Selected quad for the lemma: book_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
book_n	demonstration_n	proposition_n	quantity_n	1,827	5	13.9195	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

abc_n draw_v the_o touch_n line_n feed_v by_o the_o 17_o of_o the_o 3d._n and_o make_v at_o the_o point_n of_o touch_v e_z the_o angle_n deh_fw-it equal_a to_o the_o angle_n b_o and_o the_o angle_n feg_n equal_a to_o the_o angle_n c_o by_o the_o 23d_o of_o the_o one_a draw_v the_o line_n gh_o the_o triangle_n egh_n shall_v be_v equi_fw-la angle_v to_o abc_n demonstration_n the_o angle_n deh_n be_v equal_a to_o the_o angle_n egh_n of_o the_o alternate_a segment_n by_o the_o 32d_o of_o the_o 3d._n now_o the_o angle_n deh_n be_v make_v equal_a to_o the_o angle_n b_o and_o by_o consequence_n the_o angle_n b_o and_o g_o be_v equal_a the_o angle_n c_o and_o h_o be_v also_o equal_a for_o the_o same_o reason_n and_o by_o the_o coral_n 2._o of_o the_o 32d_o of_o the_o one_a the_o angle_n a_o and_o geh_a shall_v be_v equal_a therefore_o the_o triangle_n egh_n abc_n be_v equiangle_v proposition_n iii_o problem_n to_o describe_v about_o a_o circle_n a_o triangle_n equiangle_v to_o another_o if_o one_o will_v describe_v about_o a_o circle_n gkh_n a_o triangle_n equiangle_v to_o abc_n one_o of_o the_o side_n bc_n must_v be_v continue_v to_o d_o and_z e_z and_o make_v the_o angle_n gih_n equal_a to_o the_o angle_n abdella_n and_o hik_n equal_a to_o the_o angle_n ace_n then_o draw_v the_o tangent_n lgm_n lkn_n nhm_o through_o the_o point_n g_o k_o h._n the_o tangent_n shall_v meet_v each_o other_o for_o the_o angles_n ikl_n igl_n be_v right_o if_o one_o shall_v draw_v the_o line_n kg_v which_o be_v not_o draw_v the_o angles_n kgl_n gkl_n will_v be_v less_o than_o two_o right_n therefore_o by_o the_o 11_o axiom_n the_o line_n gl_n kl_n aught_o to_o concur_v demonstration_n all_o the_o angle_n of_o the_o quadrilateral_a gihm_n be_v equal_a to_o four_o right_n see_v it_o may_v be_v reduce_v into_o two_o triangle_n the_o angel_n igm_n ihm_fw-ge which_o be_v make_v by_o the_o tangent_n be_v right_o thence_o the_o angle_n m_o an_o i_o be_v equivalent_a to_o two_o right_a as_o well_o as_o the_o angel_n abc_n abdella_n now_o the_o angle_n gih_n be_v equal_a to_o the_o angle_n abdella_n by_o construction_n therefore_o the_o angle_n m_o shall_v be_v equal_a to_o the_o angle_n abc_n for_o the_o same_o reason_n the_o angle_n n_n and_o acb_n be_v equal_a and_o so_o the_o triangle_n lmn_n abc_n be_v equiangle_v proposition_n iv_o problem_n to_o inscribe_v a_o circle_n in_o a_o triangle_n if_o you_o will_v inscribe_v a_o circle_n in_o a_o triangle_n abc_n divide_v into_o two_o equal_o the_o angel_n abc_n acb_n by_o the_o 9th_o of_o the_o one_a draw_v the_o line_n cd_o bd_o which_o concur_v in_o the_o point_n d._n then_o draw_v from_o the_o point_n d_o the_o perpendicular_n de_fw-fr df_n dg_n which_o shall_v be_v equal_a so_o that_o the_o circle_n describe_v on_o the_o point_n d_o at_o the_o open_n de_fw-fr shall_v pass_v through_o f_o and_o g._n demonstration_n the_o triangle_n deb_n dbf_n have_v the_o angles_n deb_n dfb_n equal_a see_v they_o be_v right_o the_o angel_n dbe_n dbf_n be_v also_o equal_a the_o angle_n abc_n have_v be_v divide_v into_o two_o equal_o the_o side_n db_n be_v common_a therefore_o by_o the_o 26_o of_o the_o one_a these_o triangle_n shall_v be_v equal_a in_o every_o respect_n and_o the_o side_n de_fw-fr df_n