o consequence_n the_o line_n nr_n np_n be_v equal_a and_o because_o the_o triangle_n nlr_n nlp_n have_v each_o of_o they_o the_o side_n nl_n common_a and_o that_o their_o base_n rn_v np_n be_v also_o equal_a the_o angel_n nlp_n nlr_n shall_v be_v equal_a by_o the_o 8_o and_o the_o angle_n nlr_n be_v a_o right_a angle_n the_o angle_z nlp_n shall_v be_v also_o a_o right_a angle_n the_o end_n of_o the_o first_o book_n the_o second_o book_n of_o euclid_n element_n euclid_n treat_v in_o this_o book_n of_o the_o power_n of_o straight_a line_n that_o be_v to_o say_v of_o their_o square_n compare_v the_o divers_a rectangle_v which_o be_v make_v on_o a_o line_n divide_v as_o well_o with_o the_o square_a as_o with_o the_o rectangle_n of_o the_o whole_a line_n this_o part_n be_v very_o useful_a see_v it_o serve_v for_o a_o foundation_n to_o the_o practical_a principle_n of_o algebra_n the_o three_o first_o proposition_n demonstrate_v the_o three_o rule_n of_o arithmetic_n the_o four_o teach_v we_o to_o find_v the_o square_a root_n of_o any_o number_n whatsoever_o those_o which_o follow_v unto_o the_o eight_o serve_v in_o several_a accident_n happen_v in_o algebra_n the_o remain_a proposition_n to_o the_o end_n of_o this_o book_n be_v conversant_a in_o trigonometry_n this_o book_n appear_v at_o the_o first_o sight_n very_o difficult_a because_o one_o do_v imagine_v that_o it_o contain_v mysterious_a or_o intricate_a matter_n notwithstanding_o the_o great_a part_n of_o the_o demonstration_n be_v found_v on_o a_o very_a evident_a principle_n viz._n that_o the_o whole_a be_v equal_a to_o all_o its_o part_n take_v together_o therefore_o one_o ought_v not_o to_o be_v discourage_v although_o one_o do_v not_o apprehend_v the_o demonstration_n of_o this_o book_n at_o the_o first_o read_v definition_n boook_v def._n 1._o of_o the_o second_o boook_v a_o rectangular_a parallelogram_n be_v comprehend_v under_o two_o right_a line_n which_o at_o their_o intersection_n contain_v a_o right_a angle_n it_o be_v to_o be_v note_v henceforward_o that_o we_o call_v that_o figure_n a_o rectangular_a parallelogram_n which_o have_v all_o its_o angle_n right_o and_o that_o the_o same_o shall_v be_v distinguish_v as_o much_o at_o be_v requisite_a if_o we_o give_v thereto_o length_n and_o breadth_n name_v only_o two_o of_o its_o line_n which_o comprehend_v any_o one_o angle_n as_o the_o line_n ab_fw-la bc_n for_o the_o rectangular_a parallelogram_n abcd_v be_v comprehend_v under_o the_o line_n ab_fw-la bc_n have_v bc_n for_o its_o length_n and_o ab_fw-la for_o its_o breadth_n whence_o it_o be_v not_o necessary_a to_o mention_v the_o other_o line_n because_o they_o be_v equal_a to_o those_o already_o speak_v of_o i_o have_v already_o take_v notice_n that_o the_o line_n ab_fw-la be_v in_o a_o perpendicular_a position_n in_o respect_n of_o bc_n produce_v the_o rectangle_n abcd_n if_o move_v along_o the_o line_n bc_n and_o that_o this_o motion_n represent_v arithmetical_a multiplication_n in_o this_o manner_n as_o the_o line_n ab_fw-la move_v along_o the_o line_n bc_n that_o be_v to_o say_v take_v as_o many_o time_n as_o there_o be_v point_n in_o bc_n compose_v the_o rectangle_n abcd_v wherefore_o multiply_v ab_fw-la by_o bc_n i_o shall_v have_v the_o rectangle_n abcd._n as_o suppose_v i_o know_v the_o number_n of_o mathematical_a point_n there_o be_v in_o the_o line_n ab_fw-la for_o example_n let_v there_o be_v 40_o and_o that_o in_o bc_n

