of_o incidence_n abe_n and_o to_o the_o angle_n ibk_n its_o vertical_a angle_n ebf_n be_v equal_a and_o therefore_o the_o angle_n abe_n be_v equal_a to_o the_o angle_n ebf_n again_o the_o angle_n ade_n be_v equal_a to_o the_o angle_n of_o reflection_n gdk_n that_o be_v to_o its_o vertical_a angle_n edf_n and_o therefore_o the_o two_o angle_n abdella_n and_o adb_n of_o the_o triangle_n abdella_n be_v one_o by_o one_o equal_a to_o the_o two_o angle_n fbd_v and_o fdb_n of_o the_o triangle_n fbd_v wherefore_o also_o the_o three_o angle_n bad_a be_v equal_a to_o the_o three_o angle_n bfd_n which_o be_v to_o be_v prove_v corollary_n 1._o if_o the_o straight_a line_n of_o be_v draw_v it_o will_v be_v perpendicular_a to_o the_o straight_a line_n eke_o for_o both_o the_o angle_n at_o e_o will_v be_v equal_a by_o reason_n of_o the_o equality_n of_o the_o two_o angle_n abe_n and_o fbe_n and_o of_o the_o two_o side_n ab_fw-la and_o fb_fw-la corollary_n 2._o if_o upon_o any_o point_n between_o b_o and_o d_o there_o fall_v a_o straight_a line_n as_o ac_fw-la who_o reflect_a line_n be_v ch_z this_o also_o produce_v beyond_o c_o will_n fall_n upon_o f_o which_o be_v evident_a by_o the_o demonstration_n above_o 3_o if_o from_o two_o point_n take_v without_o a_o circle_n two_o straight_a parallel_n line_n draw_v not_o opposite_o but_o from_o the_o same_o part_n fall_v upon_o the_o circumference_n the_o line_n reflect_v from_o they_o if_o produce_v they_o meet_v within_o the_o circle_n will_v make_v a_o angle_n double_a to_o that_o which_o be_v make_v by_o two_o straight_a line_n draw_v from_o the_o centre_n to_o the_o point_n of_o incidence_n let_v the_o two_o straight_a parallel_n ab_fw-la and_o dc_o in_o the_o 3d_o figure_n fall_v upon_o the_o circumference_n bc_n at_o the_o point_n b_o and_o c_o and_o let_v the_o centre_n of_o the_o circle_n be_v e_o and_o let_v ab_fw-la reflect_v be_v bf_n and_o dc_o reflect_v be_v cg_n and_o let_v the_o line_n fb_n and_o gc_n produce_v meet_v within_o the_o circle_n in_o h_n and_o let_v ebb_n and_o aec_fw-la be_v connect_v i_o say_v the_o angle_n fhg_n be_v double_a to_o the_o angle_n bec_n for_o see_v ab_fw-la and_o dc_o be_v parallel_n and_o ebb_n cut_v ab_fw-la in_o b_o the_o same_o ebb_n produce_v will_v cut_v dc_o somewhere_o let_v it_o cut_v it_o in_o d_o &_o let_v dc_o be_v produce_v howsoever_o to_o i_o and_o let_v the_o intersection_n of_o dc_o &_o bf_o be_v at_o k._n the_o angle_n therefore_o ich_n be_v external_a to_o the_o triangle_n ckh_n will_v be_v equal_a to_o the_o two_o opposite_a angle_n ckh_n and_o chk_n again_o ice_n be_v external_a to_o the_o triangle_n cde_v be_v equal_a to_o the_o two_o angle_n at_o d_o and_o e._n wherefore_o the_o angle_n ich_n be_v double_a to_o the_o angle_n ice_n be_v equal_a to_o the_o angle_n at_o d_o and_o e_o twice_o take_v and_o therefore_o the_o two_o angle_n ckh_n and_o chk_n be_v equal_a to_o the_o two_o angle_n at_o d_o and_o e_o twice_o take_v but_o the_o angle_n ckh_n be_v equal_a to_o the_o angle_n d_o and_o abdella_n that_o be_v d_o twice_o take_v for_o ab_fw-la and_o dc_o be_v parallel_n the_o altern_a angle_n d_o and_o abdella_n be_v equal_a wherefore_o chk_n that_o be_v the_o angle_n fhg_n be_v also_o equal_a to_o the_o angle_n at_o e_o twice_o take_v which_o be_v to_o be_v prove_v corollary_n if_o from_o two_o point_n take_v within_o a_o circle_n two_o straight_a parallel_n fall_v upon_o the_o circumference_n the_o line_n reflect_v from_o they_o shall_v meet_v in_o a_o angle_n double_a to_o that_o which_o be_v make_v by_o two_o straight_a line_n draw_v from_o the_o centre_n to_o the_o point_n of_o incidence_n for_o the_o parallel_v lb_n and_o i_fw-mi falling_z upon_o the_o point_n b_o and_o c_o be_v reflect_v in_o the_o line_n bh_n and_o ch_z and_o make_v the_o angle_n at_o h_n double_a to_o the_o angle_n at_o e_o as_o be_v but_o now_o demonstrate_v 4_o if_o two_o straight_a line_n draw_v from_o the_o same_o point_n without_o a_o circle_n fall_v upon_o the_o circumference_n and_o the_o line_n reflect_v from_o they_o be_v produce_v meet_v within_o the_o circle_n they_o will_v make_v a_o angle_n equal_a to_o twice_o that_o angle_n which_o be_v make_v by_o two_o straight_a line_n draw_v from_o the_o centre_n to_o the_o point_n of_o incidence_n together_o with_o the_o angle_n which_o the_o incident_a line_n themselves_o make_v let_v the_o two_o straight_a line_n ab_fw-la and_o ac_fw-la in_o the_o four_o figure_n be_v draw_v from_o the_o point_n a_o to_o the_o circumference_n of_o the_o circle_n who_o centre_n be_v d_o and_o let_v the_o line_n reflect_v from_o they_o be_v be_v and_o cg_n and_o be_v produce_v make_v within_o the_o circle_n the_o angle_n h_n also_o let_v the_o two_o straight_a line_n db_a and_o dc_o be_v draw_v from_o the_o centre_n d_o to_o the_o point_n of_o incidence_n b_o and_o c._n i_o say_v the_o angle_n h_n be_v equal_a to_o twice_o the_o angle_n at_o d_o together_o with_o the_o angle_n at_o a._n for_o let_v ac_fw-la be_v produce_v howsoever_o to_o i._o therefore_o the_o angle_n ch_z which_o be_v external_a to_o the_o triangle_n ckh_n will_v be_v equal_a to_o the_o two_o angle_n gkh_n and_o chk_n again_o the_o angle_n icd_n which_o be_v external_a to_o the_o triangle_n cld_v will_v be_v equal_a to_o the_o two_o angle_n cld_a and_o cdl_o but_o the_o angle_n ich_n be_v double_a to_o the_o angle_n icd_n and_o be_v therefore_o equal_a to_o the_o angle_n cld_a and_o cdl_o twice_o take_v wherefore_o the_o angle_v ckh_n and_o chk_n be_v equal_a to_o the_o angle_n cld_a and_o cdl_o twice_o take_v but_o the_o angle_n cld_v be_v external_a to_o the_o triangle_n alb_n be_v equal_a to_o the_o two_o angle_n labernele_n &_o lba_n &_o consequent_o cld_v twice_o take_v be_v equal_a to_o labernele_n &_o labernele_n twice_o take_v wherefore_o ckh_n &_o chk_n be_v equal_a to_o the_o angle_n cdl_o together_o with_o labernele_n and_o lba_n twice_o take_v also_o the_o angle_n ckh_n be_v equal_a to_o the_o angle_n labernele_n once_o and_o abk_n that_o be_v lba_n twice_o take_v wherefore_o the_o angle_n chk_n be_v equal_a to_o the_o remain_a angle_n cdl_o that_o be_v to_o the_o angle_n at_o d_o twice_o take_v and_o the_o angle_n labernele_n that_o be_v the_o angle_n at_o a_o once_o take_v which_o be_v to_o be_v prove_v corollary_n if_o two_o straight_a converge_a line_n as_o i_fw-mi and_o mb_a fall_n upon_o the_o concave_a circumference_n of_o a_o circle_n their_o reflect_a line_n as_o ch_z and_o bh_n will_v meet_v in_o the_o angle_n h_n equal_a to_o twice_o the_o angle_n d_o together_z with_o the_o angle_n at_o a_o make_v by_o the_o incident_a line_n produce_v or_o if_o the_o incident_a line_n be_v hb_a and_o i_fw-mi who_o reflect_v line_n ch_n and_o bm_n meet_v in_o the_o point_n n_o the_o angle_n cnb_n will_v be_v equal_a to_o twice_o the_o angle_n d_o together_z with_o the_o angle_n ckh_n make_v by_o the_o line_n of_o incidence_n for_o the_o angle_n cnb_n be_v equal_a to_o the_o angle_n h_n that_o be_v to_o twice_o the_o angle_n d_o together_o with_o the_o two_o angle_n a_o and_o nbh_n that_o be_v kba_n but_o the_o angle_v kba_n and_o a_o be_v equal_a to_o the_o angle_n ckh_n wherefore_o the_o angle_n cnb_n be_v equal_a to_o twice_o the_o angle_n d_o together_z with_o the_o angle_n ckh_n make_v by_o the_o line_n of_o incidence_n i_fw-mi and_o hb_v produce_v to_o k._n 5_o if_o two_o straight_a line_n draw_v from_o one_o point_n fall_v upon_o the_o concave_a circumference_n of_o a_o circle_n and_o the_o angle_n they_o make_v be_v less_o than_o twice_o the_o angle_n at_o the_o centre_n the_o line_n reflect_v from_o they_o and_o meet_v within_o the_o circle_n will_v make_v a_o angle_n which_o be_v add_v to_o the_o angle_n of_o the_o incident_a line_n will_v be_v equal_a to_o twice_o the_o angle_n at_o the_o centre_n let_v the_o two_o line_n ab_fw-la and_o ac_fw-la in_o the_o 5_o figure_n draw_v from_o the_o point_n a_o fall_v upon_o the_o concave_a circumference_n of_o the_o circle_n who_o centre_n be_v d_o &_o let_v their_o reflect_a line_n be_v and_o ce_fw-fr meet_v in_o the_o point_n e_o also_o let_v the_o angle_v a_o be_v less_o than_o twice_o the_o angle_n d._n i_o say_v the_o angle_v a_o and_o e_o together_o take_v be_v equal_a to_o twice_o the_o angle_n d._n for_o let_v the_o straight_a line_n ab_fw-la and_o aec_fw-la cut_v the_o straight_a line_n dc_o and_o db_n in_o the_o point_n g_o and_o h_o and_o the_o angle_n bhc_n will_v be_v equal_a to_o the_o two_o angle_n ebh_n and_o e_o also_o the_o same_o angle_n bhc_n will_v be_v equal_a to_o the_o two_o angle_n d_o and_o dch_n
and_o in_o like_a manner_n the_o angle_n bgc_n will_v be_v equal_a to_o the_o two_o angle_n acd_v &_o a_o &_o the_o same_o angle_n bgc_n will_v be_v also_o equal_a to_o the_o two_o angle_n dbg_n and_o d._n wherefore_o the_o four_o angle_n ebh_n e_o acd_a and_o a_o be_v equal_a to_o the_o four_o angle_n d_o dch_n dbg_n and_o d._n if_o therefore_o equal_v be_v take_v away_o on_o both_o side_n namely_o on_o one_o side_n acd_v and_o ebh_n and_o on_o the_o other_o side_n dch_n and_o dbg_n for_o the_o angle_n ebh_n be_v equal_a to_o the_o angle_n dbg_n and_o the_o angle_n acd_v equal_a to_o the_o angle_n dch_o the_o remainder_n on_o both_o side_n will_v be_v equal_a namely_o on_o one_o side_n the_o angle_v a_o and_o e_o and_o on_o the_o other_o the_o angle_n d_o twice_z take_v wherefore_o the_o angle_n a_o and_o e_o be_v equal_a to_o twice_o the_o angle_n d._n corollary_n if_o the_o angle_n a_o be_v great_a than_o twice_o the_o angle_n d_o their_o reflect_a â—â—ines_n will_v diverge_n for_o by_o the_o corollary_n of_o the_o three_o proposition_n if_o the_o angle_n a_o be_v equal_a to_o twice_o the_o angle_n d_o the_o reflect_a line_n be_v and_o ce_fw-fr will_v be_v parallel_n and_o if_o it_o be_v less_o they_o will_v concur_v as_o have_v now_o be_v demonstrate_v and_o therefore_o if_o it_o be_v great_a the_o reflect_a line_n be_v and_o ce_fw-fr will_n diverge_n and_o consequent_o if_o they_o be_v produce_v the_o other_o way_n they_o will_v concur_v and_o make_v a_o angle_n equal_a to_o the_o excess_n of_o the_o angle_n a_o above_o twice_o the_o angle_n d_o as_o be_v evident_a by_o the_o four_o article_n 6_o if_o through_o any_o one_o point_n two_o unequal_a chord_n be_v draw_v cut_v one_o another_o either_o within_o the_o circle_n or_o if_o they_o be_v produce_v without_o it_o and_o the_o centre_n of_o the_o circle_n be_v not_o place_v between_o they_o and_o the_o line_n reflect_v from_o they_o concur_v wheresoever_o there_o can_v through_o the_o point_n through_o which_o the_o former_a line_n be_v draw_v be_v draw_v another_o straight_a line_n who_o reflect_a line_n shall_v pass_v through_o the_o point_n where_o the_o two_o former_a reflect_a line_n concur_v let_v any_o two_o unequal_a chord_n as_o bk_n and_o ch_n in_o the_o 6_o figure_n be_v draw_v through_o the_o point_n a_o in_o the_o circle_n bc_n and_o let_v their_o reflect_a line_n bd_o and_o ce_fw-fr meet_v in_o f_o and_o let_v the_o centre_n not_o be_v between_o ab_fw-la and_o ac_fw-la and_o from_o the_o point_n a_o let_v any_o other_o straight_o line_n as_o agnostus_n be_v draw_v to_o the_o circumference_n between_o b_o and_o c._n i_o say_v gn_n which_o pass_v through_o the_o point_n f_o where_o the_o reflect_v line_n bd_o and_o ce_fw-fr meet_a will_v not_o be_v the_o reflect_a line_n of_o ag._n for_o let_v the_o arch_n bl_n be_v take_v equal_a to_o the_o arch_n bg_n and_o the_o straight_a line_n bm_n equal_a to_o the_o straight_a line_n basilius_n and_o lm_o be_v draw_v let_v it_o be_v produce_v to_o the_o circummference_n in_o o._n see_v therefore_o basilius_n and_o bm_n be_v equal_a and_o the_o arch_n bl_n equal_a to_o the_o arch_n bg_n and_o the_o angle_n mbl_n equal_a to_o the_o angle_n abg_n agnostus_n and_o ml_o will_n also_o be_v equal_a and_o produce_v give_v to_o the_o circumference_n in_o i_o the_o whole_a line_n lo_o and_o give_v will_n in_o like_a manner_n be_v equal_a but_o lo_o be_v great_a than_o gfn_n as_o shall_v present_o be_v demonstrate_v and_o therefore_o also_o give_v be_v great_a than_o gn_n wherefore_o the_o angle_n ngc_a and_o igb_n be_v not_o equal_a wherefore_o the_o line_n gfn_n be_v not_o reflect_v from_o the_o line_n of_o incidence_n agnostus_n and_o consequent_o no_o other_o straight_a line_n beside_o ab_fw-la and_o ac_fw-la which_o be_v draw_v through_o the_o point_n a_o and_o faâ—ls_n upon_o the_o circumference_n bc_n can_v be_v reflect_v to_o the_o point_n f_o which_o be_v to_o be_v demonstrate_v it_o remain_v that_o i_o prove_v lo_o to_o be_v great_a than_o gn_n which_o i_o shall_v do_v in_o this_o manner_n lo_o and_o gn_v cut_v one_o another_o in_o p_o and_o pl_z be_v great_a than_o pg._n see_v now_o lp_v pg_n pn_n po_n be_v proportional_n therefore_o the_o two_o extreme_n lp_v and_o po_n together_o take_v that_o be_v lo_o be_v great_a than_o pg_n and_o pn_n together_o take_v that_o be_v gn_n which_o remain_v to_o be_v prove_v 7_o but_o if_o two_o equal_a chord_n be_v draw_v through_o one_o point_n within_o a_o circle_n and_o the_o line_n reflect_v from_o they_o meet_v in_o another_o point_n than_o another_o straight_a line_n may_v be_v draw_v between_o they_o through_o the_o former_a point_n who_o reflect_a line_n shall_v pass_v through_o the_o late_a point_n let_v the_o two_o equal_a chord_n bc_n and_o ed_z in_o the_o seven_o figure_n cut_v one_o another_o in_o the_o point_n a_o within_o the_o circle_n bcd_v and_o let_v their_o reflect_a line_n ch_n and_o diego_n meet_v in_o the_o point_n f._n then_o divide_v the_o arch_n cd_o equal_o in_o g_z let_v the_o two_o chord_n gk_n and_o gl_n be_v draw_v through_o the_o point_v a_o and_o f._n i_o say_v gl_n will_v be_v the_o line_n reflect_v from_o the_o chord_n kg_v for_o the_o four_o chord_n bc_n ch_z ed_z and_o di_z be_v by_o supposition_n all_o equal_a to_o one_o another_o and_o therefore_o the_o arch_a bch_n be_v equal_a to_o the_o arch_n edi_n as_o also_o the_o angle_n bch_o to_o the_o angle_n edi_n &_o the_o angle_n amc_n to_o its_o vertical_a angle_n fmd_v and_o the_o straight_a line_n dm_o to_o the_o straight_a line_n cm_o and_o in_o like_a manner_n the_o straight_a line_n ac_fw-la to_o the_o straight_a line_n fd_n and_o the_o chord_n cg_n and_o gd_n be_v draw_v will_v also_o be_v equal_a as_o also_o the_o angle_n fdg_n and_o acg_n in_o the_o equal_a segment_n gdi_n and_o gcb_n wherefore_o the_o straight_a line_n fg_v and_o agnostus_n be_v equal_a and_o therefore_o the_o angle_n fgd_v be_v equal_a to_o the_o angle_n agc_n that_o be_v the_o angle_n of_o incidence_n equal_a to_o the_o angle_n of_o reflection_n wherefore_o the_o line_n gl_n be_v reflect_v from_o the_o incident_a line_n kg_v which_o be_v to_o be_v prove_v corollary_n by_o the_o very_a sight_n of_o the_o figure_n it_o be_v manifest_a that_o if_o g_z be_v not_o the_o middle_a point_n between_o c_o and_o d_o the_o reflect_a line_n gl_n will_v not_o pass_v through_o the_o point_n f._n 8_o two_o point_n in_o the_o circumference_n of_o a_o circle_n be_v give_v to_o draw_v two_o straight_a line_n to_o they_o so_o as_o that_o their_o reflect_a line_n may_v be_v parallel_n or_o contain_v any_o angle_n give_v in_o the_o circumference_n of_o the_o circle_n who_o centre_n be_v a_o in_o the_o 8_o figure_v let_v the_o two_o point_n b_o and_o c_o be_v give_v and_o let_v it_o be_v require_v to_o draw_v to_o they_o from_o two_o point_n take_v without_o the_o circle_n two_o incident_a line_n so_o that_o their_o reflect_a line_n may_v first_o be_v parallel_n let_v ab_fw-la and_o ac_fw-la be_v draw_v as_o also_o any_o incident_a line_n dc_o with_o its_o reflect_a line_n cf_n and_o let_v the_o angle_n ecd_v be_v make_v double_a to_o the_o angle_n a_o and_o let_v hb_n be_v draw_v parallel_n to_o aec_fw-la and_o produce_v till_o it_o meet_v with_o dc_o produce_v in_o i._n last_o produce_v ab_fw-la indefinite_o to_o k_o let_v gb_n be_v draw_v so_o that_o the_o angle_n gbk_n may_v be_v equal_a to_o the_o angle_n hbk_n and_o then_o gb_n will_v be_v the_o reflect_a line_n of_o the_o incident_a line_n hb_n i_o say_v dc_o and_o hb_n be_v two_o incident_a line_n who_o reflect_a line_n cf_n and_o bg_n be_v parallel_n for_o see_v the_o angle_n ecd_n be_v double_a to_o the_o angle_n bac_n the_o angle_n hic_fw-la be_v also_o by_o reason_n of_o the_o parallel_n aec_fw-la and_o he_o double_a to_o the_o same_o bac_n therefore_o also_o fc_a and_o gb_n namely_o the_o line_n reflect_v from_o the_o incident_a line_n dc_o and_o hb_n be_v parallel_n wherefore_o the_o first_o thing_n require_v be_v do_v second_o let_v it_o be_v require_v to_o draw_v to_o the_o point_n b_o &_o c_o two_o straight_a line_n of_o incidence_n so_o that_o the_o line_n reflect_v from_o they_o may_v contain_v the_o give_v angle_n z._n to_o the_o angle_n ecd_v make_v at_o the_o point_n c_o let_v there_o be_v add_v on_o one_o side_n the_o angle_n dcl_o equal_a to_o half_a z_o and_o on_o the_o other_o side_n the_o angle_n ecm_n equal_a to_o the_o angle_n dcl_o and_o let_v the_o straight_a line_n bn_n be_v draw_v parallel_n to_o the_o straight_a line_n cm_o and_o let_v the_o angle_n kbo_n be_v make_v equal_a to_o the_o
