Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	line_n	point_n	superficies_n	1,287	5	14.6915	5	false

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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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

under_o ab_fw-la and_o ac_fw-la shall_v be_v three_o time_n 8_o or_o 24_o the_o square_a of_o ac_fw-la 3_o be_v 9_o the_o rectangle_n comprehend_v under_o ac_fw-la 3_o and_o cb_n 5_o be_v 3_o time_n 5_o or_o 15._o it_o be_v evident_a that_o 15_o and_o 9_o be_v 24._o use_v at_fw-fr  _fw-fr 43_o c_z 40._o 3_o b_o  _fw-fr 3_o 120._o  _fw-fr 9_o 129_o  _fw-fr  _fw-fr this_o proposition_n serve_v likewise_o to_o demonstrate_v the_o ordinary_a practice_n of_o multiplication_n for_o example_n if_o one_o will_v multiply_v the_o number_n 43_o by_o 3_o have_v separate_v the_o number_n of_o 43_o into_o two_o part_n in_o 40_o and_o 3_o three_o time_n 43_o shall_v be_v as_o much_o as_o three_o time_n 3_o which_o be_v nine_o the_o square_a of_o three_o and_o three_o time_n forty_o which_o be_v 120_o for_o 129_o be_v three_o time_n 43._o those_o which_o be_v young_a beginner_n ought_v not_o to_o be_v discourage_v if_o they_o do_v not_o conceive_v immediate_o these_o proposition_n for_o they_o be_v not_o difficult_a but_o because_o they_o do_v imagine_v they_o contain_v some_o great_a mystery_n proposition_n iu_o theorem_fw-la if_o a_o line_n be_v divide_v into_o two_o part_n the_o square_a of_o the_o whole_a line_n shall_v be_v equal_a to_o the_o two_o square_n make_v of_o its_o part_n and_o to_o two_o rectangle_v comprehend_v under_o the_o same_o part_n let_v the_o line_n ab_fw-la be_v divide_v in_o c_o and_o let_v the_o square_a thereof_o abde_n be_v make_v let_v the_o diagonal_a ebb_n be_v draw_v and_o the_o perpendicular_a cf_n cut_v the_o same_o and_o through_o that_o point_n let_v there_o be_v draw_v gl_n parallel_n to_o ab_fw-la it_o be_v evident_a that_o the_o square_a abde_n be_v equal_a to_o the_o four_o rectangle_v gf_n cl_n cg_n lf_n the_o two_o first_o be_v the_o square_a of_o ac_fw-la and_o of_o cb_n the_o two_o compliment_n be_v comprehend_v under_o ac_fw-la cb._n demonstration_n the_o side_n ae_n ab_fw-la be_v equal_a thence_o the_o angle_n aeb_fw-mi abe_n be_v half_a right_n and_o because_o of_o the_o parallel_n gl_n ab_fw-la the_o angle_n of_o the_o triangle_n of_o the_o square_a ge_z by_o the_o 29_o shall_v be_v equal_a as_o also_o the_o side_n by_o the_o 6_o of_o the_o 1._o thence_o gf_n be_v the_o square_a of_o ac_fw-la in_o like_a manner_n the_o rectangle_n cl_n be_v the_o square_a of_o cb_n the_o rectangle_n gc_n be_v comprehend_v under_o ac_fw-la and_o agnostus_n equal_a to_o bl_v or_o bc_n the_o rectangle_n lf_o be_v comprehend_v under_o ld_n equal_a to_o ac_fw-la and_o under_o fd_n equal_a to_o bc._n coral_n if_o a_o diagonal_a be_v draw_v in_o a_o square_a the_o rectangle_v through_o which_o it_o pass_v be_v square_n use_v a_o 144_o b_o 22_o c_o 12_o this_o proposition_n give_v we_o the_o practical_a way_n of_o find_v or_o extract_v the_o square_a root_n of_o a_o number_n propound_v let_v the_o same_o be_v the_o number_n a_o 144_o represent_v by_o the_o square_a ad_fw-la and_o its_o root_n by_o the_o line_n ab_fw-la moreover_o i_o 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find_v and_o i_o say_v how_o many_o time_n 20_o in_o 44_o i_o find_v it_o 2_o time_n for_o the_o side_n bl_n but_o because_o 20_o be_v not_o the_o whole_a side_n kb_n but_o only_a kc_n this_o 2_o which_o