Selected quad for the lemma: power_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
power_n	great_a	length_n	line_n	3,131	5	11.3194	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A05115	Via regia ad geometriam. = The vvay to geometry Being necessary and usefull, for astronomers. Geographers. Land-meaters. Sea-men. Engineres. Architecks. Carpenters. Paynters. Carvers, &c. Written in Latine by Peter Ramus, and now translated and much enlarged by the learned Mr. William Bedvvell.; Via regia ad geometriam. English	Ramus, Petrus, 1515-1572.; Bedwell, William, ca. 1561-1632.; Clarke, John, d. 1658.	1636 (1636)	STC 15251; ESTC S108337	93,096	205

in_o a_o periphery_n and_o do_v differ_v only_o in_o base_a 14_o the_o angle_n in_o opposite_a section_n be_v equal_a to_o two_o right_a angle_n 22._o p_o iij._o the_o reason_n or_o rate_n of_o a_o section_n be_v thus_o the_o similitude_n do_v follow_v 15_o if_o section_n do_v receive_v or_o contain_v equal_v angle_n they_o be_v alike_o e_fw-la 10._o d_o iij._o 16_o if_o like_a section_n be_v upon_o a_o equal_a base_a they_o be_v equal_a and_o contrariwise_o 23,24_o p_o iij._o in_o the_o first_o figure_n let_v the_o base_a be_v the_o same_o and_o if_o they_o shall_v be_v say_v to_o unequal_a section_n and_o one_o of_o they_o great_a than_o another_o the_o angle_n in_o that_o a_o o_o e_o shall_v be_v less_o than_o the_o angle_n a_o i_o e_o in_o the_o lesser_a section_n by_o the_o 16_o e_fw-la uj._o which_o notwithstanding_o by_o the_o grant_n be_v equal_a in_o the_o second_o figure_n if_o one_o section_n be_v put_v upon_o another_o it_o will_v agree_v with_o it_o otherwise_o against_o the_o first_o part_n like_o section_n upon_o the_o same_o base_a shall_v not_o be_v equal_a but_o congruency_n be_v here_o sufficient_a by_o the_o former_a two_o proposition_n and_o by_o the_o 9_o e_fw-la x_o v._n one_o may_v find_v a_o section_n like_a unto_o another_o assign_v or_o else_o from_o a_o circle_n give_v to_o cut_v off_o one_o like_a unto_o it_o 17_o a_o angle_n of_o a_o section_n be_v that_o which_o be_v comprehend_v of_o the_o bound_n of_o a_o section_n 18_o a_o section_n be_v either_o a_o semicircle_n or_o that_o which_o be_v unequal_a to_o a_o semicircle_n a_o section_n be_v two_o fold_n a_o semicircle_n to_o wit_n when_o it_o be_v cut_v by_o the_o diameter_n or_o unequal_a to_o a_o semicircle_n when_o it_o be_v cut_v by_o a_o line_n lesser_a than_o the_o diameter_n 19_o a_o semicircle_n be_v the_o half_a section_n of_o a_o circle_n or_o it_o be_v that_o which_o be_v make_v the_o diameter_n therefore_o 20_o a_o semicircle_n be_v comprehend_v of_o a_o periphery_a and_o the_o diameter_n 18_o dj_o 21_o the_o angle_n in_o a_o semicircle_n be_v a_o right_a angle_n the_o angle_n of_o a_o semicircle_n be_v lesser_a than_o a_o rectilineall_a right_a angle_n but_o great_a than_o any_o acute_a angle_n the_o angle_n in_o a_o great_a section_n be_v lesser_a than_o a_o right_a angle_n of_o a_o great_a it_o be_v a_o great_a in_o a_o lesser_a it_o be_v great_a of_o a_o lesser_a it_o be_v lesser_a ê_fw-la 31_o and_o 16._o p_o iij._o or_o