in_o a_o straight_a line_n perpendicular_a to_o its_o superficies_n in_o that_o point_n in_o which_o it_o be_v press_v let_v abcd_n in_o the_o first_o figure_n be_v a_o hard_a body_n and_o let_v another_o body_n fall_v upon_o it_o in_o the_o straight_a line_n ea_fw-la with_o any_o inclination_n or_o without_o inclination_n press_v it_o in_o the_o point_n a._n i_o say_v the_o body_n so_o press_v &_o not_o penetrate_v it_o will_v give_v to_o the_o part_n a_o a_o endeavour_n to_o yield_v or_o recede_v in_o a_o straight_a line_n perpendicular_a to_o the_o line_n ad._n for_o let_v ab_fw-la be_v perpendicular_a to_o ad_fw-la and_o let_v basilius_n be_v produce_v to_o f._n if_o therefore_o of_o be_v coincident_a with_o ae_n it_o be_v of_o itself_o manifest_v that_o the_o motion_n in_o ea_fw-la will_v make_v a_o to_o endeavour_v in_o the_o line_n ab_fw-la let_v now_o ea_z be_v oblique_a to_o ad_fw-la and_o from_o the_o point_n e_o let_v the_o straight_a line_n aec_fw-la be_v draw_v cut_v ad_fw-la at_o right_a angle_n in_o d_o and_o let_v the_o rectangle_v abcd_v and_o adef_n be_v complete_v i_o have_v show_v in_o the_o 8_o article_n of_o the_o 16_o chapter_n that_o the_o body_n will_v be_v carry_v from_o e_z to_o a_o by_o the_o concourse_n of_o two_o uniform_a motion_n the_o one_o in_o of_o and_o its_o parallel_n the_o other_o in_o ed_z and_o its_o parallel_n but_o the_o motion_n in_o of_o and_o its_o parallel_n whereof_o da_fw-la be_v one_o contribute_v nothing_o to_o the_o body_n in_o a_o to_o make_v it_o endeavour_n or_o press_v towards_o b_o and_o therefore_o the_o whole_a endeavour_n which_o the_o body_n have_v in_o the_o incline_v line_n ea_fw-la to_o pass_v or_o press_v the_o straight_a line_n ad_fw-la it_o have_v it_o all_o from_o the_o perpendicular_a motion_n or_o endeavour_n in_o fa._n wherefore_o the_o body_n e_o after_o it_o be_v in_o a_o will_v have_v only_o that_o perpendicular_a endeavour_n which_o proceed_v from_o the_o motion_n in_o favorina_n that_o be_v in_o ab_fw-la which_o be_v to_o be_v prove_v 7_o if_o a_o hard_a body_n fall_v upon_o or_o press_v another_o body_n penetrate_v the_o same_o its_o endeavour_n after_o its_o first_o penetration_n will_v be_v neither_o in_o the_o incline_v line_n produce_v nor_o in_o the_o perpendicular_a but_o sometime_o betwixt_o both_o sometime_o without_o they_o let_v eag_n in_o the_o same_o â—_o figure_n be_v the_o incline_v line_n produce_v and_o first_o let_v the_o passage_n through_o the_o medium_fw-la in_o which_o ea_z be_v be_v easy_a than_o the_o passage_n through_o the_o medium_fw-la in_o which_o agnostus_n be_v as_o soon_o therefore_o as_o the_o body_n be_v within_o the_o medium_fw-la in_o which_o be_v agnostus_n it_o will_v find_v great_a resistance_n to_o its_o motion_n in_o da_fw-la and_o its_o parallel_n than_o it_o do_v while_o it_o be_v above_o ad_fw-la and_o therefore_o below_o ad_fw-la it_o will_v proceed_v with_o slow_a motion_n in_o the_o parallel_n of_o da_fw-la then_o above_o it_o wherefore_o the_o motion_n which_o be_v compound_v of_o the_o two_o motion_n in_o of_o and_o ed_z will_v be_v slow_a below_n ad_fw-la then_o above_o it_o and_o therefore_o also_o the_o body_n will_v not_o proceed_v from_o a_o in_o ea_fw-la produce_v but_o below_o it_o see_v therefore_o the_o endeavour_n in_o ab_fw-la be_v generate_v by_o the_o endeavour_n in_o favorina_n if_o to_o the_o endeavour_n in_o favorina_n there_o be_v add_v the_o endeavour_n in_o da_fw-la which_o be_v not_o all_o take_v away_o by_o the_o immersion_n of_o the_o point_n a_o into_o the_o low_a medium_fw-la the_o body_n will_v not_o proceed_v from_o a_o in_o the_o perpendicular_a ab_fw-la but_o beyond_o it_o namely_o in_o some_o straight_a line_n between_o ab_fw-la and_o agnostus_n as_o in_o the_o line_n ah_o second_o let_v the_o passage_n through_o the_o medium_fw-la ea_fw-la be_v less_o easy_a than_o that_o through_o ag._n the_o motion_n therefore_o which_o be_v make_v by_o the_o concourse_n of_o the_o motion_n in_o of_o and_o fb_v be_v slow_a above_n ad_fw-la then_o below_o it_o and_o consequent_o the_o endeavour_n will_v not_o proceed_v from_o a_o in_o ea_fw-la produce_v but_o beyond_o it_o as_o in_o ai._n wherefore_o if_o a_o hard_a body_n fall_v which_o be_v to_o be_v prove_v this_o divergency_n of_o the_o straight_a line_n ah_o from_o the_o straight_a line_n agnostus_n be_v that_o which_o the_o writer_n of_o optic_n common_o call_v refraction_n which_o when_o the_o passage_n be_v easy_a in_o the_o first_o then_o in_o the_o second_o medium_fw-la be_v make_v by_o diverge_v from_o the_o line_n of_o inclination_n towards_o the_o perpendicular_a and_o contrary_o when_o the_o passage_n be_v not_o so_o easy_a in_o the_o first_o medium_fw-la by_o depart_v far_o from_o the_o perpendicular_a 8_o by_o the_o 6_o theorem_a it_o be_v manifest_a that_o the_o force_n of_o the_o movent_fw-la may_v be_v so_o place_v as_o that_o the_o body_n move_v by_o it_o may_v proceed_v in_o a_o way_n almost_o direct_o contrary_a to_o that_o of_o the_o movent_fw-la as_o we_o see_v in_o the_o motion_n of_o ship_n for_o let_v ab_fw-la in_o the_o 2d_o figure_n represent_v a_o ship_n who_o length_n from_o the_o prow_n to_o the_o poop_n be_v ab_fw-la and_o let_v the_o wind_n lie_v upon_o it_o in_o the_o straight_a parallel_n line_n cb_n de_fw-fr and_o fg_a and_o let_v de_fw-fr and_o fg_n be_v cut_v in_o e_z and_o g_z by_o a_o straight_a line_n draw_v from_o b_o perpendicular_a to_o ab_fw-la also_o let_v be_v and_o eglantine_n be_v equal_a and_o the_o angle_n abc_n any_o angle_n how_o small_a soever_o then_o between_o bc_n and_o basilius_n let_v the_o straight_a line_n by_o be_v draw_v and_o let_v the_o sail_n be_v conceive_v to_o be_v spread_v in_o the_o same_o line_n by_o and_o the_o wind_n to_o fall_v upon_o it_o in_o the_o point_n l_o m_o and_o b_o from_o which_o point_n perpendicular_a to_o by_o let_v bk_n mq_n and_o lp_n be_v draw_v last_o let_v en_fw-fr and_o go_v be_v draw_v perpendicular_a to_o bg_n and_o cut_v bk_n in_o h_n and_o k_n and_o let_v hn_n and_o quoth_fw-mi be_v make_v equal_a to_o one_o another_o and_o several_o equal_a to_o basilius_n i_o say_v the_o ship_n basilius_n by_o the_o wind_n fall_v upon_o it_o in_o cb_n de_fw-fr fg_n and_o other_o line_n parallel_v to_o they_o will_v be_v carry_v forward_o almost_o opposite_a to_o the_o wind_n that_o be_v to_o say_v in_o a_o way_n almost_o contrary_a to_o the_o way_n of_o the_o movent_fw-la for_o the_o wind_n that_o blow_v in_o the_o line_n cb_n will_v as_o have_v be_v show_v in_o the_o 6_o article_n give_v to_o the_o point_n b_o a_o endeavour_n to_o proceed_v in_o a_o straight_a line_n perpendicular_a to_o the_o straight_a line_n by_o that_o be_v in_o the_o straight_a line_n bk_n and_o to_o the_o point_n m_o and_o l_o a_o endeavour_n to_o proceed_v in_o the_o straight_a line_n mq_n and_o lp_n which_o be_v parallel_v to_o bk_n let_v now_o the_o measure_n of_o the_o time_n be_v bg_n which_o be_v divide_v in_o the_o middle_n in_o e_z &_o let_v the_o point_n b_o be_v carry_v to_o h_n in_o the_o time_n be._n in_o the_o same_o time_n therefore_o by_o the_o wind_n blow_v in_o dm_o &_o fl_fw-mi and_o as_o many_o other_o line_n as_o may_v be_v draw_v parallel_n to_o they_o the_o whole_a ship_n will_v be_v apply_v to_o the_o straight_a line_n hn._n also_o at_o the_o end_n of_o the_o second_o time_n eglantine_n it_o will_v be_v apply_v to_o the_o straight_a line_n quoth_fw-mi wherefore_o the_o ship_n will_v always_o go_v forward_o and_o the_o angle_n it_o make_v with_o the_o wind_n will_v be_v equal_a to_o the_o angle_n abc_n how_o small_a soever_o that_o angle_n be_v and_o the_o way_n it_o make_v will_n in_o every_o time_n be_v equal_a to_o the_o straight_a line_n eh_n i_o say_v thus_o it_o will_v be_v if_o the_o ship_n may_v be_v move_v with_o as_o great_a celerity_n sidewayes_o from_o basilius_n towards_o quoth_fw-mi as_o it_o may_v be_v move_v forward_o in_o the_o line_n basilius_n but_o this_o be_v impossible_a by_o reason_n of_o the_o resistance_n make_v by_o the_o great_a quantity_n of_o water_n which_o press_v the_o side_n much_o exceed_v the_o resistance_n make_v by_o the_o much_o small_a quantity_n which_o press_v the_o prow_n of_o the_o ship_n so_o that_o the_o way_n the_o ship_n make_v sidewayes_o be_v scarce_o sensible_a and_o therefore_o the_o point_n b_o will_v proceed_v almost_o in_o the_o very_a line_n basilius_n make_v with_o the_o wind_n the_o angle_n abc_n how_o acute_a soever_o that_o be_v to_o say_v it_o will_v proceed_v almost_o in_o the_o straight_a line_n bc_n that_o be_v in_o a_o way_n almost_o contrary_a to_o the_o way_n of_o the_o movent_fw-la which_o be_v to_o be_v demonstrate_v but_o the_o sail_n in_o by_o must_v
the_o same_o quality_n as_o every_o man_n be_v a_o live_a creature_n some_o man_n be_v a_o live_a creature_n or_o no_o man_n be_v wise_a some_o man_n be_v not_o wise._n of_o these_o iâ—_n the_o universal_a be_v true_a the_o particular_a will_v be_v true_a also_o contrary_a be_v universal_a proposition_n of_o different_a quality_n as_o every_o man_n be_v happy_a no_o man_n be_v happy_a and_o of_o these_o if_o one_o be_v true_a the_o other_o be_v false_a also_o they_o may_v both_o be_v false_a as_o in_o the_o example_n give_v subcontrary_n be_v particular_a proposition_n of_o different_a quality_n as_o some_o man_n be_v learned_a some_o man_n be_v not_o learned_a which_o can_v be_v both_o false_a but_o they_o may_v be_v both_o true_a contradictory_n be_v those_o that_o differ_v both_o in_o quantity_n and_o quality_n as_o every_o man_n be_v a_o live_a creature_n some_o man_n be_v not_o a_o live_a creature_n which_o can_v neither_o be_v both_o true_a nor_o both_o false_a 18_o a_o proposition_n be_v say_v to_o follow_v from_o two_o other_o proposition_n when_o these_o be_v grant_v to_o be_v true_a it_o can_v be_v deny_v but_o the_o other_o be_v true_a also_o for_o example_n let_v these_o two_o proposition_n every_o man_n be_v a_o live_a creature_n and_o every_o live_a creature_n be_v a_o body_n be_v suppose_v true_a that_o be_v that_o body_n be_v the_o name_n of_o every_o live_a creature_n and_o live_v creature_n the_o name_n of_o every_o man._n see_v therefore_o if_o these_o be_v understand_v to_o be_v true_a it_o can_v be_v understand_v that_o body_n be_v not_o the_o name_n of_o every_o man_n that_o be_v that_o every_o man_n be_v a_o body_n be_v false_a this_o proposition_n will_v be_v say_v to_o follow_v from_o those_o two_o or_o to_o be_v necessary_o infer_v from_o they_o 19_o that_o a_o true_a proposition_n may_v follow_v from_o false_a proposition_n may_v happen_v sometime_o but_o false_a from_o true_a never_o for_o if_o these_o every_o man_n be_v a_o stone_n and_o every_o stone_n be_v a_o live_a creature_n which_o be_v both_o false_a be_v grant_v to_o be_v true_a it_o be_v grant_v also_o that_o live_a creature_n be_v the_o name_n of_o every_o stone_n and_o stone_n of_o every_o man_n that_o be_v that_o live_v creature_n be_v the_o name_n of_o every_o man_n that_o be_v to_o say_v this_o proposition_n every_o man_n be_v a_o live_a creature_n be_v true_a as_o it_o be_v indeed_o true_a wherefore_o a_o true_a proposition_n may_v sometime_o follow_v from_o false_a but_o if_o any_o two_o proposition_n be_v true_a a_o false_a one_o can_v never_o follow_v from_o they_o for_o if_o true_a follow_v from_o false_a for_o this_o reason_n only_o that_o the_o false_a be_v grant_v to_o be_v true_a than_o truth_n from_o two_o truth_n grant_v will_v follow_v in_o the_o same_o manner_n 20_o now_o see_v none_o but_o a_o true_a proposition_n will_v follow_v from_o true_a and_o that_o the_o understanding_n of_o two_o proposition_n to_o be_v true_a be_v the_o cause_n of_o understanding_n that_o also_o to_o be_v true_a which_o be_v deduce_v from_o they_o the_o two_o antecedent_n proposition_n be_v common_o call_v the_o cause_n of_o the_o infer_v proposition_n or_o conclusion_n and_o from_o hence_o it_o be_v that_o logician_n say_v the_o premise_n be_v cause_n of_o the_o conclusion_n which_o may_v pass_v though_o it_o be_v not_o proper_o speak_v for_o though_o understanding_n be_v the_o cause_n of_o understanding_n yet_o speech_n be_v not_o the_o cause_n of_o speech_n but_o when_o they_o say_v the_o cause_n of_o the_o property_n of_o any_o thing_n be_v the_o thing_n itself_o they_o speak_v absurd_o eor_fw-la example_n if_o a_o figure_n be_v propound_v which_o be_v triangular_a see_v every_o triangle_n have_v all_o its_o angle_n together_o equal_a to_o two_o right_a angle_n from_o whence_o it_o follow_v that_o all_o the_o angle_n of_o that_o figure_n be_v equal_a to_o two_o right_a angle_n they_o say_v for_o this_o reason_n that_o that_o figure_n be_v the_o cause_n of_o that_o equality_n but_o see_v the_o figure_n do_v not_o itself_o make_v its_o angle_n and_o therefore_o can_v be_v say_v to_o be_v the_o efficient-cause_n they_o call_v it_o the_o formall-cause_n whereas_o in_o deed_n it_o be_v no_o cause_n at_o all_o nor_o do_v the_o property_n of_o any_o figure_n follow_v the_o figure_n but_o have_v its_o be_v at_o the_o same_o time_n with_o it_o only_o the_o knowledge_n of_o the_o figure_n go_v before_o the_o knowledge_n of_o the_o property_n and_o one_o knowledge_n be_v true_o the_o cause_n of_o another_o knowledge_n namely_o the_o efficient-cause_n and_o thus_o much_o concern_v proposition_n which_o in_o the_o progress_n of_o philosophy_n be_v the_o first_o step_v like_o the_o move_n towards_o of_o one_o foot_n by_o the_o due_a addition_n of_o another_o step_v i_o shall_v proceed_v to_o syllogism_n and_o make_v a_o complete_a pace_n of_o which_o in_o the_o next_o chapter_n chap._n iu._n of_o syllogism_n 1_o the_o definition_n of_o syllogism_n 2_o in_o a_o syllogism_n there_o be_v but_o three_o term_n 3_o major_n minor_n and_o middle_a term_n also_o major_a and_o minor_a proposition_n what_o they_o be_v 4_o the_o middle_a term_n in_o every_o syllogism_n ought_v to_o be_v determine_v in_o both_o the_o proposition_n to_o one_o and_o the_o same_o thing_n 5_o from_o two_o particular_a proposition_n nothing_o can_v be_v conclude_v 6_o a_o syllogism_n be_v the_o collection_n of_o two_o proposition_n into_o one_o sum_n 7_o the_o figure_n of_o a_o syllogism_n what_o it_o be_v 8_o what_o be_v in_o the_o mind_n answer_v to_o a_o syllogism_n 9_o the_o first_o indirect_a figure_n how_o it_o be_v make_v 10_o the_o second_o indirect_a figure_n how_o make_v 11_o how_o the_o three_o indirect_a figure_n be_v make_v 12_o there_o be_v many_o mood_n in_o every_o figure_n but_o most_o of_o they_o useless_a in_o philosophy_n 13_o a_o hypotheticall_a syllogism_n when_o equipollent_a to_o a_o categoricall_a 1._o a_o speech_n consist_v of_o three_o proposition_n from_o two_o of_o which_o the_o three_o follow_v be_v call_v a_o syllogism_n and_o that_o which_o follow_v be_v call_v the_o conclusion_n the_o other_o two_o premise_n for_o example_n this_o speech_n every_o man_n be_v a_o live_a creature_n every_o live_a creature_n be_v a_o body_n therefore_o every_o man_n be_v a_o body_n be_v a_o syllogism_n because_o the_o three_o proposition_n follow_v from_o the_o two_o first_o that_o be_v if_o those_o be_v grant_v to_o be_v true_a this_o must_v also_o be_v grant_v to_o be_v true_a 2_o from_o two_o proposition_n which_o have_v not_o one_o term_n common_a no_o conclusion_n can_v follow_v and_o therefore_o no_o syllogism_n can_v be_v make_v of_o they_o for_o let_v any_o two_o premise_n a_o man_n be_v a_o live_a creature_n a_o tree_n be_v a_o plant_n be_v both_o of_o they_o true_a yet_o because_o it_o can_v be_v collect_v from_o they_o that_o plant_n be_v the_o name_n of_o a_o man_n or_o man_n the_o name_n of_o a_o plant_n it_o be_v not_o necessary_a that_o this_o conclusion_n a_o man_n be_v a_o plant_n shall_v be_v true_a corollary_n therefore_o in_o the_o premise_n of_o a_o syllogism_n there_o can_v be_v but_o three_o term_n beside_o there_o can_v be_v no_o term_n in_o the_o conclusion_n which_o be_v not_o in_o the_o premise_n for_o let_v any_o two_o premise_n be_v a_o man_n be_v a_o live_a creature_n a_o live_a creature_n be_v a_o body_n yet_o if_o any_o other_o term_n be_v put_v in_o the_o conclusion_n as_o man_n be_v two_o footed_a though_o it_o be_v true_a it_o can_v follow_v from_o the_o premise_n because_o from_o they_o it_o can_v be_v collect_v that_o the_o name_n two_o footed_a belong_v to_o a_o man_n and_o therefore_o again_o in_o every_o syllogism_n there_o can_v be_v but_o three_o term_n 3_o of_o these_o term_n that_o which_o be_v the_o predicate_a in_o the_o conclusion_n be_v common_o call_v the_o major_a that_o which_o be_v the_o subject_a in_o the_o conclusion_n the_o minor_a and_o the_o other_o be_v the_o middle_a term_n as_o in_o this_o syllogism_n a_o man_n be_v a_o live_a creature_n a_o live_a creature_n be_v a_o body_n therefore_o a_o man_n be_v a_o body_n body_n be_v the_o major_n man_n the_o minor_a and_o live_a creature_n the_o middle_a term._n also_o of_o the_o premise_n that_o in_o which_o the_o major_a term_n be_v find_v be_v call_v the_o major_a proposition_n and_o that_o which_o have_v the_o minor_a term_n the_o minor_a proposition_n 4_o if_o the_o middle_a term_n be_v not_o in_o both_o the_o premise_n determine_v to_o one_o and_o the_o same_o singular_a thing_n no_o conclusion_n will_v follow_v nor_o syllogism_n be_v make_v for_o let_v the_o minor_a term_n be_v man_n the_o middle_a term_n live_v creature_n and_o the_o major_a term_n
