Selected quad for the lemma: book_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
book_n	great_a	part_n	time_n	3,297	5	3.1639	3	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

for_o pentagon_n be_v the_o most_o ordinary_a you_o must_v also_o take_v notice_n that_o these_o way_n of_o describe_v a_o pentagon_n about_o a_o circle_n may_v be_v apply_v to_o the_o other_o polygon_n i_o have_v give_v another_o way_n to_o inscribe_v a_o regular_a pentagon_n in_o a_o circle_n in_o military_a architecture_n proposition_n xv._o problem_n to_o inscribe_v a_o regular_a hexagon_n in_o a_o circle_n to_o inscribe_v a_o regular_a hexagon_n in_o the_o circle_n abcdef_n draw_v the_o diameter_n ad_fw-la and_o put_v the_o foot_n of_o the_o compass_n in_o the_o point_n d_o describe_v a_o circle_n at_o the_o open_a dg_n which_o shall_v intersect_v the_o circle_n in_o the_o point_n aec_fw-la then_o draw_v the_o diameter_n egb_n cgf_n and_o the_o line_n ab_fw-la of_o and_o the_o other_o demonstration_n it_o be_v evident_a that_o the_o triangle_n cdg_n dge_n be_v equilateral_a wherefore_o the_o angle_n cgd_v dge_n and_o their_o opposite_n bga_n agf_n be_v each_o of_o they_o the_o 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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

to_o efg_v demonstration_n we_o have_v already_o demonstrate_v that_o there_o be_v a_o great_a reason_n of_o a_o b_o c_o to_z e_z f_o g_o than_o of_o the_o part_n bc_n to_o the_o part_n fg_v there_o shall_v thence_o be_v a_o great_a reason_n of_o a_o to_o e_o than_o of_o a_o b_o c_o to_z e_z f_o g_o by_o the_o 32d_o the_o end_n of_o the_o five_o book_n the_o sixth_z book_n of_o euclid_n element_n this_o book_n explain_v and_o begin_v to_o apply_v particular_a matter_n of_o the_o doctrine_n of_o proportion_n which_o the_o precede_a book_n explain_v but_o in_o general_n it_o begin_v with_o the_o most_o easy_a figure_n that_o be_v to_o say_v triangle_n give_v rule_n to_o determine_v not_o only_o the_o proportion_n of_o their_o side_n but_o also_o that_o of_o their_o capacity_n area_n or_o superficies_n then_o it_o teach_v to_o find_v proportional_a line_n and_o to_o augment_v or_o diminish_v any_o figure_n whatever_o according_a to_o a_o give_v ratio_fw-la it_o demonstrate_v the_o rule_n of_o three_o it_o appli_v the_o forty_o seven_o of_o the_o first_o to_o all_o sort_n of_o figure_n in_o fine_a it_o give_v we_o the_o most_o easy_a and_o most_o certain_a principle_n to_o conduct_v we_o in_o all_o sort_n of_o measure_v the_o definition_n 1._o 6._o def._n 1._o fig._n i._o plate_n 6._o right_o line_v figure_n be_v like_a when_o that_o they_o have_v all_o their_o angle_n equal_a and_o the_o side_n which_o form_v those_o angel_n proportional_a i._n fig._n i._n as_o the_o triangle_n abc_n def_n shall_v be_v like_a if_o the_o angle_n a_o and_o d_o b_o and_z e_z c_z and_z f_o be_v equal_a and_o if_o there_o be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o ac_fw-la as_o of_o de_fw-fr to_o df_n and_o of_o ab_fw-la to_o cb_n as_o of_o de_fw-fr to_o ef._n 2._o ii_o fig._n ii_o figure_n be_v reciprocal_a when_o they_o may_v be_v compare_v after_o such_o sort_n that_o the_o antecedent_n of_o one_o reason_n and_o the_o consequent_a of_o the_o other_o be_v find_v in_o the_o same_o figure_n that_o be_v when_o the_o analogy_n begin_v in_o one_o figure_n and_o end_v in_o the_o same_o