the_o line_n ab_fw-la bc_n you_o will_v have_v divide_v they_o equal_o and_o perpendicular_o by_o so_o do_v this_o be_v very_o necessary_a to_o describe_v astrolabe_fw-la and_o to_o complete_a circle_n of_o which_o we_o have_v but_o a_o part_n that_o astronomical_a proposition_n which_o teach_v to_o find_v the_o apogeum_n and_o the_o excentricity_n of_o the_o sun_n circle_n require_v this_o proposition_n we_o often_o make_v use_n there_o of_o in_o the_o treatise_n of_o cut_v of_o stone_n proposition_n xxvi_o theorem_fw-la the_o equal_a angle_n which_o be_v at_o the_o centre_n or_o at_o the_o circumference_n of_o equal_a circle_n have_v for_o base_a equal_a arks._n if_o the_o equal_a angle_n d_o and_o i_o be_v in_o the_o centre_n of_o equal_a circle_n abc_n efg_n the_o ark_n bc_n fg_v shall_v be_v equal_a for_o if_o the_o ark_n bc_n be_v great_a or_o lesser_a than_o the_o ark_n fg_v see_v that_o the_o ark_n be_v the_o measure_n of_o the_o angle_n the_o angle_n d_o will_v be_v great_a or_o lesser_a than_o the_o angle_n 1._o and_o if_o the_o equal_a angle_n a_o and_o e_o be_v in_o the_o circumference_n of_o the_o equal_a circle_n the_o angel_n d_o and_o i_o which_o be_v the_o double_a of_o the_o angle_n a_o and_o e_o be_v also_o equal_a the_o ark_n bc_n fg_v shall_v be_v also_o equal_a proposition_n xxvii_o theorem_fw-la the_o angle_n which_o be_v either_o in_o the_o centre_n or_o in_o the_o circumference_n of_o equal_a circle_n and_o which_o have_v equal_a ark_n for_o base_a be_v also_o equal_a if_o the_o angle_n d_o an_o i_o be_v in_o the_o centre_n of_o equal_a circle_n and_o if_o they_o have_v for_o base_a equal_a arcks_n bc_n fg_v they_o shall_v be_v equal_a because_o that_o their_o measure_n bc_n fg_v be_v equal_a and_o if_o the_o angle_n a_o and_o e_o be_v in_o the_o circumference_n of_o equal_a circle_n have_v for_o base_a equal_a ark_n bc_n eglantine_n the_o angle_n in_o the_o centre_n shall_v be_v equal_a and_o they_o be_v their_o half_n by_o the_o 20_o shall_v be_v also_o equal_a proposition_n xxviii_o theorem_fw-la equal_a line_n in_o equal_a circle_n correspond_v to_o equal_a arks._n if_o the_o line_n bc_n of_o be_v apply_v in_o equal_a circle_n abc_n def_n they_o shall_v be_v chord_n of_o equal_a ark_n bc_n ef._n draw_v the_o line_n ab_fw-la ac_fw-la de_fw-fr ef._n demonstration_n in_o the_o triangle_n abc_n def_n the_o side_n ab_fw-la ac_fw-la de_fw-fr of_o be_v equal_a be_v the_o semidiameters_a of_o equal_a circle_n the_o base_n bc_n of_o be_v suppose_v equal_a thence_o by_o the_o 8_o of_o the_o 1_o the_o angle_n a_o and_o d_o shall_v be_v equal_a and_o by_o the_o 16_o the_o ark_n bc_n of_o shall_v be_v also_o equal_a proposition_n xxix_o theorem_fw-la line_n which_o subtend_v equal_a arck_n in_o equal_a circle_n be_v equal_a if_o the_o line_n bc_n of_o subtend_v equal_a ark_n bc_n of_o in_o equal_a circle_n those_o line_n be_v equal_a demonstration_n the_o ark_n bc_n of_o be_v equal_a and_o part_n of_o equal_a circle_n therefore_o by_o the_o 27_o the_o angle_n a_o and_o d_o shall_v be_v equal_a so_o then_o in_o the_o triangle_n cab_n edf_n the_o side_n ab_fw-la ac_fw-la de_fw-fr df_n be_v equal_a as_o also_o the_o angle_n a_o and_o d_o the_o base_n bc_n of_o shall_v be_v equal_a by_o the_o 4_o of_o the_o 1_o use_v theodosius_n demonstrate_v by_o the_o 28_o and_o 29_o that_o the_o ark_n of_o the_o circle_n of_o the_o italian_a and_o babylonian_a hour_n comprehend_v between_o two_o parallel_n be_v equal_a we_o demonstrate_v also_o after_o the_o same_o manner_n