come_v in_o the_o quotient_n be_v to_o be_v add_v to_o the_o divisor_n which_o then_o will_v be_v 22._o so_o i_o find_v the_o same_o 2_o time_n precise_o in_o 44_o the_o square_a root_n then_o shall_v be_v 12._o you_o see_v that_o the_o square_a of_o 144_o be_v equal_a to_o the_o square_n of_o 10_o to_o the_o square_n of_o 2_o which_o be_v 4_o and_o to_o twice_o 20_o which_o be_v two_o rectangle_v comprehend_v under_o 2_o and_o under_o 10._o proposition_n v._o theorem_fw-la if_o a_o right_a line_n be_v cut_v into_o equal_a part_n and_o into_o unequal_a part_n the_o rectangle_n comprehend_v under_o the_o unequal_a part_n together_o with_o the_o square_n which_o be_v of_o the_o middle_a part_n or_o difference_n of_o the_o part_n be_v equal_a to_o the_o square_n of_o half_a the_o line_n if_o the_o line_n ab_fw-la be_v divide_v equal_o in_o c_z and_o unequal_o in_o d_o the_o rectangle_n ah_o comprehend_v under_o the_o segment_n ad_fw-la db_fw-la together_o with_o the_o square_n of_o cd_o shall_v be_v equal_a to_o the_o square_a cf_n that_o be_v of_o half_a of_o ab_fw-la viz._n cb._n make_v a_o end_n of_o the_o figure_n as_o you_o see_v it_o the_o rectangle_v lg_n diego_n shall_v be_v square_n by_o the_o coral_n of_o the_o 4_o i_o prove_v that_o the_o rectangle_n ah_o comprehend_v under_o ad_fw-la and_o dh_n equal_a to_o db_v with_o the_o square_a lg_n be_v equal_a to_o the_o square_a cf._n demonstration_n the_o rectangle_n all_o be_v equal_a to_o the_o rectangle_n df_n the_o one_o and_o the_o other_o be_v comprehend_v under_o half_a the_o line_n ab_fw-la and_o under_o bd_o or_o dh_n equal_a thereto_o add_v to_o both_o the_o rectangle_n ch_z the_o rectangle_n ah_o shall_v be_v equal_a to_o the_o gnomon_n lbg_n again_o to_o both_o add_v the_o square_a lg_n the_o rectangle_n ah_o with_o the_o square_a lg_n shall_v be_v equal_a to_o the_o square_a cf._n arithmetical_o let_v ab_fw-la be_v 10_o ac_fw-la be_v 5_o as_o also_o cb._n let_v cd_o be_v 2_o and_o db_n 3_o the_o rectangle_n comprehend_v under_o ad_fw-la 7_o and_o db_n 3_o that_o be_v to_o say_v 21_o with_o the_o square_n of_o cd_o 2_o which_o be_v 4_o shall_v be_v equal_a to_o the_o square_n of_o cb_n 5_o which_o be_v 25._o use_v this_o proposition_n be_v very_o useful_a in_o the_o three_o book_n we_o make_v use_v thereof_o in_o algebra_n to_o demonstrate_v the_o way_n of_o find_v the_o root_n of_o a_o affect_a square_a or_o equation_n proposition_n vi_o theorem_fw-la if_o one_o add_v a_o line_n to_o another_o which_o be_v divide_v into_o two_o equal_a part_n the_o rectangle_n comprehend_v under_o the_o line_n compound_v of_o both_o and_o under_o the_o line_n add_v together_o with_o the_o square_n of_o half_a the_o divide_a line_n be_v equal_a to_o the_o square_n of_o a_o line_n compound_v of_o half_a the_o divide_a line_n and_o the_o line_n add_v if_o one_o add_v the_o line_n bd_o to_o the_o line_n ab_fw-la which_o be_v equal_o divide_v in_o c_o the_o rectangle_n a_fw-la comprehend_v under_o ad_fw-la and_o under_o dn_n or_o db_n with_o the_o square_n of_o cb_n be_v equal_a to_o the_o square_n of_o cd_o make_v the_o square_n of_o cd_o and_o have_v draw_v the_o diagonal_a fd_n draw_v bg_n parallel_v to_o fc_n which_o cut_v fd_v in_o the_o point_n h_n through_o which_o pass_v hn_n parallel_v to_o ab_fw-la kg_n shall_v be_v the_o square_n of_o bc_n and_o bn_n that_o of_o bd._n demonstration_n the_o rectangle_v ak_v ch_z on_o equal_a base_n ac_fw-la bc_n be_v equal_a by_o the_o 38_o of_o the_o 1_o the_o compliment_n ch_z