shall_v be_v equal_a one_o may_v demonstrate_v after_o the_o same_o manner_n that_o the_o side_n df_n dg_n be_v equal_a one_o may_v therefore_o describe_v a_o circle_n which_o shall_v pass_v through_o the_o point_n e_o f_o g_o and_o see_v the_o angle_n e_o f_o g_o be_v right_o the_o side_n ab_fw-la ac_fw-la bc_n touch_v the_o circle_n which_o shall_v by_o consequence_n be_v inscribe_v in_o the_o triangle_n proposition_n v._o problem_n to_o describe_v a_o circle_n about_o a_o triangle_n if_o you_o will_v describe_v a_o circle_n about_o a_o triangle_n abc_n divide_v the_o side_n ab_fw-la bc_n into_o two_o equal_o in_o d_o and_o e_o drawing_z the_o perpendicular_o df_n of_o which_o concur_v in_o the_o point_n f._n if_o you_o describe_v a_o circle_n on_o the_o centre_n f_o at_o the_o open_a fb_n it_o shall_v pass_v through_o a_o and_o c_o that_o be_v to_o say_v that_o the_o line_n favorina_n fb_n fc_fw-la be_v equal_a demonstration_n the_o triangle_n adf_n bdf_n have_v the_o side_n df_n common_a and_o the_o side_n ad_fw-la db_fw-la equal_a see_v the_o side_n ab_fw-la have_v be_v divide_v equal_o and_o the_o angle_n in_o d_o be_v equal_a be_v right_o thence_o by_o the_o four_o of_o the_o one_a the_o base_n of_o bf_n be_v equal_a and_o for_o the_o same_o reason_n the_o base_n bf_n cf._n use_v we_o have_v often_o need_v to_o inscribe_v a_o triangle_n in_o a_o circle_n as_o in_o the_o first_o proposition_n of_o the_o three_o book_n of_o trigonometry_n this_o practice_n be_v necessary_a for_o to_o measure_v the_o area_n of_o a_o triangle_n and_o upon_o several_a other_o occasion_n proposition_n vi_o problem_n to_o inscribe_v a_o square_a in_o a_o circle_n to_o inscribe_v a_o square_a in_o the_o circle_n abcd_v draw_v to_o the_o diameter_n ab_fw-la the_o perpendicular_a dc_o which_o may_v pass_v through_o the_o centre_n e._n draw_v also_o the_o line_n ac_fw-la cb_n bd_o ad_fw-la and_o you_o will_v have_v inscribe_v in_o the_o circle_n the_o square_n acbd_v demonstration_n the_o triangle_n aec_fw-la ceb_fw-mi have_v their_o side_n equal_a and_o the_o angle_n aec_fw-la ceb_fw-mi equal_a see_v they_o be_v right_o therefore_o the_o base_n ac_fw-la cb_n be_v equal_a by_o the_o four_o of_o the_o one_a moreover_o see_v the_o side_n ae_n ce_fw-fr be_v equal_a and_o the_o angle_n e_o being_n right_n they_o shall_v each_o of_o they_o be_v semi-right_a by_o the_o 32d_o of_o the_o one_a so_o then_o the_o angle_n ecb_n be_v semi-right_a and_o by_o consequence_n the_o angle_n acb_n shall_v be_v right_o it_o be_v the_o same_o of_o all_o the_o other_o angles_n therefore_o the_o figure_n acdb_n be_v a_o square_a proposition_n vii_o problem_n to_o describe_v a_o square_a about_o a_o circle_n have_v draw_v the_o two_o diameter_n ab_fw-la cd_o which_o cut_v each_o other_o perpendicular_o in_o the_o centre_n e_o draw_v the_o touch_n line_n fg_v gh_o he_o fi_n through_o the_o point_v a_o d_o b_o c_o and_o you_o will_v have_v describe_v a_o square_a fghi_n about_o the_o circle_n acbd_v demonstration_n the_o angle_n e_o and_o a_o be_v right_o thence_o by_o the_o 29_o of_o the_o one_a the_o line_n fg_v cd_o be_v parallel_n i_o prove_v after_o the_o same_o manner_n that_o cd_o he_o fi_n ab_fw-la ab_fw-la gh_n be_v parallel_n thence_o the_o figure_n fcdg_n be_v a_o parallelogram_n and_o by_o the_o 34th_o of_o the_o one_a the_o line_n fg_v cd_o be_v equal_a as_o also_o cd_o ih_o fi_n ab_fw-la ab_fw-la gh_n and_o by_o consequence_n the_o