which_o make_v equal_a angle_n on_o either_o side_n of_o the_o perpendicular_a be_v produce_v to_o the_o tangent_fw-la they_o will_v be_v equal_a 12_o there_o be_v in_o euclid_n a_o definition_n of_o straight-lined_n parallel_n but_o i_o do_v not_o find_v that_o parallel_n in_o general_a be_v any_o where_o define_v and_o therefore_o for_o a_o universal_a definition_n of_o they_o i_o say_v that_o any_o two_o line_n whatsoever_o straight_o or_o crooked_a as_o also_o any_o two_o superficies_n be_v parallel_n when_o two_o equal_a straight_a line_n wheresoever_o they_o fall_v upon_o they_o make_v always_o equal_a angle_n with_o each_o of_o they_o from_o which_o definition_n it_o follow_v first_o that_o any_o two_o straight_a line_n not_o incline_v opposite_a way_n fall_v upon_o two_o other_o straight_a line_n which_o be_v parallel_n and_o intercept_n equal_a part_n in_o both_o of_o they_o be_v themselves_o also_o equal_a and_o parallel_v as_o if_o ab_fw-la and_o cd_o in_o the_o three_o figure_n incline_v both_o the_o same_o way_n fall_n upon_o the_o parallel_n ac_fw-la and_o bd_o and_o ac_fw-la and_o bd_o be_v equal_a ab_fw-la and_o cd_o will_n also_o be_v equal_a and_o parallel_v for_o the_o perpendicular_o be_v and_o df_n be_v draw_v the_o right_a angle_n ebb_v and_o fdh_n will_v be_v equal_a wherefore_o see_v of_o and_o bd_o be_v parallel_n the_o angle_v eba_n and_o fdc_n will_v be_v equal_a now_o if_o dc_o be_v not_o equal_a to_o basilius_n let_v any_o other_o straight_o line_n equal_a to_o basilius_n be_v draw_v from_o the_o point_n d_o which_o see_v it_o can_v fall_v upon_o the_o point_n c_o let_v it_o fall_v upon_o g._n whereore_n agnostus_n will_v be_v either_o great_a or_o less_o than_o bd_o and_o therefore_o the_o angle_n eba_n and_o fdg_n be_v not_o equal_a as_o be_v suppose_v wherefore_o ab_fw-la and_o cd_o be_v equal_a which_o be_v the_o first_o again_o because_o they_o make_v equal_a angle_n with_o the_o perpendicular_o be_v and_o df_n therefore_o the_o angle_n cdh_n will_v be_v equal_a to_o the_o angle_n abdella_n and_o by_o the_o definition_n of_o parallel_n ab_fw-la and_o cd_o will_v be_v parallel_n which_o be_v the_o second_o that_o plain_n which_o be_v include_v both_o way_n within_o parallel_a line_n be_v call_v a_o parallelogram_n 1_o corollary_n from_o this_o last_o it_o follow_v that_o the_o angles_n abdella_n and_o cdh_n be_v equal_a that_o be_v that_o a_o straight_a line_n as_o bh_n fall_v upon_o two_o parallel_n as_o ab_fw-la and_o cd_o make_v the_o internal_a angle_n abdella_n equal_a to_o the_o external_a and_o opposite_a angle_n cdh_n 2_o coral_n and_o from_o hence_o again_o it_o follow_v that_o a_o straight_a line_n fall_v upon_o two_o parallel_n make_v the_o alternate_a angle_n equal_a that_o be_v the_o angle_n agf_n in_o the_o four_o figure_n equal_a to_o the_o angle_n gfd_n for_o see_v gfd_n be_v equal_a to_o the_o external_a opposite_a angle_n egb_n it_o will_v be_v also_o equal_a to_o its_o vertical_a angle_n agf_n which_o be_v alternate_a to_o gfd_n 3_o coral_n that_o the_o internal_a angle_n on_o the_o same_o side_n of_o the_o line_n fg_v be_v equal_a to_o two_o right_a angle_n for_o the_o angle_n at_o f_o namely_o gfc_a and_o gfd_n be_v equal_a to_o two_o right_a angle_n but_o gfd_n be_v equal_a to_o its_o alternate_a angle_n agf_n wherefore_o both_o the_o angel_n gfc_n and_o agf_n which_o be_v internal_a on_o the_o same_o side_n of_o the_o line_n fg_v be_v equal_a to_o two_o right_a angle_n 4_o coral_n that_o the_o three_o angle_n of_o a_o straight-lined_n plain_a triangle_n be_v equal_a to_o two_o right_a angle_n and_o any_o side_n be_v produce_v the_o external_a angle_n will_v be_v equal_a to_o the_o two_o opposite_a internal_a angle_n for_o if_o there_o be_v draw_v by_o the_o vertex_fw-la of_o the_o plain_a triangle_n abc_n figure_n 5._o a_o parallel_n to_o any_o of_o the_o side_n as_o to_o ab_fw-la the_o angle_n a_o and_o b_o will_v be_v equal_a to_o their_o alternate_a angle_n e_z &_o f_o &_o the_o angle_n c_o be_v common_a but_o by_o the_o 10_o article_n the_o three_o angle_n e_o c_o and_o f_o be_v equal_a to_o two_o right_a angle_n and_o therefore_o the_o three_o angle_n of_o the_o triangle_n be_v equal_a to_o the_o same_o which_o be_v the_o first_o again_o the_o two_o angle_n b_o and_o d_o be_v equal_a to_o two_o right_a angle_n by_o the_o 10_o article_n wherefore_o take_v away_o b_o there_o will_v remain_v the_o angle_n a_o and_o c_o equal_v to_o the_o angle_n d_o which_o be_v the_o second_o 5_o coral_n if_o the_o angle_n a_o and_o b_o be_v equal_a the_o side_n ac_fw-la and_o cb_n will_v also_o be_v equal_a because_o ab_fw-la and_o of_o be_v parallel_n and_o on_o the_o contrary_a if_o the_o side_n ac_fw-la and_o cb_n be_v equal_a the_o angle_n a_o and_o b_o will_v also_o be_v equal_a for_o if_o they_o be_v not_o equal_a let_v the_o angle_n b_o and_o g_o be_v equal_a wherefore_o see_v gb_n and_o of_o be_v parallel_n and_o the_o angle_n g_o and_o b_o equal_v the_o side_n gc_a and_o cb_n will_v also_o be_v equal_a and_o because_o cb_n and_o ac_fw-la be_v equal_a by_o supposition_n cg_n and_o ca_n will_v also_o be_v equal_a which_o can_v be_v by_o the_o 11_o article_n 6_o coral_n from_o hence_o it_o be_v manifest_a that_o if_o two_o radii_fw-la of_o a_o circle_n be_v connect_v by_o a_o straight_a line_n the_o angle_n they_o make_v with_o that_o connect_a line_n will_v be_v equal_a to_o one_o another_o and_o if_o there_o be_v add_v that_o segment_n of_o the_o circle_n which_o be_v subtend_v by_o the_o same_o line_n which_o connect_v the_o radii_fw-la than_o the_o angle_n which_o those_o radii_fw-la make_v with_o the_o circumference_n will_v also_o be_v equal_a to_o one_o another_o for_o a_o straight_a line_n which_o subtend_v any_o arch_n make_v equal_a angle_n with_o the_o same_o because_o if_o the_o arch_n and_o the_o subtense_n be_v divide_v in_o the_o middle_n the_o two_o half_n of_o the_o segment_n will_v be_v congruous_a to_o one_o another_o by_o reason_n of_o the_o uniformity_n both_o of_o the_o circumference_n of_o the_o circle_n and_o of_o the_o straight_a line_n 13_o perimeter_n of_o circle_n be_v to_o one_o another_o as_o their_o semidiameter_n be_v for_o let_v there_o be_v any_o two_o circle_n as_o in_o the_o first_o figure_n bcd_v the_o great_a and_o efg_v the_o lesser_a have_v their_o common_a centre_n at_o a_o and_o let_v their_o semidiameter_n be_v ac_fw-la and_o ae_n i_o say_v ac_fw-la have_v the_o same_o proportion_n to_o ae_n which_o the_o perimeter_n bcd_n have_v to_o the_o perimeter_n efg_v for_o the_o magnitude_n of_o the_o semidiameter_n ac_fw-la and_o ae_n be_v determine_v by_o the_o distance_n of_o the_o point_n c_o and_o e_o from_o the_o centre_n a_o and_o the_o same_o distance_n be_v acquire_v by_o the_o uniform_a motion_n of_o a_o point_n from_o a_o to_o c_o in_o such_o manner_n that_o in_o equal_a time_n the_o distance_n acquire_v be_v equal_a but_o the_o perimeter_n bcd_v and_o efg_n be_v also_o determine_v by_o the_o same_o distance_n of_o the_o point_n c_o and_o e_o from_o the_o centre_n a_o and_o therefore_o the_o perimeter_n bcd_v and_o efg_a as_o well_o as_o the_o semidiameter_n ac_fw-la and_o ae_n have_v their_o magnitude_n determine_v by_o the_o same_o cause_n which_o cause_n make_v in_o equal_a time_n equal_a space_n wherefore_o by_o the_o 13_o chapter_n and_o 6_o article_n the_o perimeter_n of_o circle_n and_o their_o semidiameter_n be_v proportional_n which_o be_v to_o be_v prove_v 14_o if_o two_o straight_a line_n w_z ch_n constitute_v a_o angle_n be_v cut_v by_o straight-lined_n parallel_n the_o intercept_a parallel_n will_v be_v to_o one_o another_o as_o the_o parts_z w_z ch_z they_o cut_v off_o from_o the_o vertex_fw-la let_v the_o straight_a line_n ab_fw-la and_o ac_fw-la in_o the_o 6_o figure_n make_v a_o angle_n at_o a_o &_o be_v cut_v by_o the_o two_o straight-lined_n parallel_n bc_n and_o de_fw-fr so_o that_o the_o part_n cut_v off_o from_o the_o vertex_fw-la in_o either_o of_o those_o line_n as_o in_o ab_fw-la may_v be_v ab_fw-la and_o ad._n i_o say_v the_o parallel_n bc_n and_o de_fw-fr be_v to_o one_o another_o as_o the_o part_n ab_fw-la and_o ad._n for_o let_v ab_fw-la be_v divide_v into_o any_o number_n of_o equal_a part_n as_o into_z of_o fd_n db_n and_o by_o the_o point_v f_o and_o d_o let_v fg_n and_o de_fw-fr be_v draw_v parallel_n to_o the_o base_a bc_n and_o cut_v ac_fw-la in_o g_z and_o e_z and_o again_o by_o the_o point_n g_o and_o e_o let_v other_o straight_a line_n be_v draw_v parallel_n to_o ab_fw-la and_o cut_v bc_n in_o h_n and_o i._n if_o now_o the_o point_n a_o be_v understand_v to_o be_v move_v uniform_o over_o ab_fw-la and_o in_o the_o
same_o time_n b_o be_v move_v to_o c_o and_o all_o the_o point_n f_o d_o and_o b_o be_v move_v uniform_o and_o with_o equal_a swiftness_n over_o fg_n de_fw-fr and_o bc_n then_o shall_v b_o pass_v over_o bh_n equal_a to_o fg_v in_o the_o same_o time_n that_o a_o pass_v over_o of_o and_o of_o and_o fg_n will_v be_v to_o one_o another_o as_o their_o velocity_n be_v and_o when_o a_o be_v in_o f_o d_o will_v be_v in_o k_o when_o a_o be_v in_o d_o d_o will_v be_v in_o e_z and_o in_o what_o manner_n the_o point_n a_o pass_n by_o the_o point_n f_o d_o and_o b_o in_o the_o same_o manner_n the_o point_n b_o will_v pass_v by_o the_o point_v h_n i_o and_o c_o &_o the_o straight_a line_n fg_v dk_n ke_n bh_n he_o &_o i_fw-mi be_v equal_a by_o reason_n of_o their_o parallelisme_n and_o therefore_o as_o the_o velocity_n in_o ab_fw-la be_v to_o the_o velocity_n 〈◊〉_d bc_n so_o be_v ad_fw-la to_o de_fw-fr but_o as_o the_o velocity_n in_o ab_fw-la be_v to_o the_o velocity_n in_o bc_n so_o be_v ab_fw-la to_o bc_n that_o be_v to_o say_v all_o the_o parallel_n will_v be_v several_o to_o all_o the_o part_n cut_v off_o from_o the_o vertex_fw-la as_o of_o be_v to_o fg._n wherefore_o af._n gf_n ad._n de_fw-fr ab_fw-la bc_n be_v proportional_n the_o subtense_n of_o equal_a angle_n in_o different_a circle_n as_o the_o straight_a line_n bc_n and_o fe_o in_o the_o 1_o figure_n be_v to_o one_o another_o as_o the_o arch_n which_o they_o subtend_v for_o by_o the_o 8_o article_n the_o arch_n of_o equal_a angle_n be_v to_o one_o another_o as_o their_o perimeter_n be_v and_o by_o the_o 13_o art_n the_o perimeter_n as_o their_o semidiameter_n but_o the_o the_o subtense_n bc_n and_o fe_o be_v parallel_n to_o one_o another_o by_o reason_n of_o the_o equality_n of_o the_o angle_n which_o they_o make_v with_o the_o semidiameter_n and_o therefore_o the_o same_o subtense_n by_o the_o last_o precedent_n article_n will_v be_v proportional_a to_o the_o semidiameter_n that_o be_v to_o the_o perimeter_n that_o be_v to_o the_o arch_n which_o they_o subtend_v 15_o if_o in_o a_o circle_n any_o number_n of_o equal_a subtense_n be_v place_v immediate_o after_o one_o another_o and_o straight_a line_n be_v draw_v from_o the_o extreme_a point_n of_o the_o first_o subtense_n to_o the_o extreme_a point_n of_o all_o the_o rest_n the_o first_o subtense_n be_v produce_v will_v make_v with_o the_o second_o subtense_n a_o external_a angle_n double_a to_o that_o which_o be_v make_v by_o the_o same_o first_o subtense_n and_o a_o tangent_fw-la to_o the_o circle_n touch_v it_o in_o the_o extreme_a point_n thereof_o and_o if_o a_o straight_a line_n which_o subtend_v two_o of_o those_o arch_n be_v produce_v it_o will_v make_v a_o external_a angle_n with_o the_o three_o subtense_n triple_a to_o the_o angle_n which_o be_v make_v by_o the_o tangent_fw-la with_o the_o first_o subtense_n and_o so_o continual_o for_o with_o the_o radius_fw-la ab_fw-la in_o the_o seven_o figure_n let_v a_o circle_n be_v describe_v &_o in_o it_o let_v any_o number_n of_o equal_a subtense_n bc_n cd_o &_o de_fw-fr be_fw-mi place_v also_o let_v bd_o &_o be_v be_v draw_v &_o by_o produce_v bc_n bd_o &_o be_v to_o any_o distance_n in_o g_z h_o and_o i_o let_v they_o make_v angle_n with_o the_o subtense_n which_o succeed_v one_o another_o namely_o the_o external_a angle_n gcd_v and_o hde_v last_o let_v the_o tangent_fw-la kb_fw-la be_v draw_v make_v with_o the_o first_o subtense_n the_o angle_n kbc_n i_o say_v the_o angle_n gcd_n be_v double_a to_o the_o angle_n kbc_n and_o the_o angle_n hde_v triple_a to_o the_o same_o angle_n kbc_n for_o if_o ac_fw-la be_v draw_v cut_v bd_o in_o m_o and_o from_o the_o point_n c_o there_o be_v draw_v lc_n perpendicular_a to_o the_o same_o ac_fw-la then_o cl_n and_o md_o will_v be_v parallel_n by_o reason_n of_o the_o right_a angle_n at_o c_o and_o m_o and_o therefore_o the_o alterne_a angle_n lcd_v and_o bdc_n will_v be_v equal_a as_o also_o the_o angel_n bdc_n and_o cbd_n will_v be_v equal_a because_o of_o the_o equality_n of_o the_o straight_a line_n bc_n and_o cd_o wherefore_o the_o angle_n gcd_n be_v double_a to_o either_o of_o the_o angel_n cbd_v or_o cdb_n and_o therefore_o also_o the_o angle_n gcd_n be_v double_a to_o the_o angle_n lcd_v that_o be_v to_o the_o angle_n kbc_n again_o cd_o be_v parallel_n to_o be_v by_o reason_n of_o the_o equality_n of_o the_o angel_n cbe_n and_o deb_n and_o of_o the_o straight_a line_n cb_n and_o de_fw-fr and_o therefore_o the_o angle_n gcd_v and_o gbe_n be_v equal_a and_o consequent_o gbe_n as_o also_o deb_n be_v double_a to_o the_o angle_n kbc_n but_o the_o external_a angle_n hde_v be_v equal_a to_o the_o two_o internal_a deb_n and_o dbe_n and_o therefore_o the_o angle_n hde_v be_v triple_a to_o the_o angle_n kbc_n etc._n etc._n which_o be_v to_o be_v prove_v 1_o corollary_n from_o hence_o it_o be_v manifest_a that_o the_o angel_n kbc_n and_o cbd_v as_o also_o that_o all_o the_o angle_n that_o be_v comprehend_v by_o two_o 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again_o in_o c_z according_a to_o a_o double_a angle_n and_o in_o d_o according_a to_o a_o angle_n which_o be_v triple_a and_o in_o e_z according_a to_o a_o angle_n which_o be_v quadruple_a â—o_o the_o first_o angle_n and_o so_o continual_o there_o will_v be_v describe_v a_o figure_n which_o will_v indeed_o be_v rectilineal_a if_o the_o break_a part_n be_v consider_v as_o haâ—ing_v magnitude_n but_o if_o they_o be_v understand_v to_o be_v the_o least_o tâ—aâ—_n can_v 〈◊〉_d â—â—at_o be_v as_o so_o many_o point_n then_o the_o figure_n describe_v will_v â—ot_n be_v rectilineal_a but_o a_o circle_n who_o circumference_n wâ—…_n break_a line_n 4_o 〈…〉_z be_v say_v in_o this_o present_a article_n it_o may_v 〈…〉_z angle_z in_o the_o centre_n be_v double_a to_o a_o angle_n in_o the_o circumference_n of_o the_o same_o circle_n if_o the_o intercept_a arch_n be_v equal_a for_o see_v that_o straight_a line_n by_o who_o motion_n a_o angle_n be_v determine_v pass_v over_o equal_a arch_n in_o equal_a 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the_o angle_n god_n insist_v upon_o the_o same_o arch_a cd_o be_v also_o double_a to_o the_o same_o angle_n cbd_v or_o kbc_n 16_o a_o angle_n of_o contingence_n if_o it_o be_v compare_v with_o a_o angle_n simple_o so_o call_v how_o little_a soever_o have_v such_o proportion_n to_o it_o as_o a_o point_n have_v to_o a_o line_n that_o be_v no_o proportion_n at_o all_o nor_o any_o quantity_n for_o first_o a_o angle_n of_o contingence_n be_v make_v by_o continual_a flexion_n so_o that_o in_o the_o generation_n of_o it_o there_o be_v no_o circular_a motion_n at_o all_o in_o which_o consist_v the_o nature_n of_o a_o angle_n simple_o so_o call_v and_o therefore_o it_o can_v be_v