thus_o the_o angle_n in_o a_o semicircle_n be_v a_o right_a angle_n the_o angle_n of_o a_o semicircle_n be_v less_o than_o a_o right_n rightline_v angle_n but_o great_a than_o any_o acute_a angle_n the_o angle_n in_o the_o great_a section_n be_v less_o than_o a_o right_a angle_n the_o angle_n of_o the_o great_a section_n be_v great_a than_o a_o right_a angle_n the_o angle_n in_o the_o lesser_a section_n be_v great_a than_o a_o right_a angle_n the_o angle_n of_o the_o lesser_a section_n be_v lesser_a than_o a_o right_a angle_n h._n the_o second_o part_n that_o the_o angle_n of_o a_o semicircle_n be_v lesser_a than_o a_o right_a angle_n be_v manifest_a out_o of_o that_o because_o it_o be_v the_o part_n of_o a_o right_a angle_n for_o the_o angle_n of_o the_o semicircle_n a_o i_o e_o be_v a_o part_n of_o the_o rectilineall_a right_a angle_n a_o i_o u._n the_o three_o part_n that_o it_o be_v great_a than_o any_o acute_a angle_n be_v manifest_a out_o of_o the_o 23._o e_fw-la x_o v._n for_o otherwise_o a_o tangent_fw-la be_v not_o on_o the_o same_o part_n one_o only_a and_o no_o more_o the_o four_o part_n be_v thus_o make_v manifest_a the_o angle_n at_o i_o in_o the_o great_a section_n a_o e_fw-it i_fw-it be_v lesser_a than_o a_o right_a angle_n because_o it_o be_v in_o the_o same_o triangle_n a_o e_fw-it i_fw-it which_o at_o a_o be_v right_a angle_n and_o if_o neither_o of_o the_o shank_n be_v by_o the_o centre_n notwithstanding_o a_o angle_n may_v be_v make_v equal_a to_o the_o assign_a in_o the_o same_o section_n the_o five_o be_v thus_o the_o angle_n of_o the_o great_a section_n e_fw-la a_o i_o be_v great_a than_o a_o right_a angle_n because_o it_o contain_v a_o rightangle_n the_o six_o be_v thus_o the_o angle_n a_o o_o e_o in_o a_o lesser_a section_n be_v great_a than_o a_o right_a angle_n by_o the_o 14_o e_fw-la x_o five_o i_o because_o that_o which_o be_v in_o the_o opposite_a section_n be_v lesser_a than_o a_o right_a angle_n the_o seven_o be_v thus_o the_o angle_n e_o a_o o_o be_v lesser_a than_o a_o rightangle_n because_o it_o be_v part_n of_o a_o right_a angle_n to_o wit_n of_o the_o outter_n angle_n if_o i_o a_o be_v draw_v out_o at_o length_n and_o thus_o much_o of_o the_o angle_n of_o a_o circle_n of_o all_o which_o the_o most_o effectual_a and_o of_o great_a power_n and_o use_n be_v the_o angle_n in_o a_o semicircle_n and_o therefore_o it_o be_v not_o without_o cause_n so_o often_o mention_v of_o aristotle_n this_o geometry_n therefore_o of_o aristotle_n let_v we_o somewhat_o more_o full_o open_a and_o declare_v for_o from_o hence_o do_v arise_v many_o thing_n therefore_o 22_o if_o two_o right_a line_n joint_o bound_v with_o the_o diameter_n of_o a_o circle_n be_v joint_o bound_v in_o the_o periphery_n they_o do_v make_v a_o right_a angle_n or_o thus_o if_o two_o right_a line_n have_v the_o same_o term_n with_o the_o diameter_n be_v join_v together_o in_o one_o point_n of_o the_o circomference_n they_o make_v a_o right_a angle_n h._n this_o corollary_n be_v draw_v out_o of_o the_o first_o part_n of_o the_o former_a element_n where_o it_o be_v say_v that_o