side_n of_o the_o figure_n fg_v gh_o he_o if_o be_v equal_a moreover_o see_v the_o line_n fg_v cd_o be_v parallel_n and_o that_o the_o angle_n fce_n be_v right_o the_o angle_n g_o shall_v be_v also_o right_o by_o the_o 29_o of_o the_o one_a i_o demonstrate_v after_o the_o same_o manner_n that_o the_o angle_n f_o h_o and_o i_o be_v right_o therefore_o the_o figure_n fghi_n be_v a_o square_a and_o its_o side_n touch_v the_o circle_n proposition_n viii_o problem_n to_o inscribe_v a_o circle_n in_o a_o square_a if_o you_o will_v inscribe_v a_o circle_n in_o the_o square_a fghi_n divide_v the_o side_n fg_v gh_o he_o fi_n in_o the_o middle_n in_o a_o d_o b_o c_o and_o draw_v the_o line_n ab_fw-la cd_o which_o cut_v each_o other_o in_o the_o point_n e._n i_o demonstrate_v that_o the_o line_n ea_fw-la ed_z aec_fw-la ebb_n be_v equal_a and_o that_o the_o angle_n in_o a_o b_o c_o d_o be_v right_o and_o that_o so_o you_o may_v describe_v a_o circle_n on_o the_o centre_n e_o which_o shall_v pass_v through_o a_o d_o b_o c_o and_o which_o touch_v the_o side_n of_o the_o square_n demonstration_n see_v the_o line_n ab_fw-la gh_n conjoyn_v the_o line_n agnostus_n bh_n which_o be_v parallel_v and_o equal_a they_o shall_v be_v also_o parallel_a and_o equal_a therefore_o the_o figure_n aghb_n be_v a_o parallelogram_n and_o the_o line_n ae_n gd_a agnostus_n ed_z be_v parallel_n and_o agnostus_n gd_v be_v equal_a ae_n ed_z shall_v be_v also_o equal_a it_o be_v the_o same_o with_o the_o other_o ae_n aec_fw-la ebb_n moreover_o agnostus_n ed_z be_v parallel_n and_o the_o angle_n g_z be_v right_o the_o angle_n d_o shall_v be_v also_o a_o right_a angle_n one_o may_v then_o on_o the_o centre_n e_o describe_v the_o circle_n adbc_n which_o shall_v pass_v through_o the_o point_v a_o d_o b_o c_o and_o which_o shall_v touch_v the_o side_n of_o the_o square_n proposition_n ix_o problem_n to_o describe_v a_o circle_n about_o a_o square_n

the_o line_n be_v in_o sub-duplicate_a ratio_fw-la shall_v be_v also_o proportional_a use_v a_o b_o c_o d._n 3._o 2._o 6._o 4._o 9_o 4._o 36._o 16_o e_z f_o g_o h._n this_o proposition_n may_v be_v easy_o apply_v to_o number_n if_o the_o number_n a_o b_o c_o d_o be_v proportional_a their_o square_n of_o gh_o shall_v be_v so_o likewise_o which_o we_o make_v use_v of_o in_o arithmetic_n and_o yet_o more_o use_n thereof_o in_o algebra_n proposition_n xxiii_o theorem_fw-la equiangular_a parallellogram_n have_v their_o ratio_fw-la compound_v of_o the_o ratio_fw-la of_o their_o side_n if_o the_o parallellogram_n l_o and_o m_o be_v equiangular_a the_o ratio_fw-la of_o l_o to_o m_n shall_v be_v compound_v of_o that_o of_o ab_fw-la to_o de_fw-fr and_o that_o of_o db_n to_o df._n join_v the_o parallellogram_n in_o such_o a_o manner_n that_o their_o side_n bd_o df_n be_v on_o a_o straight_a line_n as_o also_o cd_o de_fw-fr which_o may_v be_v if_o they_o be_v equiangular_a complete_a the_o parallelogram_n bdeh_n demonstration_n the_o parallelogram_n l_o have_v the_o same_o ratio_fw-la to_o the_o parallelogram_n bdeh_n as_o the_o base_a ab_fw-la have_v to_o the_o base_a bh_n or_o de_fw-fr by_o the_o first_o the_o parallelogram_n bdeh_n have_v the_o same_o ratio_fw-la to_o the_o parallelogram_n dfge_n that_o be_v to_o say_v m_n as_o the_o base_a bd_o have_v to_o the_o base_a df._n now_o the_o ratio_fw-la of_o the_o parallelogram_n l_o to_o the_o parallelogram_n m_o be_v compound_v of_o that_o of_o l_o to_o the_o