angle_n nbk_n which_o be_v do_v boy_n will_v be_v the_o line_n of_o reflection_n from_o the_o line_n of_o incidence_n nb._n last_o from_o the_o incident_a line_n lc_n let_v the_o reflect_a line_n co_n be_v draw_v cut_v boy_n at_o o_o and_o make_v the_o angle_n cob_n i_o say_v the_o angle_n cob_n be_v equal_a to_o the_o angle_n z._n let_v nb_n be_v produce_v till_o it_o meet_v with_o the_o straight_a line_n lc_n produce_v in_o p._n see_v therefore_o the_o angle_n lcm_n be_v by_o construction_n equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n z_o the_o angle_n npl_n which_o be_v equal_a to_o lcm_n by_o reason_n of_o the_o parallel_n np_n and_o mc_n will_v also_o be_v equal_a to_o twice_o the_o same_o angle_n bac_n together_o with_o the_o angle_n z._n and_o see_v the_o two_o straight_a line_n oc_n and_o ob_fw-la fall_n from_o the_o point_n o_o upon_o the_o point_n c_o and_o b_o and_o their_o reflect_a line_n lc_a and_o nb_n meet_v in_o the_o point_n p_o the_o angle_n npl_n will_v be_v equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n cop_z but_o i_o have_v already_o prove_v the_o angle_n npl_n to_o be_v equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n z._n therefore_o the_o angle_n cop_z be_v equal_a to_o the_o angle_n z_o wherefore_o two_o point_n in_o the_o circumference_n of_o a_o circle_n be_v give_v i_o have_v draw_v etc._n etc._n which_o be_v to_o be_v do_v but_o if_o it_o be_v require_v to_o draw_v the_o incident_a line_n from_o a_o point_n within_o the_o circle_n so_o that_o the_o line_n reflect_v from_o they_o may_v contain_v a_o angle_n equal_a to_o the_o angle_n z_o the_o same_o method_n be_v to_o be_v use_v save_v that_o in_o this_o case_n the_o angle_n z_o be_v not_o to_o be_v add_v to_o twice_o the_o angle_n bac_n but_o to_o be_v take_v from_o it_o 9_o if_o a_o straight_a line_n fall_v upon_o the_o circumference_n of_o a_o circle_n be_v produce_v till_o it_o reach_v the_o semidiameter_n and_o that_o part_n of_o it_o which_o be_v intercept_v between_o the_o circumference_n and_o the_o semidiameter_n be_v equal_a to_o that_o part_n of_o the_o semidiameter_n which_o be_v between_o the_o point_n of_o concourse_n and_o the_o centre_n the_o reflect_a line_n will_v be_v parallel_n to_o the_o semidiameter_n let_v any_o line_n ab_fw-la in_o the_o 9th_o figure_n be_v the_o semidiameter_n of_o the_o circle_n who_o centre_n be_v a_o and_o upon_o the_o circumference_n bd_o let_v the_o straight_a line_n cd_o fall_n and_o be_v produce_v till_o it_o cut_v ab_fw-la in_o e_z so_o that_z ed_z and_o ea_z may_v be_v equal_a &_o from_o the_o incident_a line_n cd_o let_v the_o line_n df_n be_v reflect_v i_o say_v ab_fw-la and_o df_n will_v be_v parallel_n let_v agnostus_n be_v draw_v through_o the_o point_n d._n see_v therefore_o ed_z and_o ea_z be_v equal_a the_o angle_v eda_n and_o ead_n will_v also_o be_v equal_a but_o the_o angle_n fdg_n and_o eda_n be_v equal_a for_o each_o of_o they_o be_v half_a the_o angle_n edh_n or_o fdc_n wherefore_o the_o angle_n fdg_n and_o ead_n be_v equal_a and_o consequent_o df_n and_o ab_fw-la be_v parallel_n which_o be_v to_o be_v prove_v corollahy_fw-fr if_o ea_fw-la be_v great_a than_o ed_z than_o df_n and_o ab_fw-la be_v produce_v will_v concur_v but_o if_o ea_z be_v less_o than_o ed_z then_z ba_z and_o dh_n be_v produce_v will_v concur_v 10_o if_o from_o a_o point_n within_o a_o circle_n two_o straight_a line_n be_v draw_v to_o the_o circumference_n and_o their_o reflect_a line_n meet_v in_o the_o circumference_n of_o the_o same_o circle_n the_o angle_n make_v by_o the_o line_n of_o reflection_n will_v be_v a_o three_o part_n of_o the_o angle_n make_v by_o the_o line_n of_o incidence_n from_o the_o point_n b_o in_o the_o 10_o figure_n take_v within_o the_o circle_n who_o centre_n be_v a_o let_v the_o two_o straight_a line_n bc_n and_o bd_o be_v draw_v to_o the_o circumference_n and_o let_v their_o reflect_a lines_n ce_fw-fr and_o de_fw-fr meet_v in_o the_o circumference_n of_o the_o same_o circle_n at_o the_o point_n e._n i_o say_v the_o angle_n ce_v will_v be_v a_o three_o part_n of_o the_o angle_n cbd_v let_v ac_fw-la and_o ad_fw-la be_v draw_v see_v therefore_o the_o angle_n ce_v and_o cbd_v together_o take_v be_v equal_a to_o twice_o the_o angle_n god_n as_o have_v be_v demonstrate_v in_o the_o 5_o article_n and_o the_o angle_n god_n twice_o take_v be_v quadruple_a to_o the_o angle_n ce_v the_o angle_n ce_v and_o cbd_v together_o take_v will_v also_o be_v equal_a to_o the_o angle_n ce_v four_o time_n take_v and_o therefore_o if_o the_o angle_n ce_v be_v take_v away_o on_o both_o side_n there_o will_v remain_v the_o angle_n cbd_v on_o one_o side_n equal_a to_o the_o angle_n ce_v thrice_o take_v on_o the_o other_o side_n which_o be_v to_o be_v demonstrate_v coral_n therefore_o a_o point_n be_v give_v within_o a_o circle_n there_o may_v be_v draw_v two_o line_n from_o it_o to_o the_o circumference_n so_o as_o their_o reflect_a line_n may_v meet_v in_o the_o circumference_n for_o it_o be_v but_o trisect_v the_o angle_n cbd_v which_o how_o it_o may_v be_v do_v shall_v be_v show_v in_o the_o follow_a chapter_n chap._n xx._n of_o the_o dimension_n of_o a_o circle_n and_o the_o division_n of_o angle_n or_o arch_n 1_o the_o dimension_n of_o a_o circle_n near_o determine_v in_o number_n by_o archimedes_n and_o other_o 2_o the_o first_o attempt_n for_o the_o find_v out_o of_o the_o dimension_n of_o a_o circle_n by_o line_n 3_o the_o second_o attempt_n for_o the_o find_v out_o of_o the_o dimension_n of_o a_o circle_n from_o the_o consideration_n of_o the_o nature_n of_o crookedness_n 4_o the_o three_o attempt_n and_o some_o thing_n propound_v to_o be_v further_o search_v into_o 5_o the_o equation_n of_o the_o spiral_n of_o archimedes_n with_o a_o straight_a line_n 6_o of_o the_o analysis_n of_o geometrician_n by_o the_o power_n of_o line_n 1_o in_o the_o compare_v of_o a_o arch_n of_o a_o circle_n with_o a_o straight_a line_n many_o and_o great_a geometrician_n even_o from_o the_o most_o ancient_a time_n have_v exercise_v their_o wit_n and_o more_o have_v do_v the_o same_o if_o they_o have_v not_o see_v their_o pain_n though_o undertake_v for_o the_o common_a good_a if_o not_o bring_v to_o perfection_n vilify_v by_o those_o that_o envy_v the_o praise_n of_o other_o man_n among_o those_o ancient_a writer_n who_o work_n be_v come_v to_o our_o hand_n archimedes_n be_v the_o first_o that_o bring_v the_o length_n of_o the_o perimeter_n of_o a_o circle_n within_o the_o limit_n of_o number_n very_o little_o differ_v from_o the_o truth_n demonstrate_v the_o same_o to_o be_v less_o than_o three_o diameter_n and_o a_o seven_o part_n but_o great_a than_o three_o diameter_n and_o ten_o seventy_o one_o part_n of_o the_o diameter_n so_o that_o suppose_v the_o radius_fw-la to_o consist_v of_o 10000000_o equal_a part_n the_o arch_n of_o a_o quadrant_n will_v be_v between_o 15714285_o and_o 15_o 04225_o of_o the_o same_o part_n in_o our_o time_n ludovicus_n van_n cullen_n &_o willebrordus_fw-la snellius_n with_o joint_a endeavour_n have_v come_v yet_o near_o to_o the_o truth_n and_o pronounce_v from_o true_a principle_n that_o the_o arch_n of_o a_o quadrant_n put_v as_o before_o 10000000_o for_o radius_fw-la differ_v not_o one_o whole_a unity_n from_o the_o number_n 15707963_o which_o if_o they_o have_v exhibit_v their_o arithmetical_a operation_n and_o no_o man_n have_v discover_v any_o error_n in_o that_o long_a work_n of_o they_o have_v be_v demonstrate_v by_o they_o this_o be_v the_o further_a progress_n that_o have_v be_v make_v by_o the_o way_n of_o number_n and_o they_o that_o have_v proceed_v thus_o far_o deserve_v the_o praise_n of_o industry_n nevertheless_o if_o we_o consider_v the_o benefit_n which_o be_v the_o scope_n at_o which_o all_o speculation_n shall_v aim_v the_o improvement_n they_o have_v make_v have_v be_v little_a or_o none_o for_o any_o ordinary_a man_n may_v much_o soon_o &_o more_o accurate_o find_v a_o straight_a line_n equal_a to_o the_o perimeter_n of_o a_o circle_n and_o consequent_o square_v the_o circle_n by_o wind_v a_o small_a thread_n about_o a_o give_v cylinder_n than_o any_o geometrician_n shall_v do_v the_o same_o by_o divide_v the_o radius_fw-la into_o 10000000_o equal_a part_n but_o though_o the_o length_n of_o the_o circumference_n be_v exact_o set_v out_o either_o by_o number_n or_o mechanical_o or_o only_o by_o chance_n yet_o this_o will_v contribute_v no_o help_n at_o all_o towards_o the_o section_n of_o angles_n unless_o happy_o these_o two_o problem_n to_o divide_v a_o give_v angle_n according_a to_o any_o proportion_n assign_v and_o to_o find_v a_o
equal_a that_o no_o inequality_n can_v be_v discover_v between_o they_o i_o will_v therefore_o leave_v this_o to_o be_v further_o search_v into_o for_o though_o it_o be_v almost_o out_o of_o doubt_n that_o the_o straight_a line_n bp_o and_o the_o arch_n bmd_v be_v equal_a yet_o that_o may_v not_o be_v receive_v without_o demonstration_n and_o mean_n of_o demonstration_n the_o circular_a line_n admit_v none_o that_o be_v not_o ground_v upon_o the_o nature_n of_o flexion_n or_o of_o angle_n but_o by_o that_o way_n i_o have_v already_o exhibit_v a_o straight_a line_n equal_a to_o the_o arch_n of_o a_o quadrant_n in_o the_o first_o and_o second_o aggression_n it_o remain_v that_o i_o prove_v dt_n to_o be_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n in_o basilius_n produce_v let_v hence_v he_o take_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n and_o draw_v and_o produce_v uh_o it_o will_v cut_v the_o arch_n of_o the_o quadrant_n cna_fw-mi in_o the_o midst_n in_o n_o and_o the_o same_o arch_n again_o in_o o_o and_o the_o straight_a line_n dc_o in_o t_o so_o that_o dt_n will_v be_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n or_o to_o the_o straight_a line_n hence_v also_o the_o straight_a line_n uh_o will_v be_v equal_a to_o the_o straight_a line_n he_o or_o the_o sine_fw-la of_o 60_o degree_n for_o the_o square_n of_o ave_fw-la be_v equal_a to_o two_o square_n of_o the_o semiradius_fw-la and_o consequent_o the_o square_a of_o uh_o be_v equal_a to_o three_o square_n of_o the_o semiradius_fw-la but_o he_o be_v a_o mean_a proportional_a between_o the_o semiradius_fw-la and_o three_o semiradii_fw-la and_o therefore_o the_o square_a of_o he_o be_v equal_a to_o three_o square_n of_o the_o semiradius_fw-la wherefore_o he_o be_v eqval_n to_o hv_v but_o because_o ad_fw-la be_v cut_v in_o the_o midst_n in_o h_n therefore_o uh_o and_o ht_n be_v equal_a and_o therefore_o also_o dt_n be_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n in_o the_o radius_fw-la basilius_n let_v bx_n be_v take_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n for_o so_o vx_n will_v be_v equal_a to_o the_o radius_fw-la and_o it_o will_v be_v as_o valerio_n to_o ah_o the_o semiradius_fw-la so_o vx_n the_o radius_fw-la to_o xn_n the_o sine_fw-la of_o 45_o degree_n wherefore_o uh_o produce_v pass_n through_o n._n last_o upon_o the_o centre_n five_o with_o the_o radius_fw-la va_o let_v the_o arch_n of_o a_o circle_n be_v draw_v cut_v uh_o in_o y_o which_o be_v do_v vy_z will_v be_v equal_a to_o ho_o for_o ho_o be_v by_o construction_n equal_a to_o the_o sine_fw-la of_o 45_o degree_n and_o yh_n will_v be_v equal_a to_o ot_fw-mi &_o therefore_o vt_fw-la pass_v through_o o._n all_o which_o be_v to_o be_v demonstrate_v i_o will_v here_o add_v certain_a problem_n of_o which_o if_o any_o analyst_n can_v make_v the_o construction_n he_o will_v thereby_o be_v able_a to_o judge_v clear_o of_o what_o i_o have_v now_o say_v concern_v the_o dimension_n of_o a_o circle_n now_o these_o problem_n be_v nothing_o else_o at_o least_o to_o sense_n but_o certain_a symptom_n accompany_v the_o construction_n of_o the_o first_o and_o three_o figure_n of_o this_o chapter_n describe_v therefore_o again_o the_o square_a abcd_n in_o the_o five_o figure_n and_o the_o three_o quadrant_n abdella_n bca_n and_o dac_n let_v the_o diagonal_n ac_fw-la &_o bd_o be_v draw_v cut_v the_o arch_n bhd_v &_o cia_n in_o the_o middle_n in_o h_n and_o i_o &_o the_o straight_a line_n of_o and_o gl_n divide_v the_o square_n abcd_v into_o four_o equal_a square_n and_o trisect_v the_o arch_n bhd_v and_o cia_n namely_o bhd_v in_o k_o and_o m_n and_o cia_n in_o m_n and_o o._n then_o divide_v the_o arch_n bk_n in_o the_o midst_n in_o p_o let_v qp_v the_o sine_fw-la of_o the_o arch_n bp_o be_v draw_v and_o produce_v to_o r_o so_o that_o qr_n be_v double_a to_o qp_v and_o connect_v kr_n let_v it_o be_v produce_v one_o way_n to_o bc_n in_o saint_n and_o the_o other_o way_n to_o basilius_n produce_v in_o t._n also_o let_v bv_o be_v make_v triple_a to_o b_n and_o consequent_o by_o the_o second_o article_n of_o this_o chapter_n equal_a to_o the_o arch_n bd._n this_o construction_n be_v the_o same_o with_o that_o of_o the_o first_o figure_n which_o i_o think_v fit_a to_o renew_v discharge_v of_o all_o line_n but_o such_o as_o be_v necessary_a for_o my_o present_a purpose_n in_o the_o first_o place_n therefore_o if_o hence_v be_v draw_v cut_v the_o arch_n bhd_v in_o x_o and_o the_o side_n dc_o in_o z_o i_o desire_v some_o analyst_n will_v if_o he_o can_v give_v a_o reason_n why_o the_o straight_a line_n te_fw-la and_o tc_n shall_v cut_v the_o arch_n bd_o the_o one_o in_o y_o the_o other_o in_o x_o so_o as_o to_o make_v the_o arch_n by_o equal_a to_o the_o arch_n yx_n or_o if_o they_o be_v not_o equal_a that_o he_o will_v determine_v their_o difference_n second_o if_o in_o the_o side_n dam_fw-ge the_o straight_a line_n dam_fw-ge be_v take_v equal_a to_o dz_n and_o valerio_n be_v draw_v why_o va_o and_o ub_z shall_v be_v equal_a or_o if_o they_o be_v not_o equal_a what_o be_v the_o difference_n three_o draw_v zb_a parallel_n and_o equal_a to_o the_o side_n cb_n cut_v the_o arch_n bhd_v in_o c_o and_o draw_v the_o straight_a line_n ac_fw-la and_o produce_v it_o to_o bv_v in_o d_o why_o admetus_n shall_v be_v equal_a and_o parallel_v to_o the_o straight_a line_n hence_v and_o consequent_o equal_a also_o to_o the_o arch_n bd._n four_o draw_v eke_o the_o sine_fw-la of_o the_o arch_n bk_n &_o take_v in_o ea_z produce_v of_o equal_a to_o the_o diagonal_a ac_fw-la and_o connect_v fc_n why_o fc_n shall_v pass_v through_o a_o which_o point_n be_v give_v the_o length_n of_o the_o arch_n bhd_v be_v also_o give_v and_o c_o and_o why_o fe_o and_o fc_n shall_v be_v equal_a or_o if_o not_o why_o unequal_a five_o draw_v fz_n i_o desire_v he_o will_v show_v why_o it_o be_v equal_a to_o bv_v or_o to_o the_o arch_n bd_o or_o if_o they_o be_v not_o equal_a what_o be_v their_o difference_n six_o grant_v fz_n to_o be_v equal_a to_o the_o arch_n bd_o i_o desire_v he_o will_v determine_v whether_o it_o fall_v all_o without_o the_o arch_n bca_n or_o cut_v the_o same_o or_o touch_v it_o and_o in_o what_o point_n seven_o the_o semicircle_n bdg_n be_v complete_v why_o give_v be_v draw_v and_o produce_v shall_v pass_v through_o x_o by_o which_o point_n x_o the_o length_n of_o the_o arch_n bd_o be_v determine_v and_o the_o same_o give_v be_v yet_o further_o produce_v to_o dc_o in_o h_z why_o add_v which_o be_v equal_a to_o the_o arch_n bd_o shall_v pass_v through_o that_o point_n h._n eight_o upon_o the_o centre_n of_o the_o square_n abcd_v which_o let_v be_v k_o the_o arch_n of_o the_o quadrant_a eil_n be_v draw_v cut_v eke_o produce_v in_o i_o why_o the_o draw_v straight_a line_n ix_o shall_v be_v parallel_n to_o the_o side_n cd_o nine_o in_o the_o side_n basilius_n and_o bc_n take_v bl_a and_o bm_n several_o equal_a to_o half_a bv_n or_o to_o the_o arch_n bh_n and_o draw_v mn_a parallel_n and_o equal_a to_o the_o side_n basilius_n cut_v the_o arch_n bd_o in_o o_o why_o the_o straight_a line_n which_o connect_v vl_a shall_v pass_v through_o the_o point_n o_o ten_o i_o will_v know_v of_o he_o why_o the_o straight_a line_n which_o connect_n ah_o shall_v be_v equal_a to_o bm_v or_o if_o not_o how_o much_o it_o differ_v from_o it_o the_o analyst_n that_o can_v solve_v these_o problem_n without_o know_v first_o the_o length_n of_o the_o arch_n bd_o or_o use_v any_o other_o know_v method_n then_o that_o which_o proceed_v by_o perpetual_a bisection_n of_o a_o angle_n or_o be_v draw_v from_o the_o consideration_n of_o the_o nature_n of_o flexion_n shall_v do_v more_o than_o ordinary_a geometry_n be_v able_a to_o perform_v but_o if_o the_o dimension_n of_o a_o circle_n can_v be_v find_v by_o any_o other_o method_n then_o i_o have_v either_o find_v it_o or_o it_o be_v not_o at_o all_o to_o be_v find_v from_o the_o know_a length_n of_o the_o arch_n of_o a_o quadrant_n and_o from_o the_o proportional_a division_n of_o the_o arch_n and_o of_o the_o tangent_fw-la bc_n may_v be_v deduce_v the_o section_n of_o a_o angle_n into_o any_o give_a proportion_n as_o also_o the_o square_n of_o the_o circle_n the_o square_n of_o a_o give_v sector_n and_o many_o the_o like_a proposition_n which_o it_o be_v not_o necessary_a here_o to_o demonstrate_v i_o will_v therefore_o only_o exhibit_v a_o straight_a line_n equal_a to_o the_o spiral_n of_o archimedes_n and_o so_o dismiss_v this_o speculation_n 5_o the_o length_n of_o the_o perimeter_n of_o a_o circle_n be_v find_v that_o straight_a line_n be_v also_o find_v which_o