as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o cd_o as_o of_o de_fw-fr to_o bf_n 3._o iii_o fig._n iii_o a_o line_n be_v divide_v into_o extreme_a and_o mean_a proportion_n or_o reason_n when_o there_o be_v the_o same_o reason_n of_o the_o whole_a line_n to_o its_o great_a part_n as_o of_o its_o great_a part_n to_o its_o lesser_a part_n as_o if_o there_o be_v the_o same_o reason_n of_o ab_fw-la to_o ac_fw-la as_o of_o ac_fw-la to_o cb_n the_o line_n ab_fw-la shall_v be_v divide_v in_o the_o point_n c_o in_o extreme_a and_o mean_a proportion_n 4._o the_o altitude_n or_o height_n of_o a_o figure_n be_v the_o length_n of_o the_o perpendicular_a draw_v from_o the_o top_n thereof_o to_o its_o base_a iu._n fig._n iu._n as_o in_o the_o triangle_n abc_n efg_v the_o perpendicular_a eh_n ad_fw-la whether_o it_o fall_v without_o or_o within_o the_o triangle_n be_v their_o height_n or_o altitude_n triangle_n and_o parallellogram_n who_o height_n be_v equal_a may_v be_v place_v between_o the_o same_o parallel_n for_o have_v place_v their_o base_n on_o the_o same_o line_n hc_n if_o the_o perpendicular_o da_fw-la he_o be_v equal_a the_o line_n ea_fw-la hc_fw-la shall_v be_v parallel_n 5._o a_o reason_n be_v compose_v of_o several_a reason_n when_o the_o quantity_n of_o those_o reason_n be_v multiply_v make_v a_o three_o reason_n it_o be_v to_o be_v take_v notice_n of_o that_o a_o reason_n at_o least_o the_o rational_a have_v its_o name_n take_v from_o some_o number_n which_o specifi_v the_o reason_n or_o habitude_n of_o the_o antecedent_n of_o that_o reason_n to_o its_o consequent_a as_o when_o there_o be_v propose_v two_o magnitude_n or_o quantity_n the_o one_o of_o twelve_o foot_n and_o the_o other_o of_o six_o we_o say_v that_o the_o reason_n of_o twelve_o to_o six_o be_v double_a in_o like_a manner_n when_o there_o be_v propose_v two_o magnitude_n four_o and_o twelve_o we_o will_v say_v it_o be_v a_o subtriple_a reason_n and_o ⅓_n be_v the_o denominator_fw-la thereof_o which_o specifi_v that_o there_o be_v the_o same_o reason_n of_o four_o to_o twelve_o as_o of_o ⅓_n to_o one_o or_o as_o of_o one_o to_o three_o one_o may_v call_v this_o denominaotr_n the_o quantity_n of_o the_o reason_n let_v there_o be_v propose_v three_o term_n twelve_o six_o two_o the_o first_o reason_n of_o twelve_o to_o six_o be_v double_a its_o denominator_fw-la be_v two_o the_o reason_n of_o six_o to_o two_o be_v triple_a its_o denominator_fw-la be_v three_o the_o reason_n of_o twelve_o to_o two_o be_v compound_v of_o the_o reason_n of_o twelve_o to_o six_o and_o of_o that_o of_o six_o to_o two_o to_o have_v th●…_n denominator_fw-la or_o the_o reason_n of_o twelve_o to_o two_o which_o be_v compound_v of_o double_a and_o of_o triple_a multiply_v three_o by_o two_o and_o you_o shall_v have_v six_o thence_o the_o reason_n of_o twelve_o to_o two_o be_v sextuple_a this_o be_v that_o which_o mathematician_n understand_v by_o composition_n of_o reason_n although_o it_o ought_v to_o be_v call_v multiplication_n of_o reason_n proposition_n i._o theorem_fw-la parallellogram_n and_o triangle_n of_o equal_a height_n have_v the_o same_o reason_n as_o their_o base_n vi._n plate_n vi._n let_v there_o be_v propose_v the_o triangle_n acg_n dem_n of_o equal_a height_n in_o such_o sort_n that_o they_o may_v be_v place_v between_o the_o same_o parallel_n ad_fw-la gm_n i_o say_v that_o there_o shall_v be_v the_o