parallelogram_n bdeh_n and_o that_o of_o bdeh_n to_o the_o parallelogram_n m._n thence_o the_o ratio_fw-la of_o l_o to_o m_n be_v compound_v of_o that_o of_o ab_fw-la to_o de_fw-fr and_o of_o that_o of_o bd_o to_o eglantine_n for_o example_n if_o ab_fw-la be_v eight_o and_o bh_n five_o bd_o four_o df_n seven_a say_v as_o four_o be_v to_o seven_o so_o be_v five_o to_o eight_o ¾_n you_o shall_v have_v these_o three_o number_n eight_o five_o eight_o ¾_n eight_o to_o five_o shall_v be_v the_o ratio_fw-la of_o the_o parallelogram_n l_o to_o bdeh_n the_o same_o as_o that_o of_o ab_fw-la to_o de_fw-fr five_o to_o eight_o ¾_n shall_v be_v that_o of_o the_o parallelogram_n bdeh_n to_o m._n so_o then_o take_v away_o the_o middle_a term_n which_o be_v five_o you_o shall_v have_v the_o ratio_fw-la of_o eight_o to_o eight_o ●_o for_o the_o ratio_fw-la compound_v of_o both_o or_o as_o 4_o time_n 8_o or_o 32_o to_o 5_o time_n 7_o or_o 35._o proposition_n xxiv_o theorem_fw-la in_o all_o sort_n of_o parallellogram_n those_o through_o which_o the_o diameter_n pass_v be_v like_a to_o the_o great_a let_v the_o diameter_n of_o the_o parallelogram_n ac_fw-la pass_v through_o the_o parallellogram_n of_o gh_o i_o say_v they_o be_v like_a unto_o the_o parallelogram_n ac_fw-la demonstration_n the_o parallelograms_n ac_fw-la of_o have_v the_o same_o angle_n b_o and_o because_o in_o the_o triangle_n bcd_v if_o be_v parallel_n to_o the_o base_a dc_o the_o triangle_n bfi_n bcd_n be_v equiangular_a there_o be_v therefore_o by_o the_o four_o the_o same_o ratio_fw-la of_o bc_n to_o cd_o as_o of_o bf_n to_o fi_n and_o consequent_o the_o side_n be_v in_o the_o same_o ratio_fw-la in_o like_a manner_n ih_o be_v parallel_n to_o bc_n there_o shall_v then_o be_v the_o same_o ratio_fw-la of_o dh_n to_o he_o as_o of_o dc_o to_o bc_n and_o the_o angle_n be_v also_o equal_a all_o the_o side_n be_v parallel_n thence_o by_o the_o 8_o def._n the_o parallellogram_n of_o gh_o be_v like_a unto_o the_o parallelogram_n ac_fw-la use_v i_o make_v use_v of_o this_o proposition_n in_o the_o ten_o proposition_n of_o the_o last_o book_n of_o perspective_n to_o show_v that_o a_o image_n be_v draw_v like_o unto_o the_o original_a with_o the_o help_n of_o a_o parallelogram_n compose_v of_o four_o line_n proposition_n xxv_o problem_n to_o describe_v a_o polygon_n like_v unto_o a_o give_v polygon_n and_o equal_a to_o any_o other_o right_o line_v figure_n if_o you_o will_v describe_v a_o polygon_n equal_a to_o the_o right_o line_a figure_n a_o and_o like_v unto_o the_o polygon_n b_o make_v a_o parallelogram_n ce_fw-fr equal_a to_o the_o polygon_n b_o by_o the_o 34th_o of_o the_o first_o and_o on_o de_fw-fr make_v a_o parallelogram_n of_o equal_a to_o the_o right_o line_a figure_n a_o by_o the_o 45th_o of_o the_o first_o then_o seek_v a_o mean_a proportional_a gh_o between_o cd_o and_o df_n by_o the_o 13_o last_o make_v on_o gh_a a_o polygon_n o_o like_v unto_o b_o by_o the_o 18_o it_o shall_v be_v equal_a to_o the_o right_o line_a figure_n a._n demonstration_n see_v that_o cd_o gh_n df_n be_v continual_o proportional_a the_o right_o line_a figure_n b_o describe_v on_o the_o first_o shall_v be_v to_o the_o right_o line_a figure_n o_o describe_v on_o the_o second_o as_o cd_o be_v to_o df_n by_o the_o coral_n of_o the_o 20_o now_o as_o cd_o be_v to_o df_n so_o be_v the_o parallelogram_n ce_fw-fr to_o the_o parallelogram_n of_o or_o as_o b_o to_o a_o see_v they_o be_v equal_a