weaken_v by_o little_a and_o little_a but_o this_o can_v be_v do_v but_o by_o the_o long_a continuance_n of_o action_n or_o by_o action_n often_o repeat_v and_o therefore_o custom_n beget_v that_o facicility_n which_o be_v common_o and_o right_o call_v habit_n and_o it_o may_v be_v define_v thus_o habit_n be_v motion_n make_v more_o easy_a and_o ready_a by_o custom_n that_o be_v to_o say_v by_o perpetual_a endeavour_n or_o by_o iterate_a endeavour_n in_o a_o way_n differ_v from_o that_o in_o which_o the_o motion_n proceed_v from_o the_o beginning_n and_o oppose_v such_o endeavour_n as_o resist_v and_o to_o make_v this_o more_o perspicuous_a by_o example_n we_o may_v observe_v that_o when_o one_o that_o have_v no_o skill_n in_o music_n first_o put_v his_o hand_n to_o a_o instrument_n he_o can_v after_o the_o first_o stroke_n carry_v to_o his_o hand_n to_o the_o place_n where_o he_o will_v make_v the_o second_o stroke_n without_o take_v it_o back_o by_o a_o new_a endeavour_n and_o as_o it_o be_v beginning_n again_o pass_v from_o the_o first_o to_o the_o second_o nor_o will_v he_o be_v able_a to_o go_v on_o to_o the_o three_o place_n without_o another_o new_a endeavour_n but_o he_o will_v be_v force_v to_o draw_v back_o his_o hand_n again_o and_o so_o successive_o by_o renew_v his_o endeavour_n at_o every_o stroke_n till_o at_o the_o last_o by_o do_v this_o often_o and_o by_o compound_v many_o interrupt_a motion_n or_o endeavour_n into_o one_o equal_a endeavour_n he_o be_v able_a to_o make_v his_o hand_n go_v ready_o on_o from_o stroke_n to_o stroke_n in_o that_o order_n and_o way_n which_o be_v at_o the_o first_o design_v nor_o be_v habit_n to_o be_v observe_v in_o live_a creature_n only_o but_o also_o in_o body_n inanimate_a for_o we_o find_v that_o when_o the_o lath_n of_o a_o crossbow_n be_v strong_o bend_v and_o will_v if_o the_o impediment_n be_v remove_v return_n again_o with_o great_a force_n if_o it_o remain_v a_o long_a time_n bend_v it_o will_v get_v such_o a_o habit_n that_o when_o it_o be_v loose_v and_o leave_v to_o its_o own_o freedom_n it_o will_v not_o only_o not_o restore_v itself_o but_o will_v require_v as_o much_o force_n for_o the_o bring_n of_o it_o back_o to_o its_o first_o posture_n as_o it_o do_v for_o the_o bend_n of_o it_o at_o the_o first_o chap._n xxiii_o of_o the_o centre_n of_o equiponderation_n of_o body_n press_v dottard_n in_o straight_o parallel_v line_n 1_o definition_n and_o supposition_n 2_o two_o plain_n of_o equiponderation_n be_v nâ—â—_n parallel_v 3_o the_o centre_n of_o equiponderation_n be_v in_o every_o plain_n of_o equiponderation_n 4_o the_o moment_n of_o equal_a ponderant_n be_v to_o one_o another_o as_o their_o distance_n from_o the_o centre_n of_o the_o scale_n 5,6_o the_o moment_n of_o unequal_a ponderant_n have_v their_o proportion_n to_o one_o another_o compound_v of_o the_o proportion_n of_o their_o weight_n and_o distance_n from_o the_o centre_n of_o the_o scale_n reciprocal_o take_v 7._o if_o two_o ponderant_n have_v their_o moment_n and_o distance_n from_o the_o centre_n of_o the_o scale_n in_o reciprocal_a proportion_n they_o be_v equal_o poise_v and_o contrary_o 8_o if_o the_o part_n of_o any_o ponderant_fw-la press_v the_o beam_n of_o the_o scale_n every_o where_o equal_o all_o the_o part_n cut_v out_o off_o reckon_v from_o the_o centre_n of_o the_o scale_n will_v have_v their_o moment_n in_o the_o same_o proportion_n with_o that_o of_o the_o part_n of_o a_o triangle_n cut_v off_o from_o the_o vertex_fw-la by_o straight_a line_n parallel_v to_o the_o base_a 9_o the_o diameter_n of_o equiponderation_n of_o figure_n which_o be_v deficient_a according_a to_o commensurable_a proportion_n of_o their_o altitude_n and_o base_n divide_v the_o axis_n so_o that_o the_o part_n take_v next_o the_o vertex_fw-la be_v to_o the_o other_o part_n as_o the_o complete_a figure_n to_o the_o deficient_a figure_n 10_o the_o diameter_n of_o equiponderation_n of_o the_o compliment_n of_o the_o half_a of_o any_o of_o the_o say_v deficient_a figure_n divide_v that_o line_n which_o be_v draw_v through_o the_o vertex_fw-la parallel_n to_o the_o base_a so_o that_o the_o part_n next_o the_o vertex_fw-la be_v to_o the_o other_o part_n as_o the_o complete_a figure_n to_o the_o compliment_n 11_o the_o centre_n of_o equiponderation_n of_o the_o half_a of_o any_o of_o the_o desicient_a figure_n in_o the_o first_o row_n of_o the_o table_n of_o the_o 3d._a article_n of_o the_o 17_o chapter_n may_v be_v find_v out_o by_o the_o number_n of_o the_o second_o row_n 12_o the_o centre_n of_o equiponderation_n of_o the_o half_a of_o any_o of_o the_o figure_n in_o the_o second_o row_n of_o the_o same_o table_n may_v be_v find_v out_o by_o the_o number_n of_o the_o four_o row_n 13_o the_o centre_n of_o equiponderation_n of_o the_o half_a of_o any_o of_o the_o figure_n in_o the_o same_o table_n be_v know_v the_o centre_n of_o the_o excess_n of_o the_o same_o figure_n above_o a_o triangle_n of_o the_o same_o altitude_n and_o base_a be_v also_o know_v 14_o the_o centre_n of_o equiponderation_n of_o a_o solid_a sector_n be_v in_o the_o axis_n so_o divide_v that_o the_o part_n next_o the_o vertex_fw-la be_v to_o the_o whole_a axis_n want_v half_a the_o axis_n of_o the_o portion_n of_o the_o sphere_n as_o 3_o to_o 4._o 1_o definition_n 1_o a_o scale_n be_v a_o straight_a line_n who_o middle_a point_n be_v immovable_a all_o the_o rest_n of_o its_o point_n be_v at_o liberty_n and_o that_o part_n of_o the_o scale_n which_o reach_v from_o the_o centre_n to_o either_o of_o the_o weight_n be_v call_v the_o beam_n 2_o equiponderation_n be_v when_o the_o endeavour_n of_o one_o body_n which_o press_v one_o of_o the_o beam_n resist_v the_o endeavour_n of_o another_o body_n press_v the_o other_o beam_n so_o that_o neither_o of_o they_o be_v move_v and_o the_o body_n when_o neither_o of_o they_o be_v move_v be_v say_v to_o be_v equal_o poise_v 3_o weight_n be_v the_o aggregate_v of_o all_o the_o endeavour_n by_o which_o all_o the_o point_n of_o that_o body_n which_o press_v the_o beam_n tend_v downward_o in_o line_n parallel_v to_o one_o another_o and_o the_o body_n which_o press_v be_v call_v the_o ponderant_fw-la 4_o moment_n be_v the_o power_n which_o the_o ponderant_fw-la have_v to_o move_v the_o beam_n by_o reason_n of_o a_o determine_a situation_n 5_o the_o plain_a of_o equiponderation_n be_v that_o by_o which_o the_o ponderant_fw-la be_v so_o divide_v that_o the_o moment_n on_o both_o side_n remain_v equal_a 6_o the_o diameter_n of_o equiponderation_n be_v the_o common_a section_n of_o the_o two_o plain_n of_o equiponderation_n and_o be_v in_o the_o straight_a line_n by_o which_o the_o weight_n be_v hang_v 7_o the_o centre_n of_o equiponderation_n be_v the_o common_a point_n of_o the_o two_o diameter_n of_o equiponderation_n supposition_n 1_o when_o two_o body_n be_v equal_o poise_v if_o weight_n be_v add_v to_o one_o of_o they_o and_o not_o to_o the_o other_o their_o equiponderation_n cease_v 2_o when_o two_o ponderant_n of_o equal_a magnitude_n and_o of_o the_o same_o species_n or_o matter_n press_v the_o beam_n on_o both_o side_n at_o equal_a distance_n from_o the_o centre_n of_o the_o scale_n their_o moment_n be_v equal_a also_o when_o two_o body_n endeavour_v at_o equal_a distance_n from_o the_o centre_n of_o the_o scale_n if_o they_o be_v of_o equal_a magnitude_n and_o of_o the_o same_o species_n their_o moment_n be_v equal_a 2_o no_o two_o plain_n of_o equiponderation_n be_v parallel_n let_v a_o b_o c_o d_o in_o the_o first_o figure_n be_v any_o ponderant_fw-la whatsoever_o and_o in_o it_o let_v e_o f_o be_v a_o plain_a of_o equiponderation_n parallel_n to_o which_o let_v any_o other_o plain_o be_v draw_v as_o g_z h._n i_o say_v g_o h_n be_v not_o a_o plain_a of_o equiponderation_n for_o see_v the_o part_n a_o e_o f_o d_o and_o e_o b_o c_o f_o of_o the_o ponderant_fw-la a_o b_o c_o d_o be_v equal_o poise_v and_o the_o weight_n e_o g_o h_o f_o be_v add_v to_o the_o part_n a_o e_o f_o d_o and_o nothing_o be_v add_v to_o the_o part_n e_o b_o c_o f_o but_o the_o weight_n e_o g_o h_o f_o be_v take_v from_o it_o therefore_o by_o the_o first_o supposition_n the_o part_n a_o g_o h_o d_o and_o g_o b_o c_o h_o will_v not_o be_v equal_o poise_v and_o consequent_o g_z h_n be_v not_o a_o plain_a of_o equiponderation_n wherefore_o no_o two_o plain_n of_o equiponderation_n etc._n etc._n which_o be_v to_o be_v prove_v 3_o the_o centre_n of_o equiponderation_n be_v in_o every_o plain_n of_o equiponderation_n for_o if_o another_o plain_n of_o equiponderation_n be_v take_v it_o will_v not_o by_o the_o last_o article_n be_v parallel_n to_o the_o former_a plain_n and_o therefore_o both_o those_o plain_n will_v
together_o take_v so_o that_o the_o part_n next_o the_o vertex_fw-la be_v triple_a to_o the_o other_o part_n or_o to_o the_o whole_a straight_a line_n as_o 3_o to_o 4._o for_o let_v a_o b_o c_o in_o the_o 9th_o fig._n be_v the_o sector_n of_o a_o sphere_n who_o vertex_fw-la be_v the_o centre_n of_o the_o sphere_n a_o who_o axis_n be_v a_o d_o and_o the_o circle_n upon_o b_o c_z be_v the_o common_a base_a of_o the_o portion_n of_o the_o sphere_n and_o of_o the_o cone_n who_o vertex_fw-la be_v a_o the_o axis_n of_o which_o portion_n be_v e_o d_o and_o the_o half_a thereof_o f_o d_o and_o the_o axis_n of_o the_o cone_n a_o e._n last_o let_v a_o g_o be_v ¾_n of_o the_o straight_a line_n a_o f._n i_o say_v g_o be_v the_o centre_n of_o equiponderation_n of_o the_o sector_n a_o b_o c._n let_v the_o straight_a line_n f_o h_n be_v draw_v of_o any_o length_n make_v right_a angle_n with_o a_o f_o at_o f_o and_o draw_v the_o straight_a line_n a_o h_n let_v the_o triangle_n a_o f_o h_n be_v make_v then_o upon_o the_o same_o centre_n a_o let_v any_o arch_n i_o k_o be_v draw_v cut_v a_o d_o in_o l_o and_o its_o chord_n cut_v a_o d_o in_o m_n and_o divide_v m_n l_o equal_o in_o n_o let_v n_o o_o be_v draw_v parallel_n to_o the_o straight_a line_n f_o h_n and_o meet_v with_o the_o straight_a line_n a_o h_n in_o o._n see_v now_o b_o d_o c_z be_v the_o spherical_a superficies_n of_o the_o portion_n cut_v off_o with_o a_o plain_a pass_v through_o b_o c_z and_o cut_v the_o axis_n at_o right_a angle_n and_o see_v f_o h_n divide_v e_o d_o the_o axis_n of_o the_o portion_n into_o two_o equal_a part_n in_o f_o the_o centre_n of_o equiponderation_n of_o the_o superficies_n b_o d_o c_o will_n be_v in_o f_o by_o the_o 8_o article_n and_o for_o the_o same_o reason_n the_o centre_n of_o equiponderation_n of_o the_o superficies_n i_o l_o k_n k_o be_v in_o the_o straight_a line_n a_o c_o will_v be_v in_o n._n and_o in_o like_a manner_n if_o there_o be_v draw_v between_o the_o centre_n of_o the_o sphere_n a_o and_o the_o outermost_a spherical_a superficies_n of_o the_o sector_n arch_n infinite_a in_o number_n the_o centre_n of_o equiponderation_n of_o the_o spherical_a superficies_n in_o which_o those_o arch_n be_v will_v be_v find_v to_o be_v in_o that_o part_n of_o the_o axis_n which_o be_v intercept_v between_o the_o superficies_n itself_o and_o a_o plain_a pass_v along_o by_o the_o chord_n of_o the_o arch_n and_o cut_v the_o axis_n in_o the_o middle_n at_o right_a angle_n let_v it_o now_o be_v suppose_v that_o the_o moment_n of_o the_o outermost_a spherical_a superficies_n b_o d_o c_o be_v f_o h._n see_v therefore_o the_o superficies_n b_o d_o c_o be_v to_o the_o superficies_n i_o l_o k_o in_o proportion_n duplicate_v to_o that_o of_o the_o arch_n b_o d_o c_o to_o the_o arch_n i_o l_o k_n that_o be_v of_o b_o e_z to_z ay_o m_o that_o be_v of_o f_o h_n to_o n_o o_o let_v it_o be_v as_o f_o h_n to_o n_o o_o so_o n_o o_o to_o another_o n_o p_o and_o again_o as_o n_o o_o to_o n_o p_o so_o n_o p_o to_o another_o n_o q_o and_o let_v this_o be_v do_v in_o all_o the_o straight_a line_n parallel_v to_o the_o base_a f_o h_n that_o can_v possible_o be_v draw_v between_o the_o base_a and_o the_o vertex_fw-la of_o the_o triangle_n a_o f_o h._n if_o then_o through_o all_o the_o point_n q_o there_o be_v draw_v the_o crooked_a line_n a_o q_o h_n the_o figure_n a_o f_o h_o q_o a_o will_v be_v the_o compliment_n of_o the_o first_o three-sided_n figure_n of_o two_o mean_n and_o the_o same_o will_v also_o be_v the_o moment_n of_o all_o the_o spherical_a superficies_n of_o which_o the_o solid_a sector_n a_o b_o c_o d_o be_v compound_v and_o by_o consequent_a the_o moment_n of_o the_o sector_n itself_o let_v now_o f_o h_n be_v understand_v to_o be_v the_o semidiameter_n of_o the_o base_a of_o a_o right_a cone_n who_o side_n be_v a_o h_n and_o axis_n a_o f._n wherefore_o see_v the_o base_n of_o the_o cones_n which_o pass_v through_o f_o and_o n_o and_o the_o rest_n of_o the_o point_n of_o the_o axis_n be_v in_o proportion_n duplicate_v to_o that_o of_o the_o straight_a line_n f_o h_n and_o n_o o_o etc._n etc._n the_o moment_n of_o all_o the_o base_n together_o that_o be_v of_o the_o whole_a cone_n will_v be_v the_o figure_n itself_o a_o f_o h_o q_o a_o and_o therefore_o the_o centre_n of_o equiponderation_n of_o the_o cone_n a_o f_o h_n be_v the_o same_o with_o that_o of_o the_o solid_a sector_n wherefore_o see_v a_o g_z be_v ¾_n of_o the_o axis_n a_o f_o the_o centre_n of_o equiponderation_n of_o the_o cone_n a_o f_o h_n be_v in_o g_z and_o therefore_o the_o centre_n of_o the_o solid_a sector_n be_v in_o g_o also_o and_o divide_v the_o part_n a_o f_o of_o the_o axis_n so_o that_o a_o g_o be_v triple_a to_o g_o f_o that_o be_v a_o g_z be_v to_o a_o f_o as_o 3_o to_o 4_o which_o be_v to_o be_v demonstrate_v note_v that_o when_o the_o sector_n be_v a_o hemisphere_n the_o axis_n of_o the_o cone_n vanish_v into_o that_o point_n which_o be_v the_o centre_n of_o the_o sphere_n and_o therefore_o it_o add_v nothing_o to_o half_a the_o axis_n of_o the_o portion_n wherefore_o if_o in_o the_o axis_n of_o the_o hemisphere_n there_o be_v take_v from_o the_o centre_n ¾_n of_o half_a the_o axis_n that_o be_v 3_o â—_o of_o the_o semidiameter_n of_o the_o sphere_n there_o will_v be_v the_o centre_n of_o equiponderation_n of_o the_o hemisphere_n chap._n xxiv_o of_o refraction_n and_o reflection_n 1_o definition_n 2_o in_o perpendicular_a motion_n there_o be_v no_o refraction_n 3_o thing_n throw_v out_o of_o a_o thin_a into_o a_o thick_a medium_fw-la be_v so_o refract_v that_o the_o angle_n refract_v be_v great_a than_o the_o angle_n of_o inclination_n 4_o endeavour_v which_o from_o one_o point_n tend_v every_o way_n will_v be_v so_o refract_v at_o that_o the_o sine_fw-la of_o the_o angle_n refract_v