he_o be_v equal_a by_o the_o 43_o of_o the_o 1_o therefore_o the_o rectangle_v ak_v he_o be_v equal_a add_v to_o both_o the_o rectangle_n cn_fw-la and_o the_o square_a kg_n the_o rectangle_v ak_v cn_fw-la that_o be_v to_o say_v the_o rectangle_n a_fw-la with_o the_o square_a kg_n shall_v be_v equal_a to_o the_o rectangle_v cn_fw-la he_o and_o to_o the_o square_a kg_n that_o be_v to_o say_v to_o the_o square_a ce._n arithmetical_o or_o by_o number_n let_v ab_fw-la be_v 8_o ac_fw-la 4_o cb_n 4_o bd_o 3_o than_o ad_fw-la shall_v be_v 11._o it_o be_v evident_a that_o the_o rectangle_n a_fw-la three_o time_n 11_o that_o be_v to_o say_v 33_o with_o the_o square_n of_o kg_v 16_o which_o together_o be_v 49_o be_v equal_a to_o the_o square_n of_o cd_o 7_o which_o be_v 49_o for_o 7_o time_n 7_o be_v 49._o use_v 6._o fig._n 6._o maurolycus_n measure_v the_o whole_a earth_n by_o one_o single_a

the_o 1_o because_o the_o side_n bc_n bd_o be_v equal_a the_o angle_n abc_n shall_v be_v double_a of_o each_o the_o second_o case_n be_v when_o a_o angle_n enclose_v the_o other_o and_o the_o line_n make_v the_o same_o angle_n not_o meet_v each_o other_o as_o you_o see_v in_o the_o second_o figure_n the_o angle_n bid_v be_v in_o the_o centre_n and_o the_o angle_n bad_a be_v at_o the_o circumference_n draw_v the_o line_n aic_n through_o the_o centre_n demonstration_n the_o angle_n bic_n be_v double_a to_o the_o angle_n bac_n and_o cid_n double_a to_o the_o angle_n god_n by_o the_o precede_a case_n therefore_o the_o angle_n bid_v shall_v be_v double_a to_o the_o angle_n bad_a use_v there_o be_v give_v ordinary_o a_o practical_a way_n to_o describe_v a_o horizontal_n dial_n by_o a_o single_a open_n of_o the_o compass_n which_o be_v ground_v in_o part_n on_o this_o proposition_n second_o when_o we_o will_v determine_v the_o apogaeon_fw-mi of_o the_o sun_n and_o the_o excentricity_n of_o his_o circle_n by_o three_o observation_n we_o suppose_v that_o the_o angle_n at_o the_o centre_n be_v double_a to_o the_o angle_n at_o the_o circumference_n ptolemy_n make_v often_o use_v of_o this_o proposition_n to_o determine_v as_o well_o the_o excentricity_n of_o the_o sun_n as_o the_o moon_n be_v epicycle_n the_o first_o proposition_n of_o the_o three_o book_n of_o trigonometry_n be_v ground_v on_o this_o proposition_n xxi_o theorem_fw-la the_o angle_n which_o be_v in_o the_o same_o segment_n of_o a_o circle_n or_o that_o have_v the_o same_o arch_n for_o base_a be_v equal_a if_o the_o angles_n bac_n bdc_n be_v in_o the_o same_o segment_n of_o a_o circle_n great_a than_o a_o semicircle_n they_o shall_v be_v equal_a draw_v the_o line_n by_o ci._n demonstration_n the_o angles_n a_o and_o d_o be_v each_o of_o they_o half_o of_o the_o angle_n bic_n by_o the_o precede_a proposition_n therefore_o they_o be_v equal_a they_o have_v also_o the_o same_o arch_a bc_n for_o base_a second_o 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figure_n to_o brass_n cauldron_n to_o the_o end_n we_o may_v work_v thereon_o and_o to_o polish_v prospective_n or_o telescope_n glass_n for_o have_v make_v in_o iron_n a_o angle_n bac_n equal_a to_o that_o which_o the_o segment_n abc_n contain_v and_o have_v put_v in_o the_o point_n b_o and_o c_o two_o small_a pin_n of_o iron_n if_o the_o triangle_n bac_n be_v make_v to_o move_v after_o such_o a_o manner_n that_o the_o side_n ab_fw-la may_v always_o touch_v the_o pin_n b_o and_o the_o side_n ac_fw-la the_o pin_n c_o the_o point_v a_o shall_v be_v always_o in_o the_o