there_o be_v thence_o the_o same_o ratio_fw-la of_o b_o to_z o_o as_o of_o b_o to_o a._n so_o then_o by_o the_o seven_o of_o the_o 5_o a_o and_o o_o be_v equal_a use_v this_o proposition_n contain_v a_o change_n of_o figure_n keep_v always_o the_o same_o area_n which_o be_v very_o useful_a principal_o in_o practical_a geometry_n to_o reduce_v a_o irregular_a figure_n into_o a_o square_n proposition_n xxvi_o theorem_fw-la if_o in_o one_o of_o the_o angle_n of_o a_o parallelogram_n there_o be_v describe_v a_o lesser_a parallelogram_n like_o unto_o the_o great_a the_o diameter_n of_o the_o great_a shall_v meet_v the_o angle_n of_o the_o lesser_a if_o in_o the_o angle_n d_o of_o the_o parallelogram_n ac_fw-la be_v describe_v another_o lesser_a dg_n like_v thereto_o the_o diameter_n db_n shall_v pass_v through_o the_o point_n g._n for_o if_o it_o do_v not_o but_o it_o pass_v through_o ay_o as_o do_v the_o line_n bid_v draw_v the_o line_n je_n parallel_n to_o hd_n demonstration_n the_o parallelogram_n diego_n be_v like_a unto_o the_o parallelogram_n ac_fw-la by_o the_o 24_o now_o it_o be_v suppose_v that_o the_o parallelogram_n dg_n be_v also_o like_a thereto_o thence_o the_o parallellogram_n dii_fw-la dg_n will_v be_v like_a which_o be_v impossible_a otherwise_o there_o will_v be_v the_o same_o ratio_fw-la of_o he_o to_o je_n or_o gf_n as_o of_o hg_v to_o gf_n and_o by_o the_o seven_o of_o the_o 5_o the_o line_n he_o hg_n will_v be_v equal_a the_o twenty_o seven_o twenty_o eight_o and_o twenty_o nine_o proposition_n be_v unnecessary_a proposition_n xxx_o problem_n to_o cut_v a_o line_n give_v into_o extreme_a and_o mean_a proportion_n it_o be_v propose_v to_o cut_v the_o line_n ab_fw-la in_o extreme_a and_o mean_a proportion_n that_o be_v to_o say_v in_o such_o a_o manner_n that_o there_o may_v be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o ac_fw-la as_o of_o ac_fw-la to_o cb._n divide_v the_o line_n ab_fw-la by_o the_o 11_o of_o the_o second_o in_o such_o a_o manner_n that_o the_o rectangle_n comprehend_v under_o ab_fw-la cb_n be_v equal_a to_o the_o square_n of_o ac_fw-la demonstration_n see_v the_o rectangle_n of_o ab_fw-la bc_n be_v equal_a to_o the_o square_n of_o ac_fw-la there_o will_v be_v the_o same_o ratio_fw-la of_o ab_fw-la to_o ac_fw-la as_o of_o ac_fw-la to_o bc_n by_o the_o 17_o use_v this_o proposition_n be_v necessary_a in_o the_o thirteen_o book_n of_o euclid_n to_o find_v the_o length_n of_o the_o side_n of_o some_o of_o the_o five_o regular_a body_n father_n lucas_n of_o st._n sepulcher_n have_v compose_v a_o book_n of_o the_o propriety_n of_o a_o line_n which_o be_v cut_v into_o extreme_a and_o mean_a proportion_n proposition_n xxxi_o theorem_fw-la a_o polygon_n which_o be_v describe_v on_o the_o base_a of_o a_o right_a angle_a triangle_n be_v equal_a to_o the_o other_o two_o like_v polygon_n describe_v on_o the_o other_o side_n of_o the_o same_o triangle_n if_o the_o triangle_n abc_n have_v a_o right_a angle_n bac_n the_o polygon_n d_o describe_v on_o the_o base_a bc_n be_v equal_a to_o the_o like_a polygon_n f_o and_o e_o describe_v on_o the_o side_n ab_fw-la ac_fw-la demonstration_n the_o polygon_n d_o e_o f_o be_v among_o themselves_o in_o duplicate_v ratio_fw-la of_o their_o homologous_n side_n bc_n ac_fw-la ab_fw-la by_o the_o 20_o if_o there_o be_v describe_v a_o square_a on_o those_o side_n they_o will_v be_v also_o in_o duplicate_v ratio_fw-la to_o their_o side_n now_o by_o the_o 47th_o of_o the_o first_o the_o square_a of_o bc_n will_v be_v equal_a to_o the_o square_n of_o ac_fw-la ab_fw-la thence_o the_o polyligon_n d_o describe_v on_o bc_n be_v equal_a to_o the_o like_a polygon_n e_o and_o f_o describe_v on_o ab_fw-la ac_fw-la use_v this_o proposition_n be_v make_v use_n