will_v be_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n as_o the_o density_n of_o the_o first_o medium_fw-la be_v to_o the_o density_n of_o the_o second_o medium_fw-la reciprocal_o take_v 5_o the_o sine_z of_o the_o refracted_a angle_n in_o one_o inclination_n be_v to_o the_o sine_fw-la of_o the_o refracted_a angle_n in_o another_o inclination_n as_o the_o sine_fw-la of_o the_o angle_n of_o that_o inclination_n be_v to_o the_o sine_fw-la of_o the_o angle_n of_o this_o inclination_n 6_o if_o two_o line_n of_o incidence_n have_v equal_a inclination_n be_v the_o one_o in_o a_o thin_a the_o other_o in_o a_o thick_a medium_fw-la the_o sine_fw-la of_o the_o angle_n of_o inclination_n will_v be_v a_o mean_a proportional_a between_o the_o two_o sin_n of_o the_o refracted_a angle_n 7_o if_o the_o angle_n of_o inclination_n be_v semirect_v and_o the_o line_n of_o inclination_n be_v in_o the_o thick_a medium_fw-la and_o the_o proportion_n of_o their_o density_n be_v the_o same_o with_o that_o of_o the_o diagonal_a to_o the_o side_n of_o a_o square_a and_o the_o separate_a superficies_n be_v plain_a the_o refracted_a line_n will_v be_v in_o the_o separate_a superficies_n 8_o if_o a_o body_n be_v carry_v in_o a_o straight_a line_n upon_o another_o body_n and_o do_v not_o penetrate_v the_o same_o but_o be_v reflect_v from_o it_o the_o angle_n of_o reflection_n will_v be_v equal_a to_o the_o angle_n of_o incidence_n 9_o the_o same_o happen_v in_o the_o generation_n of_o motion_n in_o the_o line_n of_o incidence_n 1_o definition_n 1_o refraction_n be_v the_o break_n of_o that_o straight_a line_n in_o which_o a_o body_n be_v move_v or_o its_o action_n will_v proceed_v in_o one_o and_o the_o same_o medium_fw-la into_o two_o straight_a line_n by_o reason_n of_o the_o different_a nature_n of_o the_o two_o medium_n 2_o the_o former_a of_o these_o be_v call_v the_o line_n of_o incidence_n the_o late_a the_o refract_v line_n 3_o the_o point_n of_o refraction_n be_v the_o common_a point_n of_o the_o line_n of_o incidence_n and_o of_o the_o refracted_a line_n 4_o the_o refract_a superficies_n which_o also_o be_v the_o separate_a superficies_n of_o the_o two_o medium_n be_v that_o in_o which_o be_v the_o point_n of_o refraction_n 5_o the_o angle_n refract_v be_v that_o which_o the_o refracted_a line_n make_v in_o the_o point_n of_o refraction_n with_o that_o line_n which_o from_o the_o same_o point_n be_v draw_v perpendicular_a to_o the_o separate_a superficies_n in_o a_o different_a medium_fw-la 6_o the_o angle_n of_o refraction_n be_v that_o which_o the_o refracted_a line_n make_v with_o the_o line_n of_o incidence_n produce_v 7_o the_o angle_n of_o inclination_n be_v
and_o the_o property_n of_o straight_a parallel_n 13_o the_o circumference_n of_o circle_n be_v to_o one_o another_o as_o their_o diameter_n be_v 14_o in_o triangle_n straight_a line_n parallel_v to_o the_o base_n be_v to_o one_o another_o as_o the_o part_n of_o the_o side_n which_o they_o cut_v off_o from_o the_o vertex_fw-la 15_o by_o what_o fraction_n of_o a_o straight_a line_n the_o circumference_n of_o a_o circle_n be_v make_v 16_o that_o a_o angle_n of_o contingence_n be_v quantity_n but_o of_o a_o different_a kind_n from_o that_o of_o a_o angle_n simple_o so_o call_v and_o that_o it_o can_v neither_o add_v nor_o take_v away_o any_o thing_n from_o the_o same_o 17_o that_o the_o inclination_n of_o plain_n be_v angle_v simple_o so_o call_v 18_o a_o solid_a angle_n what_o it_o be_v 19_o what_o be_v the_o nature_n of_o asymptotes_n 20_o situation_n by_o what_o it_o be_v determine_v 21_o what_o be_v like_o situation_n what_o be_v figure_n and_o what_o be_v like_o figure_n 1_o 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understand_v to_o be_v draw_v further_o asunder_o and_o a_o straight_a line_n to_o be_v that_o who_o extreme_a point_n can_v be_v draw_v further_o asunder_o and_o comparative_o a_o more_o crooked_a to_o be_v that_o line_n who_o extreme_a point_n be_v near_o to_o one_o another_o then_o those_o of_o the_o other_o suppose_v both_o the_o line_n to_o be_v of_o equal_a length_n now_o howsoever_o a_o line_n be_v bow_v it_o make_v always_o a_o sinus_n or_o cavity_n sometime_o on_o one_o side_n sometime_o on_o another_o so_o that_o the_o same_o crooked_a line_n may_v either_o have_v its_o whole_a cavity_n on_o one_o side_n only_o or_o it_o may_v have_v it_o part_n on_o one_o side_n and_o part_n on_o other_o side_n which_o be_v well_o understand_v it_o will_v be_v easy_a to_o understand_v the_o follow_a comparison_n of_o straight_a and_o crooked_a line_n first_o if_o a_o straight_a &_o a_o crooked_a line_n have_v their_o extreme_a point_n common_a the_o crooked_a line_n be_v long_a than_o the_o 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the_o least_o of_o all_o those_o that_o have_v the_o same_o terminate_a line_n and_o also_o every_o part_n of_o the_o same_o superficies_n be_v the_o least_o of_o all_o those_o that_o be_v terminate_v with_o the_o same_o line_n if_o every_o part_n shall_v not_o constitute_v a_o plain_a superficies_n all_o the_o part_n put_v together_o will_v not_o be_v equal_a to_o the_o whole_a 3_o of_o straightness_n whether_o it_o be_v in_o line_n or_o in_o superficies_n there_o be_v but_o one_o kind_n but_o of_o crookedness_n there_o be_v many_o kind_n for_o of_o crooked_a magnitude_n some_o be_v congruous_a that_o be_v be_v coincident_a when_o they_o be_v apply_v to_o one_o another_o other_o be_v incongruous_a again_o some_o be_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d or_o uniform_a that_o be_v have_v their_o part_n howsoever_o take_v congruous_a to_o one_o
same_o chapter_n i_o have_v show_v that_o whatsoever_o be_v at_o rest_n will_v always_o be_v at_o rest_n unless_o there_o be_v some_o other_o body_n beside_o it_o which_o by_o get_v into_o its_o place_n suffer_v it_o no_o long_o to_o remain_v at_o rest._n and_o that_o whatsoever_o be_v move_v will_v always_o be_v move_v unless_o there_o be_v some_o other_o body_n beside_o it_o which_o hinder_v its_o motion_n ten_o in_o the_o 9_o chapter_n and_o 7_o article_n i_o have_v demonstrate_v that_o when_o any_o body_n be_v move_v which_o be_v former_o at_o rest_n the_o immediate_a efficient_a cause_n of_o that_o motion_n be_v in_o some_o other_o move_v and_o contiguous_a body_n eleven_o i_o have_v show_v in_o the_o same_o place_n that_o whatsoever_o be_v move_v will_v always_o be_v move_v in_o the_o same_o way_n and_o with_o the_o same_o swiftness_n if_o it_o be_v not_o hinder_v by_o some_o other_o move_v and_o contiguous_a body_n 2_o to_o which_o principle_n i_o shall_v here_o add_v these_o that_o follow_v first_o i_o define_v endeavour_n to_o be_v motion_n make_v in_o less_o space_n and_o time_n then_o can_v be_v give_v that_o be_v less_o than_o can_v be_v determine_v or_o assign_v by_o exposition_n or_o number_n that_o be_v motion_n make_v through_o the_o length_n of_o a_o point_n and_o in_o a_o instant_a or_o point_n of_o time_n for_o the_o explayn_v of_o which_o definition_n it_o must_v be_v remember_v that_o by_o a_o point_n be_v not_o to_o be_v understand_v that_o which_o have_v no_o quantity_n or_o which_o can_v by_o any_o mean_n be_v divide_v for_o there_o be_v no_o such_o thing_n in_o nature_n but_o that_o who_o quantity_n be_v not_o at_o all_o consider_v that_o be_v whereof_o neither_o quantity_n nor_o any_o part_n be_v compute_v in_o demonstration_n so_o that_o a_o point_n be_v not_o to_o be_v take_v for_o a_o indivisible_a but_o for_o a_o undivided_a thing_n as_o also_o a_o instant_a be_v to_o be_v take_v for_o a_o undivided_a and_o not_o for_o a_o indivisible_a time_n in_o like_a manner_n endeavour_n be_v to_o be_v conceive_v as_o motion_n but_o so_o as_o that_o neither_o the_o quantity_n of_o the_o time_n in_o which_o nor_o of_o the_o line_n in_o which_o it_o be_v make_v may_v in_o demonstration_n be_v at_o all_o bring_v into_o comparison_n with_o the_o quantity_n of_o that_o time_n or_o of_o that_o line_n of_o which_o it_o be_v a_o part_n and_o yet_o as_o a_o point_n may_v be_v compare_v with_o a_o point_n so_o one_o endeavour_n may_v be_v compare_v with_o another_o endeavour_n and_o one_o may_v be_v find_v to_o be_v great_a or_o less_o than_o another_o for_o if_o the_o vertical_a point_n of_o two_o angle_n be_v compare_v they_o will_v be_v equal_a or_o unequal_a in_o the_o same_o proportion_n which_o the_o angle_n themselves_o have_v to_o one_o another_o or_o if_o a_o straight_a line_n cut_v many_o circumference_n of_o concentric_a circle_n the_o inequality_n of_o the_o point_n of_o intersection_n will_v be_v in_o the_o same_o proportion_n which_o the_o perimeter_n have_v to_o one_o another_o and_o in_o the_o same_o manner_n if_o two_o motion_n begin_v and_o end_n both_o together_o their_o endeavour_n will_v be_v equal_a or_o unequal_a according_a to_o the_o proportion_n of_o their_o velocity_n as_o we_o see_v a_o bullet_n of_o lead_n descend_v with_o great_a endeavour_n than_o a_o ball_n of_o wool_n second_o i_o define_v impetus_fw-la or_o quickness_n of_o motion_n to_o be_v the_o swiftness_n or_o velocity_n of_o the_o body_n move_v but_o consider_v in_o the_o several_a point_n of_o that_o time_n in_o which_o it_o be_v move_v in_o which_o sense_v impetus_fw-la be_v nothing_o else_o but_o the_o quantity_n or_o velocity_n of_o endeavour_n but_o consider_v with_o the_o whole_a time_n it_o be_v the_o whole_a velocity_n of_o the_o body_n move_v take_v together_o throughout_o all_o the_o time_n and_o equal_a to_o the_o product_n of_o a_o line_n represent_v the_o time_n multiply_v into_o a_o line_n represent_v the_o arithmetical_o mean_v impetus_fw-la or_o quickness_n which_o arithmetical_a mean_n what_o it_o be_v be_v define_v in_o the_o 29_o article_n of_o the_o 13_o chapter_n and_o because_o in_o equal_a time_n the_o way_n that_o be_v pass_v be_v as_o the_o velocity_n and_o the_o impetus_fw-la be_v the_o velocity_n they_o go_v withal_o reckon_v in_o all_o the_o several_a point_n of_o the_o time_n it_o follow_v that_o during_o any_o time_n whatsoever_o howsoever_o the_o impetus_fw-la be_v increase_v or_o decrease_v the_o length_n of_o the_o way_n pass_v over_o shall_v be_v increase_v or_o decrease_v in_o the_o same_o proportion_n and_o the_o same_o line_n shall_v represent_v both_o the_o way_n of_o the_o body_n move_v and_o the_o several_a impetus_fw-la or_o degree_n of_o swiftness_n wherewith_o the_o way_n be_v pass_v over_o and_o if_o the_o body_n move_v be_v not_o a_o point_n but_o a_o straight_a line_n move_v so_o as_o that_o every_o point_n thereof_o make_v a_o several_a straight_a line_n the_o plain_a describe_v by_o its_o motion_n whether_o uniform_a accelerate_a or_o retard_v shall_v be_v great_a or_o less_o the_o time_n be_v the_o same_o in_o the_o same_o proportion_n with_o that_o of_o the_o impetus_fw-la reckon_v in_o one_o motion_n to_o the_o impetus_fw-la reckon_v in_o the_o other_o for_o the_o reason_n be_v the_o same_o in_o parallellogram_n and_o their_o side_n for_o the_o same_o cause_n also_o if_o the_o body_n move_v be_v a_o plain_a the_o solid_a describe_v shall_v be_v still_o great_a or_o less_o in_o the_o proportion_n of_o the_o several_a impetus_fw-la or_o quickness_n reckon_v through_o one_o line_n to_o the_o several_a impetus_fw-la reckon_v through_o another_o this_o understand_v let_v abcd_v in_o the_o first_o figure_n of_o the_o 17_o chapter_n be_v a_o parallelogram_n in_o which_o suppose_v the_o side_n ab_fw-la to_o be_v move_v parallel_o to_o the_o opposite_a side_n cd_o decrease_v all_o the_o way_n till_o it_o vanish_v in_o the_o point_n c_o and_o so_o describe_v the_o figure_n abefc_n the_o point_n b_o as_o ab_fw-la decrease_v will_v therefore_o describe_v the_o line_n befc_n and_o suppose_v the_o time_n of_o this_o motion_n design_v by_o the_o line_n cd_o and_o in_o the_o same_o time_n cd_o suppose_v the_o side_n ac_fw-la to_o be_v move_v parallelly_n and_o uniform_o to_o bd._n from_o the_o point_n o_o take_v at_o adventure_n in_o the_o line_n cd_o draw_v or_o parallel_v to_o bd_o cut_v the_o line_n befc_n in_o e_o and_o the_o side_n ab_fw-la in_o r._n and_o again_o from_o the_o point_n qui_fw-fr take_v also_o at_o adventure_n in_o the_o line_n cd_o draw_v q_n parallel_v to_o bd_o cut_v the_o line_n befc_n in_o f_o and_o the_o side_n ab_fw-la in_o saint_n and_o draw_v eglantine_n and_o fh_n parallel_n to_o cd_o cut_v ac_fw-la in_o g_z and_o h._n last_o suppose_v the_o same_o construction_n do_v in_o all_o the_o point_n possible_a of_o the_o line_n befc_n i_o saâ—_n that_o as_o the_o proportion_n of_o the_o swiftness_n wherewith_o qf_n oe_o db_n and_o all_o the_o rest_n supopse_v to_o be_v draw_v parallel_n to_o db_n and_o terminate_v in_o the_o line_n befc_n be_v to_o the_o proportion_n of_o their_o several_a time_n design_v by_o the_o several_a parallel_n hf_n ge_n ab_fw-la and_o all_o the_o rest_n suppose_v to_o be_v draw_v parallel_n to_o the_o line_n of_o time_n cd_o and_o terminate_v in_o the_o line_n befc_n the_o aggregate_v to_o the_o aggregate_v so_o be_v the_o area_n or_o plain_a dbefc_n to_o the_o area_n or_o plain_a acfeb_n for_o as_o ab_fw-la decrease_v continual_o by_o the_o line_n befc_n vanish_v in_o the_o time_n cd_o into_o the_o point_n c_o so_o in_o the_o same_o time_n the_o line_n dc_o continual_o decrease_v vanish_v by_o the_o same_o line_n cfeb_n into_o the_o point_n b_o and_o the_o point_n d_o describe_v in_o that_o decrease_a motion_n the_o line_n db_n equal_a to_o the_o line_n ac_fw-la describe_v by_o the_o point_n a_o in_o the_o decrease_a motion_n of_o a_o &_o b_o &_o their_o swiftness_n be_v therefore_o equal_a again_o because_o in_o the_o time_n ge_n the_o point_n o_o describe_v the_o line_n oe_v and_o in_o the_o same_o time_n the_o point_n r_o describe_v the_o line_n re_fw-mi the_o line_n oe_n shall_v be_v to_o the_o line_n re_fw-mi as_o the_o swiftness_n wherewith_o oe_n be_v describe_v to_o the_o swiftness_n wherewith_o re_n be_v describe_v in_o like_a manner_n because_o in_o the_o same_o time_n hf_n the_o point_n qui_fw-fr describe_v the_o line_n qf_n and_o the_o point_n s_o the_o line_n sf_n it_o shall_v be_v as_o the_o swiftness_n by_o which_o qf_n be_v describe_v to_o the_o swiftness_n by_o which_o sf_n be_v describe_v so_o the_o line_n itself_o qf_o to_o the_o line_n itself_o sf_n and_o so_o in_o all_o the_o line_n that_o can_v possible_o be_v