circumference_n of_o the_o circle_n abcd._n this_o way_n of_o describe_v a_o circle_n may_v also_o serve_v to_o make_v large_a astrolabe_fw-la proposition_n xxii_o theorem_fw-la qvadrilateral_a figure_n describe_v in_o a_o circle_n have_v their_o opposite_a angle_n equal_a to_o two_o right_a let_v a_o quadrilateral_a or_o four_o side_v figure_n be_v describe_v in_o a_o circle_n in_o such_o manner_n that_o all_o its_o angle_n touch_v the_o circumference_n of_o the_o circle_n abcd_v i_o say_v that_o its_o opposite_a angle_n bad_a bcd_v be_v equal_a to_o two_o right_a draw_v the_o diagonal_n ac_fw-la bd._n demonstration_n all_o the_o angle_n of_o the_o triangle_n bad_a be_v equal_a to_o two_o right_a in_o the_o place_n of_o its_o angle_n abdella_n put_v the_o angle_n acd_v which_o be_v equal_a thereto_o by_o the_o 21_o as_o be_v in_o the_o same_o segment_n abcd_v and_o in_o the_o place_n of_o its_o angle_n adb_n put_v the_o angle_n acb_n which_o be_v in_o the_o same_o segment_n of_o the_o circle_n bcda_n so_o then_o the_o angle_n bad_a and_o the_o angle_n acd_v acb_n that_o be_v to_o say_v the_o whole_a angle_n bcd_n be_v equal_a to_o two_o right_a use_v ptolomy_n make_v use_v of_o this_o proposition_n to_o make_v the_o table_n of_o chord_n or_o subtendent_o i_o have_v also_o make_v use_v thereof_o in_o trigonometry_n in_o the_o three_o book_n to_o prove_v that_o the_o side_n of_o a_o obtuse_a angle_a triangle_n have_v the_o same_o reason_n among_o themselves_o as_o the_o sin_n of_o their_o opposite_a angle_n proposition_n xxiii_o theorem_fw-la two_o like_a segment_n of_o a_o circle_n describe_v on_o the_o same_o line_n be_v equal_a i_o call_v like_o segment_n of_o a_o circle_n those_o which_o contain_v equal_a angle_n and_o i_o say_v that_o if_o they_o be_v describe_v on_o the_o same_o line_n ab_fw-la they_o shall_v fall_v one_o on_o the_o other_o and_o shall_v not_o surpass_v each_o other_o in_o any_o part_n for_o if_o they_o do_v surpass_v each_o other_o as_o do_v the_o segment_n acb_n the_o segment_n adb_n they_o will_v not_o be_v like_a and_o to_o demonstrate_v it_o draw_v the_o line_n adc_a db_n and_o bc._n demonstration_n the_o angle_n adb_n be_v exterior_a in_o respect_n of_o the_o triangle_n bdc_n thence_o by_o the_o 21_o of_o the_o 1_o it_o be_v great_a than_o the_o angle_n acb_n and_o by_o consequence_n the_o segments_n adb_n acb_n contain_v unequal_a angle_n which_o i_o call_v unlike_a proposition_n xxiv_o theorem_fw-la two_o like_a segment_n of_o circle_n describe_v on_o equal_a line_n be_v equal_a if_o the_o segment_n of_o circle_n aeb_fw-mi cfd_n be_v like_a and_o if_o the_o line_n ab_fw-la cd_o be_v equal_a they_o shall_v be_v equal_a demonstration_n let_v it_o be_v imagine_v that_o the_o line_n cd_o be_v place_v on_o ab_fw-la they_o shall_v not_o surpass_v each_o other_o see_v they_o be_v suppose_v equal_a and_o then_o the_o segment_n aeb_fw-mi ce_v shall_v be_v describe_v on_o the_o same_o line_n and_o they_o shall_v then_o be_v equal_a by_o the_o precede_a proposition_n use_v 24._o use_v 24._o cvrve_v line_a figure_n be_v often_o reduce_v to_o right_o line_v by_o this_o proposition_n as_o if_o one_o shall_v describe_v two_o like_a segment_n of_o circle_n aec_fw-la adb_fw-la on_o the_o equal_a side_n ab_fw-la ac_fw-la of_o the_o triangle_n abc_n it_o be_v evident_a that_o transpose_v the_o segment_n aec_fw-la on_o adb_n the_o triangle_n abc_n 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propose_v the_o point_n a_o b_o c_o put_v the_o point_n of_o