same_o reason_n of_o the_o base_a gc_n to_o the_o base_a em_n as_o of_o the_o triangle_n agc_n to_o the_o triangle_n dem_n let_v the_o base_a em_n be_v divide_v into_o as_o many_o equal_a part_n as_o you_o please_v and_o let_v there_o be_v draw_v to_o each_o division_n the_o line_n df_n dh_n etc._n etc._n let_v also_o the_o line_n gc_fw-mi be_v divide_v into_o part_n equal_a to_o those_o of_o em_n and_o let_v be_v draw_v line_n to_o each_o division_n from_o the_o top_n a_o all_o those_o little_a triangle_n which_o be_v make_v in_o the_o two_o great_a triangle_n be_v between_o the_o same_o parallel_n and_o they_o have_v equal_a base_n they_o be_v thence_o equal_a by_o the_o 28_o demonstration_n the_o base_a gc_n contain_v as_o many_o aliquot_fw-la part_n of_o the_o line_n em_n as_o can_v be_v find_v part_n equal_a to_o of_o now_o as_o many_o time_n as_o there_o be_v in_o the_o base_a gc_a part_n equal_a to_o of_o so_o many_o time_n the_o triangle_n agc_n contain_v the_o little_a triangle_n equal_a to_o those_o which_o be_v in_o the_o triangle_n dem_n which_o be_v equal_a among_o themselves_o be_v its_o aliquot_fw-la part_n thence_o as_o many_o time_n as_o the_o base_a gc_n contain_v aliquot_fw-la part_n of_o em_n so_o many_o time_n the_o triangle_n agc_n contain_v aliquot_fw-la part_n of_o the_o triangle_n dem_n which_o will_v happen_v in_o all_o manner_n of_o division_n there_o be_v thence_o the_o same_o reason_n of_o the_o base_a gc_n to_o the_o base_a em_n as_o of_o the_o triangle_n agc_n to_o the_o triangle_n dem_n coral_n parallellogram_n draw_v on_o the_o same_o base_n and_o that_o be_v between_o the_o same_o parallel_n line_n be_v double_a to_o the_o triangle_n by_o the_o 41st_o they_o be_v thence_o in_o the_o same_o reason_n as_o triangle_n that_o be_v to_o say_v in_o the_o same_o reason_n as_o their_o base_n use_v i._n fig._n i._n this_o proposition_n be_v not_o only_o necessary_a to_o demonstrate_v those_o which_o follow_v but_o we_o may_v make_v use_n thereof_o in_o divide_v of_o land_n let_v there_o be_v propose_v a_o trapezium_fw-la abcd_n which_o have_v its_o side_n ad_fw-la bc_n parallel_n and_o admit_v one_o will_v cut_v of_o a_o three_o part_n let_v ce_fw-fr be_v make_v equal_a to_o ad_fw-la and_o bg_n the_o three_o part_n of_o be._n draw_v ag._n i_o say_v that_o the_o triangle_n abg_n be_v the_o three_o of_o the_o trapezium_fw-la abcd._n demonstration_n the_o triangle_n adf_n fce_n be_v equiangle_v because_o of_o the_o parallel_n ad_fw-la ce_fw-fr and_o they_o have_v their_o sides_n ad_fw-la ce_fw-fr equal_a they_o be_v thence_o equal_a by_o the_o 26_o of_o the_o one_a and_o by_o consequence_n the_o triangle_n abe_n be_v equal_a to_o the_o trapezium_fw-la now_o the_o triangle_n abg_n be_v the_o three_o part_n of_o the_o triangle_n abe_n by_o the_o precede_a proposition_n thence_o the_o triangle_n abg_n be_v the_o three_o part_n of_o the_o trapezium_fw-la abcd._n proposition_n ii_o theorem_fw-la a_o line_n be_v draw_v in_o a_o triangle_n parallel_n to_o its_o base_a divide_v its_o side_n proportional_o and_o if_o a_o line_n divide_v proportional_o the_o side_n of_o a_o triangle_n it_o shall_v be_v parallel_n to_o its_o base_a if_o in_o the_o