every_o where_o triplicate_v of_o the_o proportion_n of_o ab_fw-la to_o ge_n and_o of_o ab_fw-la to_o hf_n etc._n etc._n the_o proportion_n of_o hf_n to_o ab_fw-la and_o of_o ge_n to_o ab_fw-la etc._n etc._n by_o the_o 16_o article_n of_o the_o 13_o chap._n be_v triplicate_v of_o the_o proportion_n of_o qf_a to_o db_v and_o of_o oe_n to_o db_fw-la etc._n etc._n and_o therefore_o the_o deficient_a figure_n abefc_n which_o be_v the_o aggregate_v of_o all_o the_o line_n hf_n ge_n ab_fw-la etc._n etc._n be_v triple_a to_o the_o compliment_n befcd_v make_v of_o all_o the_o line_n qf_n oe_o db_n etc._n etc._n which_o be_v to_o be_v prove_v it_o follow_v from_o hence_o that_o the_o same_o compliment_n befcd_n be_v ¼_n of_o the_o whole_a parallelogram_n and_o by_o the_o same_o method_n may_v be_v calculate_v in_o all_o other_o deficient_a figure_n generate_v as_o above_o declare_v the_o proportion_n of_o the_o parallelogram_n to_o either_o of_o its_o part_n as_o that_o when_o the_o parallel_n increase_v from_o a_o point_n in_o the_o same_o proportion_n the_o parallelogram_n will_v be_v divide_v into_o two_o equal_a triangle_n when_o one_o increase_n be_v double_a to_o the_o other_o it_o will_v be_v divide_v into_o a_o semiparabola_fw-la and_o its_o compliment_n or_o into_o 2_o and_o 1._o the_o same_o construction_n stand_v the_o same_o conclusion_n may_v otherwise_o be_v demonstrate_v thus_o let_v the_o straight_a line_n cb_n be_v draw_v cut_v gk_n in_o l_o &_o through_o l_o let_v mn_n be_v draw_v parallel_n to_o the_o straight_a line_n ac_fw-la wherefore_o the_o parallellogram_n gm_n and_o ld_n will_v be_v equal_a then_o let_v lk_n be_v divide_v into_o three_o equal_a part_n so_o that_o it_o may_v be_v to_o one_o of_o those_o part_n in_o the_o same_o proportion_n which_o the_o proportion_n of_o ac_fw-la to_o gc_fw-la or_o of_o gk_n to_o gl_n have_v to_o the_o proportion_n of_o gk_n to_o ge._n therefore_o lk_n will_v be_v to_o one_o of_o those_o three_o part_n as_o the_o arithmetical_a proportion_n between_o gk_n and_o gl_n be_v to_o the_o arithmetical_a proportion_n between_o gk_n and_o the_o same_o gk_n want_v the_o three_o part_n of_o lk_n and_o ke_n will_v be_v somewhat_o great_a than_o a_o three_o of_o lk_n see_v now_o the_o altitude_n agnostus_n or_o ml_o be_v by_o reason_n of_o the_o continual_a decrease_n to_o be_v suppose_v less_o than_o any_o quantity_n that_o can_v be_v give_v lk_v which_o be_v intercept_v between_o the_o diagonal_a bc_n and_o the_o side_n bd_o will_v be_v also_o less_o than_o any_o quantity_n that_o can_v be_v give_v and_o consequent_o if_o g_z be_v put_v so_o near_o to_o a_o in_o g_o as_o that_o the_o difference_n between_o cg_n and_o ca_n be_v less_o than_o any_o quantity_n that_o can_v be_v assign_v the_o difference_n also_o between_o cl_n remove_v l_o to_o l_o and_o cb_n will_v be_v less_o than_o any_o quantity_n that_o can_v be_v assign_v and_o the_o line_n gl_o be_v draw_v &_o produce_v to_o the_o line_n bd_o in_o k_o cut_v the_o crooked_a line_n in_o e_o the_o proportion_n of_o gk_n to_o gl_n will_v still_o be_v triplicate_v to_o the_o proportion_n of_o gk_n to_o ge_n and_o the_o difference_n between_o k_o and_o e_o the_o three_o part_n of_o kl_n will_v be_v less_o than_o any_o quantity_n that_o can_v be_v give_v and_o therefore_o the_o parallelogram_n ed_z will_v differ_v from_o a_o three_o part_n of_o the_o parallelogram_n ae_n by_o a_o less_o difference_n than_o any_o quantity_n that_o can_v be_v assign_v again_o let_v he_o be_v draw_v parallel_n and_o equal_a to_o ge_fw-mi cut_v cb_n in_o p_o the_o crooked_a line_n in_o f_o and_o bd_o in_o ay_o and_o the_o proportion_n of_o cg_n to_o ch_z will_v be_v triplicate_v to_o the_o proportion_n of_o hf_n to_o hp_n and_o if_o will_v be_v great_a than_o the_o three_o part_n of_o pi._n but_o again_o set_v h_n in_o h_z so_o near_o to_o g_o as_o that_o the_o difference_n between_o ch_z and_o cg_n may_v be_v but_o as_o a_o point_n the_o point_n p_o will_v also_o in_o p_o be_v so_o near_o to_o l_o as_o that_o the_o difference_n between_o cp_n and_o cl_n will_v be_v but_o as_o a_o point_n and_o draw_v hp_n till_o it_o meet_v with_o gk_n in_o i_o cut_v the_o crooked_a line_n in_o f_o and_o have_v draw_v eo_fw-la parallel_n to_o bd_o cut_v dc_o in_o o_o the_o parallelogram_n foe_fw-mi will_v differ_v less_o from_o the_o three_o part_n of_o the_o parallelogram_n gf_n then_o by_o any_o quantity_n that_o can_v be_v give_v and_o so_o it_o will_v be_v in_o all_o other_o space_n generate_v in_o the_o same_o manner_n wherefore_o the_o difference_n of_o the_o arithmetical_a and_o geometrical_a mean_n which_o be_v but_o as_o so_o many_o point_n b_o e_o f_o etc._n etc._n see_v the_o whole_a figure_n be_v make_v up_o of_o so_o many_o indivisible_a space_n will_v constitute_v a_o certain_a line_n such_o as_o be_v the_o line_n befc_n which_o will_v divide_v the_o complete_a figure_n ad_fw-la into_o two_o part_n whereof_o one_o namely_o abefc_a which_o i_o call_v a_o deficient_a figure_n be_v triple_a to_o the_o other_o namely_o bdcef_a which_o i_o call_v the_o compliment_n thereof_o and_o whereas_o the_o proportion_n of_o the_o altitude_n to_o one_o another_o be_v in_o this_o case_n everywhere_o triplicate_v to_o that_o of_o the_o decrease_a quantity_n to_o one_o another_o in_o the_o same_o manner_n if_o the_o proportion_n of_o the_o altitude_n have_v be_v every_o where_o quadruplicate_v to_o that_o of_o the_o decrease_a quantity_n it_o may_v have_v be_v demonstrate_v that_o the_o deficient_a figure_n have_v be_v quadruple_a to_o its_o compliment_n and_o so_o in_o any_o other_o proportion_n wherefore_o a_o deficient_a figure_n which_o be_v make_v etc._n etc._n which_o be_v to_o be_v demonstrate_v the_o same_o rule_n fold_v also_o in_o the_o diminution_n of_o the_o base_n of_o cylinder_n as_o be_v demonstrate_v chap._n 15._o art_n 2._o â—_o by_o this_o proposition_n the_o magnitude_n of_o all_o deficient_a figure_n when_o the_o proportion_n by_o which_o their_o base_n decrease_v continual_o be_v proportional_a to_o those_o by_o which_o their_o altitude_n decrease_v may_v be_v compare_v with_o the_o magnitude_n of_o their_o compliment_n and_o consequent_o with_o the_o magnitude_n of_o their_o complete_a figure_n and_o they_o will_v be_v find_v to_o be_v as_o i_o have_v set_v they_o down_o in_o the_o follow_a table_n in_o which_o i_o compare_v a_o parallelogram_n with_o threesided_n figure_n and_o first_o with_o a_o straight_o line_v triangle_n make_v by_o the_o base_a of_o the_o parallelogram_n continual_o decrease_v in_o such_o manner_n that_o the_o altitude_n be_v always_o in_o proportion_n to_o one_o another_o as_o the_o base_n be_v and_o so_o the_o triangle_n will_v be_v equal_a to_o its_o compliment_n or_o the_o proportion_n of_o the_o altitude_n and_o base_n will_v be_v as_o 1_o to_o 1_o and_o then_o the_o triangle_n will_v be_v half_a the_o parallelogram_n second_o with_o that_o threesided_n figure_n which_o be_v make_v by_o the_o continual_a decrease_a of_o the_o base_n in_o subduplicate_v proportion_n to_o that_o of_o the_o altitude_n and_o so_o the_o deficient_a figure_n will_v be_v double_a to_o its_o compliment_n and_o to_o the_o parallelogram_n as_o 2_o to_o 3._o then_o with_o that_o where_o the_o proportion_n of_o the_o altitude_n be_v triplicate_v to_o that_o of_o the_o base_n and_o then_o the_o deficient_a figure_n will_v be_v triple_a to_o its_o compliment_n and_o to_o the_o parallelogram_n as_o 3_o to_o 4._o also_o the_o proportion_n of_o the_o altitude_n to_o that_o of_o the_o base_n may_v be_v as_o 3_o to_o 2_o and_o then_o the_o deficient_a figure_n will_v be_v to_o its_o compliment_n as_o 3_o to_o 2_o &_o to_o the_o parallelogram_n as_o 3_o to_z 5_o and_o so_o forward_o according_a as_o more_o mean_a proportional_n be_v take_v or_o as_o the_o proportion_n be_v more_o multiply_v as_o may_v be_v see_v in_o the_o follow_a table_n for_o example_n if_o the_o base_n decrease_v so_o that_o the_o proportion_n of_o the_o altitude_n to_o that_o of_o the_o base_n be_v always_o as_o 5_o to_o 2_o and_o it_o be_v demand_v what_o proportion_n the_o figure_n make_v have_v to_o the_o parallelogram_n which_o be_v suppose_v to_o be_v unity_n then_o see_v that_o where_o the_o proportion_n be_v take_v five_o time_n there_o must_v be_v four_o mean_n look_v in_o the_o table_n among_o the_o threesided_n figure_v of_o four_o mean_n and_o see_v the_o proportion_n be_v as_o 5_o to_o 2_o look_v in_o the_o uppermost_a row_n for_o the_o number_n 2_o and_o descend_v in_o the_o 2d_o column_n till_o you_o meet_v with_o that_o threesided_n figure_n you_o will_v find_v 5_o 7_o which_o show_v that_o the_o deficient_a figure_n be_v to_o the_o parallelogram_n as_o 5_o â—_o
a_o f_o and_o a_o f_o be_v draw_v a_o f_o therefore_o will_v be_v the_o sine_fw-la of_o the_o angle_n of_o inclination_n of_o the_o straight_a line_n a_o b_o and_o a_o f_z the_o sine_z of_o the_o angle_n of_o inclination_n of_o the_o straight_a line_n a_o h_z which_o two_o inclination_n be_v by_o construction_n make_v equal_a i_o say_v as_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v b_o c_z and_o b_o c_o be_v to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v b_o d_o and_o b_o d_o so_o be_v the_o sine_fw-la of_o the_o angle_n refract_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n let_v the_o straight_a line_n f_o g_z be_v draw_v parallel_n to_o the_o straight_a line_n a_o b_o meet_v with_o the_o straight_a line_n b_o b_o produce_v in_o g._n see_v therefore_o a_o f_o and_o b_o g_o be_v also_o parallel_n they_o will_v be_v equal_a and_o consequent_o the_o endeavour_n in_o a_o f_o be_v propagate_v in_o the_o same_o time_n in_o which_o the_o endeavour_n in_o b_o g_o will_v be_v propagate_v if_o the_o medium_fw-la be_v of_o the_o same_o density_n but_o because_o b_o g_z be_v in_o a_o thick_a medium_fw-la that_o be_v in_o a_o medium_fw-la which_o resist_v the_o endeavour_n more_o than_o the_o medium_fw-la in_o which_o a_o f_o be_v the_o endeavour_n will_v be_v propagate_v less_o in_o b_o g_z than_o in_o a_o f_o according_a to_o the_o proportion_n which_o the_o density_n of_o the_o medium_fw-la in_o which_o a_o f_o be_v have_v to_o the_o density_n of_o the_o medium_fw-la in_o which_o b_o g_z be_v let_v therefore_o the_o density_n of_o the_o medium_fw-la in_o which_o b_o g_z be_v be_v to_o the_o density_n of_o the_o medium_fw-la in_o which_o a_o f_o be_v as_o b_o g_z be_v to_z b_o h_n and_o let_v the_o measure_n of_o the_o time_n be_v the_o radius_fw-la of_o the_o circle_n let_v h_n i_o be_v draw_v parallel_n to_o b_o d_o meet_v with_o the_o circumference_n in_o i_o and_o from_o the_o point_n i_o let_v i_o k_o be_v draw_v perpendicular_a to_o b_o d_o which_o be_v do_v b_o h_n and_o i_o k_o will_v be_v equal_a and_o i_o k_o will_v be_v to_o a_o f_o as_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v a_o f_o be_v to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v i_o k._n see_v therefore_o in_o the_o time_n a_o b_o which_o be_v the_o radius_fw-la of_o the_o circle_n the_o endeavour_n be_v propagate_v in_o a_o f_o in_o the_o thin_a medium_fw-la it_o will_v be_v propagate_v in_o the_o same_o time_n that_o be_v in_o the_o time_n b_o i_o in_o the_o thick_a medium_fw-la from_o k_o to_o i._o therefore_o b_o i_o be_v the_o refracted_a line_n of_o the_o line_n of_o incidence_n a_o b_o and_o i_o k_o be_v the_o sine_fw-la of_o the_o angle_n refract_v and_o a_o f_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n wherefore_o see_v i_o k_o be_v to_o a_o f_o as_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v a_o f_o to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v i_o k_o it_o will_v be_v as_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v a_o f_o or_o b_o c_z to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v i_o k_o or_o b_o d_o so_o the_o sine_fw-la of_o the_o angle_n refract_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n and_o by_o the_o same_o reason_n it_o may_v be_v show_v that_o as_o the_o density_n of_o the_o thin_a medium_fw-la be_v to_o the_o density_n of_o the_o thick_a medium_fw-la so_o will_v k_o i_o the_o sine_fw-la of_o the_o angle_n refract_v be_v to_o a_o f_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n second_o let_v the_o body_n which_o endeavour_v every_o way_n be_v in_o the_o thick_a medium_fw-la at_o i._n if_o therefore_o both_o the_o medium_n be_v of_o the_o same_o density_n the_o endeavour_n of_o the_o body_n in_o i_o b_o will_v tend_v direct_o to_o l_o and_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n l_o m_o will_v be_v equal_a to_o i_o k_o or_o b_o h._n but_o because_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v ik_fw-mi to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v l_o m_o be_v as_o bh_n to_o b_o g_z that_o be_v to_o a_o f_o the_o endeavour_n will_v be_v propagate_v further_o in_o the_o medium_fw-la in_o which_o l_o m_o be_v then_o in_o the_o medium_fw-la in_o which_o i_o k_o be_v in_o the_o proportion_n of_o density_n to_o density_n that_o be_v of_o m_o l_o to_o a_o f._n wherefore_o b_o a_o be_v draw_v the_o angle_n refract_v will_v be_v c_o b_o a_o and_o its_z sine_z a_o f._n but_o l_o m_o be_v the_o sine_fw-la of_o the_o angle_n of_o inclination_n and_o therefore_o again_o as_o the_o density_n of_o one_o medium_fw-la be_v to_o the_o density_n of_o the_o different_a medium_fw-la so_o reciprocal_o be_v the_o sine_fw-la of_o the_o angle_n refract_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n which_o be_v to_o be_v demonstrate_v in_o this_o demonstration_n i_o have_v make_v the_o separate_a superficies_n b_o b_o plain_a by_o construction_n but_o though_o it_o be_v concave_a or_o convex_a the_o theorem_a will_v nevertheless_o be_v true_a for_o the_o refraction_n be_v make_v in_o the_o point_n b_o of_o the_o plain_a separate_a superficies_n if_o a_o crooked_a line_n as_o p_o qui_fw-fr be_v draw_v touch_v the_o separate_a line_n in_o the_o point_n b_o neither_o the_o refracted_a line_n b_o i_o nor_o the_o perpendicular_a b_o d_o will_v be_v alter_v and_o the_o refracted_a angle_n k_o b_o i_o as_o also_o its_o sine_fw-la k_o i_o will_v be_v still_o the_o same_o they_o be_v 5_o the_o sine_z of_o the_o angle_n refract_v in_o one_o inclination_n be_v to_o the_o sine_fw-la of_o the_o angle_n refract_v in_o another_o inclination_n as_o the_o sine_fw-la of_o the_o angle_n of_o that_o inclination_n to_o the_o sine_fw-la of_o the_o angle_n of_o this_o inclination_n for_o see_v the_o sine_fw-la of_o the_o refracted_a angle_n be_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n whatsoever_o that_o inclination_n be_v as_o the_o density_n of_o one_o medium_fw-la to_o the_o density_n of_o the_o other_o medium_fw-la the_o proportion_n of_o the_o sine_fw-la of_o the_o refracted_a angle_n to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n will_v be_v compound_v of_o the_o proportion_n of_o density_n to_o density_n and_o of_o the_o sine_fw-la of_o the_o angle_n of_o one_o inclination_n to_o the_o sine_fw-la of_o the_o angle_n of_o the_o other_o inclination_n but_o the_o proportion_n of_o the_o density_n in_o the_o same_o homogeneous_a body_n be_v suppose_v to_o be_v the_o same_o wherefore_o refracted_a angle_n in_o different_a inclination_n be_v as_o the_o sin_n of_o the_o angle_n of_o those_o inclination_n which_o be_v to_o be_v demonstrate_v 6_o if_o two_o line_n of_o incidence_n have_v equal_a inclination_n be_v the_o one_o in_o a_o thin_a the_o other_o in_o a_o thick_a medium_fw-la the_o sine_fw-la of_o the_o angle_n of_o their_o inclination_n will_v be_v a_o mean_a proportional_a between_o the_o two_o sin_n of_o their_o angle_n refract_v for_o let_v the_o straight_a line_n ab_fw-la in_o the_o same_o 3d_o figure_n have_v its_o inclination_n in_o the_o thin_a medium_fw-la and_o be_v refract_v in_o the_o thick_a medium_fw-la in_o b_o i_o and_o let_v e_o b_o have_v as_o much_o inclination_n in_o the_o thick_a medium_fw-la and_o be_v refract_v in_o the_o thin_a medium_fw-la in_o b_o saint_n and_o let_v r_o s_o the_o sine_fw-la of_o the_o angle_n refract_v be_v draw_v i_o say_v the_o straight_a line_n r_o saint_n a_o f_o and_o i_o k_o be_v in_o continual_a proportion_n for_o it_o be_v as_o the_o density_n of_o the_o thick_a medium_fw-la to_o the_o density_n of_o the_o thin_a medium_fw-la so_o r_o s_o to_o a_o f._n but_o it_o be_v also_o as_o the_o density_n of_o the_o same_o thick_a medium_fw-la to_o that_o of_o the_o same_o thin_a medium_fw-la so_o of_o to_o ik_fw-mi wherefore_o r_o s._n a_o f_o a_o f._n i_o k_o be_v propoortional_n that_o be_v r_o s_o a_o f_o and_o i_o k_o be_v in_o continual_a proportion_n and_o a_o f_o be_v the_o mean_a proportional_a which_o be_v to_o be_v prove_v 7_o if_o the_o angle_n of_o inclination_n be_v semirect_v and_o the_o line_n of_o inclination_n be_v in_o the_o thick_a medium_fw-la and_o the_o proportion_n of_o the_o density_n be_v as_o that_o of_o a_o diagonal_a to_o the_o side_n of_o its_o square_n and_o the_o separate_a superficies_n be_v plain_a the_o refracted_a line_n will_v be_v in_o that_o separate_a superficies_n for_o in_o the_o circle_n a_o c_o in_o the_o four_o figure_n let_v the_o angle_n