the_o compass_n in_o c_o and_o at_o what_o open_v soever_o describe_v two_o ark_n f_o and_o e._n transport_v the_o foot_n of_o the_o compass_n to_o b_o and_o with_o the_o same_o open_n describe_z two_o arck_n to_o cut_v the_o former_a in_o e_z and_z f._n describe_v on_o b_o as_o centre_n at_o what_o open_v soever_o the_o arch_n h_n and_o g_o and_o at_o the_o same_o open_n of_o the_o compass_n describe_v on_o the_o centre_n a_o two_o ark_n to_o cut_v the_o same_o in_o g_z and_z h._n draw_v the_o line_n fe_o and_o gh_a which_o will_v cut_v each_o other_o in_o the_o point_n d_o the_o centre_n of_o the_o circle_n the_o demonstration_n be_v evident_a enough_o for_o if_o you_o have_v draw_v

the_o diameter_n basilius_n shall_v divide_v each_o other_o equal_o in_o o._n draw_v the_o line_n bg_n gk_n ah_o lh_n i_o prove_v in_o the_o first_o place_n that_o they_o make_v but_o one_o strait_a line_n for_o the_o triangle_n dbg_n kmg_n have_v their_o side_n bd_o km_n equal_a see_v they_o be_v the_o half_n of_o equal_a side_n as_o also_o gd_v gm_n moreover_o db_n km_fw-la be_v parallel_n the_o alternate_a angel_n bdg_n gmk_n shall_v be_v equal_a by_o the_o 29_o of_o the_o first_o so_o then_o by_o the_o four_o of_o the_o first_o the_o triangle_n dbg_n kgm_n shall_v be_v equal_a in_o every_o respect_n and_o consequent_o the_o angle_n bgd_v kgm_n now_o by_o the_o 16_o of_o the_o one_a bg_n gk_n be_v but_o one_o line_n as_o also_o lh_n ha_o thence_o all_o bk_n be_v but_o one_o plane_n in_o which_o be_v find_v the_o diameter_n ab_fw-la and_o gh_n the_o common_a section_n of_o the_o plane_n the_o plane_n all_o bk_n cut_v the_o parallel_n planes_n a_o cd_o shall_v have_v its_o common_a 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circle_n be_v like_a they_o shall_v be_v in_o the_o same_o ratio_fw-la as_o be_v the_o square_n of_o the_o diameter_n be_o fn_n draw_v the_o line_n bm_n gn_n fh_o ac_fw-la demonstration_n it_o be_v suppose_v that_o the_o polygon_n be_v like_a that_o be_v to_o say_v that_o the_o angle_n b_o and_z g_z be_v equal_a and_o that_o there_o be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o bc_n as_o of_o fg_a to_o gh_v whence_o i_o conclude_v by_o the_o 6_o of_o the_o 6_o that_o the_o triangle_n abc_n fgh_a be_v equiangular_a and_o that_o the_o angles_n acb_n fhg_n be_v equal_a so_o then_o by_o the_o 21_o of_o the_o 3d._n the_o angel_n ambiguity_n fng_v be_v equal_a now_o the_o angel_n abm_n fgn_n be_v in_o a_o semicircle_n be_v right_o by_o the_o 31st_o of_o the_o 3d._n and_o consequent_o the_o triangle_n abm_n fgn_n be_v equiangular_a therefore_o by_o the_o 5_o of_o the_o 6_o there_o shall_v be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o fg_n as_o of_o be_o to_o fn_n and_o by_o the_o 22d_o 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than_o the_o one_o half_a of_o the_o circle_n it_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o in_o describe_v the_o octagon_n you_o take_v away_o more_o than_o the_o half_a of_o the_o remainder_n that_o be_v to_o say_v the_o four_o segment_n chd_v die_v ekf_n clf_n for_o the_o triangle_n chd_n be_v the_o half_a of_o the_o rectangle_n co_n by_o the_o 34th_o of_o the_o one_a it_o be_v then_o more_o than_o the_o half_a of_o the_o segment_n chd_v it_o be_v the_o same_o of_o the_o other_o arks._n in_o like_a manner_n in_o describe_v a_o polygon_n of_o sixteen_o side_n you_o take_v away_o more_o than_o one_o half_a of_o that_o which_o remain_v in_o the_o circle_n and_o so_o of_o the_o rest_n you_o