a_o angle_n in_o a_o semicircle_n be_v a_o right_a angle_n and_o 23_o if_o a_o infinite_a right_a line_n be_v cut_v of_o a_o periphery_a of_o a_o external_a centre_n in_o a_o point_n assign_v and_o contingent_a and_o the_o diameter_n be_v draw_v from_o the_o contingent_a point_n a_o right_a line_n from_o the_o point_n assign_v knit_v it_o with_o the_o diameter_n shall_v be_v perpendicular_a unto_o the_o infinite_a line_n give_v let_v the_o infinite_a right_a line_n be_v a_o e_fw-es from_o who_o point_n a_o a_o perpendicular_a be_v to_o be_v raise_v and_o 24_o if_o a_o right_a line_n from_o a_o point_n give_v make_v a_o acute_a angle_n with_o a_o infinite_a line_n be_v make_v the_o diameter_n of_o a_o periphery_a cut_v the_o infinite_a a_o right_a line_n from_o the_o point_n assign_v knit_v the_o segment_n shall_v be_v perpendicular_a upon_o the_o infinite_a line_n as_o in_o the_o same_o example_n have_v a_o external_a point_n give_v let_v a_o perpendicular_a unto_o the_o infinite_a right_a line_n a_o e_o be_v seek_v let_v the_o right_a line_n i_o o_o e_o be_v make_v the_o diameter_n of_o the_o peripherie_n and_o withal_o let_v it_o make_v with_o the_o infinite_a right_a line_n giyen_v a_o acute_a angle_n in_o e_o from_o who_o bisection_n for_o the_o centre_n let_v a_o periphery_n cut_v the_o infinite_a etc._n etc._n and_o 25_o if_o of_o two_o right_a line_n the_o great_a be_v make_v the_o diameter_n of_o a_o circle_n and_o the_o lesser_a joint_o bound_v with_o the_o great_a an_o inscribe_v be_v knit_v together_o the_o power_n of_o the_o great_a shall_v be_v more_o than_o the_o power_n of_o the_o lesser_a by_o the_o quadrate_n of_o that_o which_o knit_v they_o both_o together_o ad_fw-la 13_o p._n x._o 26_o if_o a_o right_a line_n continue_v or_o continual_o make_v of_o two_o right_a line_n give_v be_v make_v the_o diameter_n of_o a_o circle_n the_o perpendicular_a from_o the_o point_n of_o their_o continuation_n unto_o the_o periphery_n shall_v be_v the_o mean_a proportional_a between_o the_o two_o line_n give_v 13_o p_o uj._o so_o if_o the_o side_n of_o a_o quadrate_n of_o 10._o foot_n content_a be_v seek_v let_v the_o side_n 1_o foot_n and_o 10_o foot_n a_o oblong_a equal_a to_o that_o same_o quadrate_n be_v continue_v the_o mean_a proportional_a shall_v be_v the_o side_n of_o the_o quadrate_n that_o be_v the_o power_n of_o it_o shall_v be_v 10._o foot_n the_o reason_n of_o the_o angle_n in_o opposite_a section_n do_v follow_v 27_o the_o angle_n in_o opposite_a section_n be_v equal_a in_o the_o alterne_a angle_n make_v of_o the_o secant_fw-la and_o touch_v line_n 32._o p_o iij._o as_o let_v the_o unequal_a section_n be_v e_o i_o o_o and_o e_z a_o o_o the_o tangent_fw-la let_v it_o be_v u._fw-mi e_fw-es y_fw-es and_o the_o angle_n in_o the_o opposite_a section_n e_z a_o o_o and_o e_z i_z o._n i_o say_v they_o be_v equal_a in_o the_o alterne_a angle_n of_o the_o secant_fw-la and_o touch_v line_n o_fw-mi e_fw-es y_fw-es and_o o_o e_o u._n first_o that_o which_o be_v at_o a_o be_v equal_a to_o the_o