nor_o can_v they_o go_v direct_o backward_o against_o the_o force_n of_o the_o movent_fw-la it_o remain_v therefore_o that_o they_o diffuse_v themselves_o upon_o the_o superficies_n of_o that_o body_n as_o towards_o o_o and_o p_o which_o be_v to_o be_v prove_v 9_o compound_v circular_a motion_n in_o which_o all_o the_o part_n of_o the_o move_v body_n do_v at_o once_o describe_v circumference_n some_o great_a other_o less_o according_a to_o the_o proportion_n of_o their_o several_a distance_n from_o the_o common_a centre_n carry_v about_o with_o it_o such_o body_n as_o be_v not_o fluid_a adhere_v to_o the_o body_n so_o move_v and_o such_o as_o do_v not_o adhere_v it_o cast_v forward_o in_o a_o straight_a line_n which_o be_v a_o tangent_fw-la to_o the_o point_n from_o which_o they_o be_v cast_v off_o for_o let_v there_o be_v a_o circle_n who_o radius_fw-la be_v ab_fw-la in_o the_o four_o figure_n and_o let_v a_o body_n be_v place_v in_o the_o circumference_n in_o b_o which_o if_o it_o be_v fix_v there_o will_v necessary_o be_v carry_v about_o with_o it_o as_o be_v manifest_a of_o itself_o but_o while_o the_o motion_n proceed_v let_v we_o suppose_v that_o body_n to_o be_v unfixed_a in_o b._n i_o say_v the_o body_n will_v continue_v its_o motion_n in_o the_o tangent_fw-la bc._n for_o let_v both_o the_o radius_fw-la ab_fw-la and_o the_o sphere_n b_o be_v conceive_v to_o consist_v of_o hard_a matter_n and_o let_v we_o suppose_v the_o radius_fw-la ab_fw-la to_o be_v strike_v in_o the_o point_n b_o by_o some_o other_o body_n which_o fall_v upon_o it_o in_o the_o tangent_fw-la db._n now_o therefore_o there_o will_v be_v a_o motion_n make_v by_o the_o concourse_n of_o two_o thing_n the_o one_o endeavour_v towards_o c_o in_o the_o straight_a line_n db_v produce_v in_o which_o the_o body_n b_o will_v proceed_v if_o it_o be_v not_o retain_v by_o the_o radius_fw-la ab_fw-la the_o other_z the_o retention_n itself_o but_o the_o retention_n alone_o cause_v no_o endeavour_n towards_o the_o centre_n and_o therefore_o the_o retention_n be_v take_v away_o which_o be_v do_v by_o the_o unfix_v of_o b_o there_o will_v remain_v but_o one_o endeavour_n in_o b_o namely_o that_o in_o the_o tangent_fw-la bc._n wherefore_o the_o motion_n of_o the_o body_n b_o unfixed_a will_v proceed_v in_o the_o tangent_fw-la bc_n which_o be_v to_o be_v prove_v by_o this_o demonstration_n it_o be_v manifest_a that_o circular_a motion_n about_o a_o unmoved_a axis_n shake_v off_o and_o put_v further_a from_o the_o centre_n of_o its_o motion_n such_o thing_n as_o touch_v but_o do_v not_o stick_v fast_o to_o its_o superficies_n and_o the_o more_o by_o how_o much_o the_o distance_n be_v great_a from_o the_o pole_n of_o the_o circular_a motion_n and_o so_o much_o the_o more_o also_o by_o how_o much_o the_o thing_n that_o be_v shake_v off_o be_v less_o drive_v towards_o the_o centre_n by_o the_o fluid_fw-la ambient_fw-la for_o other_o cause_n 10_o if_o in_o a_o fluid_fw-la medium_fw-la a_o spherical_a body_n be_v move_v with_o simple_a circular_a motion_n and_o in_o the_o same_o medium_fw-la there_o float_v another_o sphere_n who_o matter_n be_v not_o fluid_a this_o sphere_n also_o shall_v be_v move_v with_o simple_a circular_a motion_n let_v bcd_n in_o the_o 5_o figure_n be_v a_o circle_n who_o centre_n be_v a_o and_o in_o who_o circumference_n there_o be_v a_o sphere_n so_o move_v that_o it_o describe_v with_o simple_a motion_n the_o perimeter_n bcd_n let_v also_o efg_o be_v another_o sphere_n of_o consistent_a matter_n who_o semidiameter_n be_v eh_n and_o centre_n h_n and_o with_o the_o radius_fw-la ah_o let_v the_o circle_n he_o be_v describe_v i_o say_v the_o sphere_n efg_v will_v by_o the_o motion_n of_o the_o body_n in_o bcd_n be_v move_v in_o the_o circumference_n he_o with_o simple_a motion_n for_o see_v the_o motion_n in_o bcd_n by_o the_o four_o article_n of_o this_o chapter_n make_v all_o the_o point_n of_o the_o fluid_fw-la medium_fw-la describe_v in_o the_o same_o time_n circular_a line_n equal_a to_o one_o another_o the_o points_z e_z h_z and_o g_z of_o the_o straight_a line_n ehg_v will_n in_o the_o same_o time_n describe_v with_o equal_a radii_fw-la equal_a circle_n let_v ebb_n be_v draw_v equal_a and_o parallel_v to_o the_o straight_a line_n ah_o and_o let_v ab_fw-la be_v connect_v which_o will_v therefore_o be_v equal_a and_o parallel_v to_o eh_n and_o therefore_o also_o if_o upon_o the_o centre_n b_o and_o radius_fw-la be_v the_o arch_n eke_o be_v draw_v equal_a to_o the_o arch_n he_o and_o the_o straight_a line_n a_z bk_n and_o ik_fw-mi be_v draw_v bk_n and_o ai_fw-fr will_v be_v equal_a and_o they_o will_v also_o be_v parallel_n because_o the_o two_o arch_n eke_o and_o he_o that_o be_v the_o two_o angle_n kbe_n and_o jah_n be_v equal_a and_o consequent_o the_o straight_a line_n ab_fw-la and_o ki_v which_o connect_v they_o will_v also_o be_v equal_a and_o parallel_v wherefore_o ki_n and_o eh_n be_v parallel_n see_v therefore_o e_z and_o h_n be_v carry_v in_o the_o same_o time_n to_o k_o and_o i_o the_o whole_a straight_a line_n ik_fw-mi will_v be_v parallel_n to_o eh_n from_o whence_o it_o depart_v and_o therefore_o see_v the_o sphere_n efg_n be_v suppose_v to_o be_v of_o consistent_a matter_n so_o as_o all_o its_o point_n keep_v always_o the_o same_o situation_n it_o be_v necessary_a that_o every_o other_o straight_a line_n take_v in_o the_o same_o sphere_n be_v carry_v always_o parallel_v to_o the_o place_n in_o which_o it_o former_o be_v wherefore_o the_o sphere_n efg_n be_v move_v with_o simple_a circular_a motion_n which_o be_v to_o be_v demonstrate_v 11_o if_o in_o a_o fluid_fw-la medium_fw-la who_o part_n be_v stir_v by_o a_o body_n move_v with_o simple_a motion_n there_o float_v another_o body_n which_o have_v its_o superficies_n either_o whole_o hard_a or_o whole_o fluid_a the_o part_n of_o this_o body_n shall_v approach_v the_o centre_n equal_o on_o all_o side_n that_o be_v to_o say_v the_o motion_n of_o the_o body_n shall_v be_v circular_a and_o concentrique_fw-la with_o the_o motion_n of_o the_o movent_fw-la but_o if_o it_o have_v one_o side_n hard_o and_o the_o other_o side_n fluid_a then_o both_o those_o motion_n shall_v not_o have_v the_o same_o centre_n nor_o shall_v the_o float_a body_n be_v move_v in_o the_o circumference_n of_o a_o perfect_a circle_n let_v a_o body_n be_v move_v in_o the_o circumference_n of_o the_o circle_n kl_n mn_v in_o the_o 2d_o figure_n who_o centre_n be_v a._n and_o let_v there_o be_v another_o body_n at_o i_o who_o superficies_n be_v either_o all_o hard_a or_o all_o fluid_a also_o let_v the_o medium_fw-la in_o which_o both_o the_o body_n be_v place_v be_v fluid_a i_o say_v the_o body_n at_o i_o will_v be_v move_v in_o the_o circle_n ib_n about_o the_o centre_n a._n for_o this_o have_v be_v demonstrate_v in_o the_o last_o article_n wherefore_o let_v the_o superficies_n of_o the_o body_n at_o i_o be_v fluid_a on_o one_o side_n and_o hard_o on_o the_o other_o and_o first_o let_v the_o fluid_a side_n be_v towards_o the_o centre_n see_v therefore_o the_o motion_n of_o the_o medium_fw-la be_v such_o as_o that_o its_o part_n do_v continual_o change_v their_o place_n as_o have_v be_v show_v in_o the_o 5_o article_n if_o this_o change_n of_o place_n be_v consider_v in_o those_o part_n of_o the_o medium_fw-la which_o be_v contiguous_a to_o the_o fluid_a superficies_n it_o must_v needs_o be_v that_o the_o small_a part_n of_o that_o superficies_n enter_v into_o the_o place_n of_o the_o small_a part_n of_o the_o medium_fw-la which_o be_v contiguous_a to_o they_o and_o the_o like_a change_n of_o place_n will_v be_v make_v with_o the_o next_o contiguous_a part_n towards_o a._n and_o if_o the_o fluid_a part_n of_o the_o body_n at_o i_o have_v any_o degree_n at_o all_o of_o tenacity_n for_o there_o be_v degree_n of_o tenacity_n as_o in_o the_o air_n and_o water_n the_o whole_a fluid_a side_n will_v be_v lift_v up_o a_o little_a but_o so_o much_o the_o less_o as_o its_o part_n have_v less_o tenacity_n whereas_o the_o hard_a part_n of_o the_o superficies_n which_o be_v contiguous_a to_o the_o fluid_a part_n have_v no_o cause_n at_o all_o of_o elevation_n that_o be_v to_o say_v no_o endeavour_n towards_o a._n second_o let_v the_o hard_a superficies_n of_o the_o body_n at_o i_o be_v towards_o a._n by_o reason_n therefore_o of_o the_o say_a change_n of_o place_n of_o the_o part_n which_o be_v contiguous_a to_o it_o the_o hard_a superficies_n must_v of_o necessity_n see_v by_o supposition_n there_o be_v no_o empty_a space_n either_o come_v near_o to_o a_o or_o else_o its_o small_a part_n must_v supply_v the_o contiguous_a place_n of_o the_o medium_fw-la which_o otherwise_o will_v be_v empty_a but_o this_o can_v be_v by_o reason_n of_o
the_o suppose_a hardness_n and_o therefore_o the_o other_o must_v needs_o be_v namely_o that_o the_o body_n come_v near_o to_o a._n wherefore_o the_o body_n at_o i_o have_v great_a endeavour_n towards_o the_o centre_n a_o when_o its_o hard_a side_n be_v next_o it_o then_o when_o it_o be_v avert_v from_o it_o but_o the_o body_n in_o i_o while_o it_o be_v move_v in_o the_o circumference_n of_o the_o circle_n ib_n have_v sometime_o one_o side_n sometime_o another_o turn_v towards_o the_o centre_n and_o therefore_o it_o be_v sometime_o near_o sometime_o further_o off_o from_o the_o centre_n a._n wherefore_o the_o body_n at_o i_o be_v not_o carry_v in_o the_o circumference_n of_o a_o perfect_a circle_n which_o be_v to_o be_v demonstrate_v chap._n xxii_o of_o other_o variety_n of_o motion_n 1_o endeavour_n and_o pressure_n how_o they_o differ_v 2_o two_o kind_n of_o medium_n in_o which_o body_n be_v move_v 3_o propagation_n of_o motion_n what_o it_o be_v 4_o what_o motion_n body_n have_v when_o they_o press_v one_o another_o 5_o fluid_a body_n when_o they_o be_v press_v together_o penetrate_v one_o another_o 6_o when_o one_o body_n press_v another_o and_o do_v not_o penetrate_v it_o the_o action_n of_o the_o press_a body_n be_v perpendicular_a to_o the_o superficies_n of_o the_o body_n press_v 7_o when_o a_o hard_a body_n press_v another_o body_n penetrate_v the_o same_o it_o do_v not_o penetrate_v it_o perpendicular_o unless_o it_o fall_v perpendicular_o upon_o it_o 8_o motion_n sometime_o opposite_a to_o that_o of_o the_o movent_fw-la 9_o in_o a_o full_a medium_fw-la motion_n be_v propagate_v to_o any_o distance_n 10_o dilatation_n and_o contraction_n what_o they_o be_v 11_o dilatation_n and_o contraction_n suppose_v mutation_n of_o the_o small_a part_n in_o respect_n of_o their_o situation_n 12_o all_o traction_n be_v pulsion_n 13_o such_o thing_n as_o be_v press_v or_o bend_v restore_v themselves_o have_v motion_n in_o their_o internal_a part_n 14_o though_o that_o which_o carry_v another_o be_v stop_v the_o body_n carry_v will_v proceed_v 15_o 16_o the_o effect_n of_o percussion_n not_o to_o be_v compare_v with_o those_o of_o waight_n 17_o 18_o motion_n can_v begin_v first_o in_o the_o internal_a part_n of_o a_o body_n 19_o action_n and_o reaction_n proceed_v in_o the_o same_o line_n 20_o habit_n what_o it_o be_v 1_o i_o have_v already_o in_o the_o 15_o chap._n at_o the_o 2d_o article_n define_v endeavour_n to_o be_v motion_n through_o some_o length_n though_o not_o consider_v as_o length_n but_o as_o a_o point_n whether_o therefore_o there_o be_v resistance_n or_o no_o resistance_n the_o endeavour_n will_v still_o be_v the_o same_o for_o simple_o to_o endeavour_v be_v to_o go._n but_o when_o two_o body_n have_v opposite_a endeavour_n press_v one_o another_o than_o the_o endeavour_n of_o either_o of_o they_o be_v that_o which_o we_o call_v pressure_n and_o be_v mutual_a when_o their_o pressure_n be_v opposite_a 2._o body_n move_v and_o also_o the_o medium_n in_o which_o they_o be_v move_v be_v of_o two_o kind_n for_o either_o they_o have_v their_o part_n coherent_a in_o such_o manner_n as_o no_o part_n of_o the_o move_v body_n will_v easy_o yield_v to_o the_o movent_fw-la except_o the_o whole_a body_n yield_v also_o and_o such_o be_v the_o thing_n we_o call_v hard_a or_o else_o their_o part_n while_o the_o whole_a remain_v unmoved_a will_v easy_o yield_v to_o the_o movent_fw-la and_o these_o we_o call_v fluid_a or_o soft_a body_n for_o the_o word_n fluid_a soft_a tough_a and_o hard_a in_o the_o same_o manner_n as_o great_a and_o little_a be_v use_v only_o comparative_o and_o be_v not_o different_a kind_n but_o different_a degree_n of_o quality_n 3_o to_o do_v and_o to_o suffer_v be_v to_o move_v and_o to_o be_v move_v and_o nothing_o be_v move_v but_o by_o that_o which_o touch_v it_o and_o be_v also_o move_v as_o have_v be_v former_o show_v and_o how_o great_a sooner_o the_o distance_n be_v we_o say_v the_o first_o movent_fw-la move_v the_o last_o move_v body_n but_o mediate_o namely_o so_o as_o that_o the_o first_o move_v the_o second_o the_o second_o the_o three_o and_o so_o on_o till_o the_o last_o of_o all_o be_v touch_v when_o therefore_o one_o body_n have_v opposite_a endeavour_n to_o another_o body_n move_v the_o same_o and_o that_o move_v a_o three_o and_o so_o on_o i_o call_v that_o action_n propagation_n of_o motion_n 4_o when_o two_o fluid_a body_n which_o be_v in_o a_o free_a and_o open_a space_n press_v one_o another_o their_o part_n will_v endeavour_v or_o be_v move_v towards_o the_o side_n not_o only_o those_o part_n which_o be_v there_o where_o the_o mutual_a contact_n be_v but_o all_o the_o other_o part_n for_o in_o the_o first_o contact_n the_o part_n which_o be_v press_v by_o both_o the_o endeavour_v body_n have_v no_o place_n either_o forward_o or_o backward_o in_o which_o they_o can_v be_v move_v and_o therefore_o they_o be_v press_v out_o towards_o the_o side_n and_o this_o expressure_n when_o the_o force_n be_v equal_a be_v in_o a_o line_n perpendicular_a to_o the_o body_n press_v but_o whensoever_o the_o foremost_a part_n of_o both_o the_o body_n be_v press_v the_o hindermost_a also_o must_v be_v press_v at_o the_o same_o time_n for_o the_o motion_n of_o the_o hindermost_a part_n can_v in_o a_o instant_n be_v stop_v by_o the_o resistance_n of_o the_o foremost_a part_n but_o proceed_v for_o some_o time_n and_o therefore_o see_v they_o must_v have_v some_o place_n in_o which_o they_o may_v be_v move_v and_o that_o there_o be_v no_o place_n at_o all_o for_o they_o forward_o it_o be_v necessary_a that_o they_o be_v move_v into_o the_o place_n which_o be_v towards_o the_o side_n every_o way_n and_o this_o effect_n follow_v of_o necessity_n not_o only_o in_o fluid_a but_o in_o consistent_a and_o hard_a body_n though_o it_o be_v not_o always_o manifest_a to_o sense_n for_o though_o from_o the_o compression_n of_o two_o stone_n we_o can_v with_o our_o eye_n discern_v any_o swell_a outward_n towards_o the_o side_n as_o we_o perceive_v in_o two_o body_n of_o wax_n yet_o we_o know_v well_o enough_o by_o reason_n that_o some_o tumour_n must_v needs_o be_v there_o though_o it_o be_v but_o little_a 5_o but_o when_o the_o space_n be_v enclose_v and_o both_o the_o body_n be_v fluid_a they_o will_v if_o they_o be_v press_v together_o penetrate_v one_o anoteer_n though_o different_o according_a to_o their_o different_a endeavour_n for_o suppose_v a_o hollow_a cylinder_n of_o hard_a matter_n well_o stop_v at_o both_o end_n but_o fill_v first_o below_z with_o some_o heavy_a fluid_a body_n as_o quicksilver_n and_o above_o with_o water_n or_o aire_n if_o now_o the_o bottom_n of_o the_o cylinder_n be_v turn_v upward_o the_o heavy_a fluid_a body_n which_o be_v now_o at_o the_o top_n have_v the_o great_a endeavour_n downward_o and_o be_v by_o the_o hard_a side_n of_o the_o vessel_n hinder_v from_o extend_v itself_o sidewayes_o must_v of_o necessity_n either_o be_v receive_v by_o the_o light_a body_n that_o it_o may_v sink_v through_o it_o or_o else_o it_o must_v open_v a_o passage_n through_o itself_o by_o which_o the_o light_a body_n may_v ascend_v for_o of_o the_o two_o body_n that_o who_o part_n be_v most_o easy_o separate_v will_v the_o first_o be_v divide_v which_o be_v do_v it_o be_v not_o necessary_a that_o the_o part_n of_o the_o other_o suffer_v any_o separation_n at_o all_o and_o therefore_o when_o two_o liquor_n which_o be_v enclose_v in_o the_o same_o vessel_n change_v their_o place_n there_o be_v no_o need_n that_o their_o small_a part_n shall_v be_v mingle_v with_o one_o another_o for_o a_o way_n be_v open_v through_o one_o of_o they_o the_o part_n of_o the_o other_o need_v not_o be_v separate_v now_o if_o a_o fluid_a body_n which_o be_v not_o enclose_v press_v a_o hard_a body_n its_o endeavour_n will_v indeed_o be_v towards_o the_o internal_a part_n of_o that_o hard_a body_n but_o be_v exclude_v by_o the_o resistance_n of_o it_o the_o part_n of_o the_o fluid_a body_n will_v be_v move_v every_o way_n according_a to_o the_o superficies_n of_o the_o hard_a body_n and_o that_o equal_o if_o the_o pressure_n be_v perpendicular_a for_o when_o all_o the_o part_n of_o the_o cause_n be_v equal_a the_o effect_n will_v be_v equal_a also_o but_o if_o the_o pressure_n be_v not_o perpendicular_a than_o the_o angle_n of_o incidence_n be_v unequal_a the_o expansion_n also_o will_v be_v unequal_a namely_o great_a on_o that_o side_n where_o the_o angle_n be_v great_a because_o that_o motion_n be_v most_o direct_a which_o proceed_v by_o the_o direct_v line_n 6_o if_o a_o body_n press_v another_o body_n do_v not_o penetrate_v it_o it_o will_v nevertheless_o give_v to_o the_o part_n it_o press_v a_o endeavour_n to_o yield_v and_o recede_v