of_o to_o make_v all_o sort_n of_o figure_n great_a or_o lesser_a for_o it_o be_v more_o universal_a then_o the_o 47th_o of_o the_o one_a which_o notwithstanding_o be_v so_o useful_a that_o it_o seem_v that_o almost_o all_o geometry_n be_v establish_v on_o this_o principle_n the_o 32d_o proposition_n be_v unnecessary_a proposition_n xxxiii_o theorem_fw-la in_o equal_a circle_n the_o angle_n as_o well_o those_o at_o the_o centre_n as_o those_o at_o the_o circumference_n as_o also_o the_o sector_n be_v in_o the_o same_o ratio_fw-la as_o the_o ark_n which_o serve_v to_o they_o as_o base_a if_o the_o circle_n anc_n dof_n be_v equal_a there_o shall_v be_v the_o same_o ratio_fw-la of_o the_o angle_n abc_n to_o the_o angle_n def_n as_o of_o the_o ark_n ac_fw-la to_o the_o ark_n df._n let_v the_o ark_n agnostus_n gh_n hc_n he_o equal_a ark_n and_o consequent_o aliquot_fw-la part_n of_o the_o ark_n ac_fw-la and_o let_v be_v divide_v the_o ark_n df_n into_o so_o many_o equal_a to_o agnostus_n as_o may_v be_v find_v therein_o and_o let_v the_o line_n ei_o eke_o and_o the_o rest_n be_v draw_v demonstration_n all_o the_o angles_n abg_n gbh_n hbc_n dei_fw-la jek_n and_o the_o rest_n be_v equal_a by_o the_o 37th_o of_o the_o three_o so_o then_o agnostus_n a_o aliquot_fw-la part_n of_o ac_fw-la be_v find_v in_o the_o ark_n df_n as_o many_o time_n as_o the_o angle_n abg_n a_o aliquot_fw-la part_n of_o the_o angle_n abc_n be_v find_v in_o def_n there_o be_v therefore_o the_o same_o ratio_fw-la of_o the_o ark_n ac_fw-la to_o the_o ark_n df_n as_o of_o the_o angle_n abc_n to_o the_o angle_n def_n and_o because_o n_n and_o o_o be_v the_o half_n of_o the_o angel_n abc_n def_n they_o shall_v be_v in_o the_o same_o ratio_fw-la as_o be_v those_o angles_n there_o be_v therefore_o the_o same_o ratio_fw-la of_o the_o angle_n n_n to_o the_o angle_n o_o as_o of_o the_o ark_n ac_fw-la to_o the_o ark_n df._n it_o be_v the_o same_o with_o sector_n for_o if_o the_o line_n agnostus_n gh_n hc_n di_fw-it ik_fw-mi and_o the_o rest_n be_v draw_v they_o will_v be_v equal_a by_o the_o 28_o of_o the_o three_o and_o each_o sector_n will_v be_v divide_v into_o a_o triangle_n and_o a_o segment_n the_o triangle_n will_v be_v equal_a by_o the_o 8_o of_o the_o first_o and_o the_o little_a segment_n will_v be_v also_o equal_a by_o the_o 24_o of_o the_o three_o thence_o all_o those_o little_a sector_n will_v be_v equal_a and_o so_o as_o many_o as_o the_o ark_n bf_n contain_v of_o aliquot_fw-la part_n of_o the_o ark_n of_o ac_fw-la so_o many_o the_o sector_n dkf_n will_v contain_v aliquot_fw-la part_n of_o the_o sector_n agc_n there_o be_v therefore_o the_o same_o reason_n of_o ark_n to_o ark_n as_o of_o sector_n to_o sector_n the_o end_n of_o the_o six_o book_n the_o eleven_o book_n of_o euclid_n element_n this_o book_n comprehend_v the_o first_o principle_n of_o solid_a body_n whence_o it_o be_v impossible_a to_o establish_v any_o thing_n touch_v the_o three_o species_n of_o quantity_n without_o know_v what_o this_o book_n teach_v we_o which_o make_v it_o very_o necessary_a in_o the_o great_a part_n of_o the_o mathematics_n in_o the_o first_o place_n theodosius_n spherics_n do_v whole_o suppose_v it_o spherical_a trigonometry_n the_o three_o part_n of_o practical_a geometry_n several_a proposition_n of_o staticks_n and_o of_o geography_n be_v establish_v on_o the_o principle_n of_o solid_a body_n gnomonic_n conical_a section_n and_o the_o treatise_n of_o stone-cutting_a be_v not_o difficult_a but_o because_o one_o be_v often_o oblige_v to_o represent_v on_o paper_n figure_n that_o be_v raise_v