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

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 three_o part_n of_o two_o right_n and_o that_o be_v 60_o degree_n now_o all_o the_o angle_n which_o can_v be_v make_v about_o one_o point_n be_v equal_a to_o four_o right_n that_o be_v to_o say_v 360._o so_o take_v away_o four_o time_n 60_o that_o be_v 240_o from_o 360_o there_o remain_v 120_o degree_n for_o bgc_n and_o fge_n whence_o they_o shall_v each_o be_v 60_o degree_n so_o all_o the_o angle_n at_o the_o centre_n be_v equal_a all_o the_o ark_n and_o all_o the_o side_n shall_v be_v equal_a and_o each_o angle_n a_o b_o c_o etc._n etc._n shall_v be_v compose_v of_o two_o angle_n of_o sixty_o that_o be_v to_o say_v one_o hundred_o and_o twenty_o degree_n they_o shall_v therefore_o be_v equal_a coral_n the_o side_n of_o a_o hexagon_n be_v equal_a to_o the_o semi_a diameter_n use_v because_o that_o the_o side_n of_o a_o hexagon_n be_v the_o base_a of_o a_o ark_n of_o sixty_o degree_n and_o that_o be_v equal_a to_o the_o semi-diameter_n its_o half_n be_v the_o sine_fw-la of_o thirty_o and_o it_o be_v with_o this_o sine_fw-la we_o begin_v the_o table_n of_o sines_n euclid_n treat_v of_o hexagon_n in_o the_o last_o book_n of_o his_o element_n proposition_n xvi_o problem_n to_o inscribe_v a_o regular_a pentadecagon_n in_o a_o circle_n inscribe_v in_o a_o circle_n a_o equilateral_a triangle_n abc_n by_o the_o 2d_o and_o a_o regular_a pentagon_n by_o the_o 11_o in_o such_o sort_n that_o the_o angle_n meet_v in_o the_o point_n a._n the_o line_n bf_n by_o je_n shall_v be_v the_o side_n of_o the_o pentadecagon_n and_o by_o inscribe_v in_o the_o other_o ark_n line_n equal_a to_o bf_n by_o you_o may_v complete_a this_o polygon_n demonstration_n see_v the_o line_n ab_fw-la be_v the_o side_n of_o the_o equilateral_a triangle_n the_o ark_n aeb_fw-mi shall_v be_v the_o three_o of_o the_o whole_a circle_n or_o 5_o fifteenth_n and_o the_o ark_n ae_n be_v the_o five_o part_n it_o shall_v contain_v 3_o 15_o thence_o ebb_n contain_v two_o and_o if_o you_o divide_v it_o in_o the_o middle_n in_o i_o each_o part_n shall_v be_v a_o fifteen_o use_v this_o proposition_n serve_v only_o to_o open_v the_o way_n for_o other_o polygon_n we_o have_v in_o the_o compass_n of_o proportion_n very_o easy_a method_n to_o inscribe_v all_o the_o ordinary_a polygon_n but_o they_o be_v ground_v on_o this_o for_o one_o can_v not_o put_v polygon_n on_o that_o instrument_n if_o one_o do_v not_o find_v their_o side_n by_o this_o proposition_n or_o such_o like_a the_o end_n of_o the_o four_o book_n the_o five_o book_n of_o euclid_n element_n this_o five_o book_n be_v absolute_o necessary_a to_o demonstrate_v the_o propoposition_n of_o the_o six_o book_n it_o contain_v a_o most_o universal_a doctrine_n and_o a_o way_n of_o argue_v by_o proportion_n which_o be_v most_o subtle_a solid_a and_o brief_a so_o that_o all_o treatise_n which_o be_v found_v on_o proportion_n can_v be_v without_o this_o mathematical_a logic_n geometry_n arithmetic_n music_n astronomy_n staticks_n and_o to_o say_v 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