that_o the_o ark_n of_o circle_n of_o astronomical_a hour_n comprehend_v between_o two_o parallel_n to_o the_o equator_fw-la be_v equal_a these_o proposition_n come_v almost_o continual_o in_o use_n in_o spherical_a trigonometry_n as_o also_o in_o gnomonic_n proposition_n xxx_o problem_n to_o divide_v a_o ark_n of_o a_o circle_n into_o two_o equal_a part_n it_o be_v propose_v to_o divide_v the_o ark_n aeb_fw-mi into_o two_o equal_a part_n put_v the_o foot_n of_o the_o compass_n in_o the_o point_n a_o make_v two_o ark_n f_o and_o g_o then_o transport_v the_o compass_n without_o open_v or_o shut_v it_o to_o the_o point_n b_o describe_v two_o ark_n cut_v the_o former_a in_o f_o and_o g_o the_o line_n gf_n will_v cut_v the_o ark_n ab_fw-la equal_o in_o the_o point_n e._n draw_v the_o line_n ab_fw-la demonstration_n you_o divide_v the_o line_n ab_fw-la equal_o by_o the_o construction_n for_o imagine_v the_o line_n of_o bf_n agnostus_n bg_n which_o i_o have_v not_o draw_v lest_o i_o shall_v imbroil_v the_o figure_n the_o triangle_n fga_n fgb_n have_v all_o their_o side_n equal_a so_o then_o by_o the_o 8_o of_o the_o 1_o the_o angles_n afd_v bfd_n be_v equal_a moreover_o the_o triangle_n dfa_n dfb_n have_v the_o sides_n df_n common_a the_o side_n of_o bf_n equal_a and_o the_o angles_n dfa_n dfb_n equal_a whence_o by_o the_o 4_o of_o the_o 1_o the_o base_n ad_fw-la db_fw-la be_v equal_a and_o the_o angles_n adf_n bdf_n be_v equal_a we_o have_v then_o divide_v the_o line_n ab_fw-la equal_o and_o perpendicular_o in_o the_o point_n d._n so_o then_o by_o the_o 1_o the_o centre_n of_o the_o circle_n be_v in_o the_o line_n eglantine_n let_v it_o be_v the_o point_n c_o and_o let_v be_v draw_v the_o line_n ca_n cb_n all_o the_o side_n of_o the_o triangle_n acd_v bcd_a be_v equal_a thence_o the_o angle_n acd_v bcd_a be_v equal_a by_o the_o 8_o of_o the_o 1_o and_o by_o the_o 27_o the_o ark_n ae_n ebb_n be_v equal_a use_v as_o we_o have_v often_o need_v to_o divide_v a_o ark_n in_o the_o middle_n the_o practice_n of_o this_o proposition_n be_v very_o ordinary_o in_o use_n it_o be_v by_o this_o mean_v we_o divide_v the_o mariner_n compass_n into_o 32_o rumb_n for_o have_v draw_v two_o diameter_n which_o cut_v each_o other_o at_o right_a angle_n we_o divide_v the_o circle_n in_o four_o and_o sub-dividing_a each_o quarter_n in_o the_o middle_n we_o have_v eight_o part_n and_o sub-dividing_a each_o part_n twice_o we_o come_v to_o thirty_o two_o part_n we_o have_v also_o occasion_n of_o the_o same_o practice_n to_o divide_v a_o semicircle_n into_o 180_o degree_n and_o because_o for_o the_o perform_v the_o same_o division_n throughout_o we_o be_v oblige_v to_o divide_v a_o ark_n into_o three_o all_o the_o ancient_a geometrician_n have_v endeavour_v to_o find_v a_o method_n to_o divide_v a_o angle_n or_o a_o ark_n into_o three_o equal_a part_n but_o it_o be_v not_o yet_o find_v proposition_n xxxi_o theorem_fw-la the_o angle_n which_o be_v in_o a_o semicircle_n be_v right_o that_o which_o be_v comprehend_v in_o a_o great_a segment_n be_v acute_a and_o that_o in_o a_o lesser_a segment_n be_v obtuse_a if_o the_o angle_n bac_n be_v in_o a_o semicircle_n i_o demonstrate_v that_o it_o be_v right_o draw_v the_o line_n da._n demonstration_n the_o angle_n adb_n exterior_a in_o respect_n of_o the_o triangle_n dac_n be_v equal_a by_o the_o 32d_o of_o the_o one_a to_o the_o two_o interiour_o dac_n dca_n and_o those_o 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