and_o which_o be_v comprise_v under_o several_a surface_n i_o leave_v out_o the_o seven_o the_o eight_o the_o nine_o and_o the_o ten_o book_n of_o euclid_n element_n because_o they_o be_v unnecessary_a in_o almost_o all_o the_o part_n of_o the_o mathematics_n i_o have_v often_o wonder_v that_o they_o have_v be_v put_v among_o the_o element_n see_v it_o be_v evident_a that_o euclid_n do_v compose_v they_o only_o to_o establish_v the_o doctrine_n of_o incommensurable_n which_o be_v only_o a_o curiosity_n ought_v not_o to_o have_v be_v place_v with_o the_o book_n of_o element_n but_o aught_o to_o make_v a_o particular_a treatise_n the_o same_o may_v be_v say_v of_o the_o thirteen_o and_o of_o the_o rest_n for_o i_o believe_v that_o one_o may_v learn_v almost_o all_o part_n of_o the_o mathematics_n provide_v one_o under●…_n well_o these_o eight_o book_n of_o euclid_n element_n definition_n 1._o i._n def._n 1._o plate_n vii_o fig._n i._n a_o solid_a be_v a_o magnitude_n which_o have_v length_n breadth_n and_o thickness_n as_o the_o figure_n lt_n its_o length_n be_v nx_n its_o breadth_n no_o its_o thickness_n ln_o 2._o the_o extremity_n or_o term_n of_o a_o solid_a be_v superficies_n 3._o a_o line_n be_v at_o right_a angle_n or_o perpendicular_a to_o a_o plane_n when_o it_o be_v perpendicular_a to_o all_o the_o line_n that_o it_o meet_v with_o in_o the_o plane_n ii_o fig._n ii_o as_o the_o line_n ab_fw-la shall_v be_v at_o right_a angle_n to_o the_o plane_n cd_o if_o it_o be_v perpendicular_a to_o the_o line_n cd_o fe_o which_o line_n be_v draw_v in_o the_o plane_n cd_o pass_v through_o the_o point_n b_o in_o such_o sort_n that_o the_o angel_n abc_n abdella_n abe_n abf_n be_v right_o 4._o a_o plane_n be_v perpendicular_a to_o another_o when_o the_o line_n perpendicular_a to_o the_o common_a section_n of_o the_o plane_n and_o draw_v in_o the_o one_o be_v also_o perpendicular_a to_o the_o other_o plane_n iii_o fig._n iii_o we_o call_v the_o common_a section_n of_o the_o plane_n a_o line_n which_o be_v in_o both_o plane_n as_o the_o line_n ab_fw-la which_o be_v also_o in_o the_o plane_n ac_fw-la as_o well_o as_o in_o the_o plane_n ad._n if_o then_o the_o line_n de_fw-fr draw_v in_o the_o plane_n ad_fw-la and_o perpendicular_a to_o ab_fw-la be_v also_o perpendicular_a to_o the_o plane_n ac_fw-la the_o plane_n ad_fw-la shall_v be_v right_o or_o perpendicular_a to_o the_o plane_n ac_fw-la 5._o iu._n fig._n iu._n if_o the_o line_n ab_fw-la be_v not_o perpendicular_a to_o the_o plane_n cd_o and_o if_o there_o be_v draw_v from_o the_o point_n a_o the_o line_n ae_n perpendicular_a to_o the_o plane_n cd_o and_o then_o the_o line_n be_v the_o angle_n abe_n be_v that_o of_o the_o inclination_n of_o the_o line_n ab_fw-la to_o the_o plane_n cd_o 6._o v._n fig._n v._n the_o inclination_n of_o one_o plane_n to_o another_o be_v the_o acute_a angle_n comprehend_v by_o the_o two_o perpendicular_o to_o the_o common_a section_n draw_v in_o each_o plane_n as_o the_o inclination_n of_o the_o plane_n ab_fw-la to_o the_o plane_n ad_fw-la be_v no_o other_o than_o the_o angle_n bcd_v comprehend_v by_o the_o line_n bccd_v drawnin_n both_o plane_n perpendicular_a to_o the_o common_a section_n ae_n 7._o plain_n shall_v be_v incline_v after_o the_o same_o manner_n if_o the_o angle_n of_o inclination_n be_v equal_a 8._o planes_n which_o be_v 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