such_o as_o the_o magnitude_n by_o the_o measure_a be_v in_o planimetry_n i_o mean_v they_o be_v plain_n in_o stereometry_n they_o be_v solid_n as_o hereafter_o we_o shall_v make_v manifest_a therefore_o in_o that_o which_o follow_v a_o inch_n be_v not_o only_o a_o length_n three_o barley-corne_n long_o but_o a_o plain_a three_o barley-corne_n long_o and_o three_o broad_a a_o foot_n be_v not_o only_o a_o length_n of_o 12._o ynche_n but_o a_o plain_a also_o of_o 12._o ynche_n square_a or_o contain_v 144._o square_a ynches●_n a_o yard_n be_v not_o only_o the_o length_n of_o three_o foot_n but_o it_o be_v also_o a_o plain_a 3._o foot_n square_v every_o way_n a_o perch_n be_v not_o only_o a_o length_n of_o 5½_n yard_n but_o it_o be_v a_o plot_n of_o ground_n 5½_n yard_n square_v every_o way_n a_o quadrate_n therefore_o or_o square_v see_v that_o it_o be_v equilater_n that_o be_v of_o equal_a side_n and_o equiangle_n by_o mean_n of_o the_o equal_a right_a angle_n of_o quandrangle_v that_o only_o be_v ordinate_a therefore_o 3_o the_o side_n of_o equal_a quadrate_n be_v equal_a and_o the_o side_n of_o equal_a quadrate_n be_v equal_o compare_v if_o therefore_o two_o or_o more_o quadrate_n be_v equal_a it_o must_v needs_o follow_v that_o their_o side_n be_v equal_a one_o to_o another_o and_o 4_o the_o power_n of_o a_o right_a line_n be_v a_o quadrate_n or_o thus_o the_o possibility_n of_o a_o right_a line_n be_v a_o square_a h._n a_o right_a line_n be_v say_v posse_fw-la quadratum_fw-la to_o be_v in_o power_n a_o square_a because_o be_v multiply_v in_o itself_o it_o do_v make_v a_o square_a 5_o if_o two_o conterminall_a perpendicular_a equal_a right_a line_n be_v close_v with_o parallel_n they_o shall_v make_v a_o quadrate_n 46._o p.j._n 6_o the_o plain_a of_o a_o quadrate_n be_v a_o equilater_n plain_a or_o thus_o the_o plain_a number_n of_o a_o square_a be_v a_o plain_a number_n of_o equal_a side_n h._n a_o quadrate_n or_o square_a number_n be_v that_o which_o be_v equal_o equal_a or_o that_o which_o be_v comprehend_v of_o two_o equal_a number_n a_o quadrate_n of_o all_o plain_n be_v especial_o rational_a and_o yet_o not_o always_o but_o that_o only_o be_v rational_a who_o number_n be_v a_o quadrate_n therefore_o the_o quadrate_n of_o number_n not_o quadrate_n be_v not_o rational_o therefore_o 7_o a_o quadrate_n be_v make_v of_o a_o number_n multiply_v by_o itself_o such_o quadrate_n be_v the_o first_o nine_o 1,4,9_o 16,25,36,49,64_o 81_o make_v of_o once_o one_o twice_z two_o thrice_o three_o four_o time_n four_o five_o time_n five_o six_o time_n six_o seven_o time_n seven_o eight_o time_n eight_o and_o nine_o time_n nine_o and_o this_o be_v the_o sum_n of_o the_o make_n and_o invention_n of_o a_o quadrate_n number_n of_o multiplication_n of_o the_o side_n give_v by_o itself_o hereafter_o diverse_a comparison_n of_o a_o quadrate_n or_o square_n with_o a_o rectangle_n with_o a_o quadrate_n and_o with_o a_o rectangle_n and_o a_o quadrate_n joint_o the_o comparison_n or_o rate_n of_o a_o quadrate_n with_o a_o rectangle_n be_v first_o 8_o if_o three_o right_a line_n be_v proportional_a the_o quadrate_n of_o the_o middle_a one_o shall_v be_v equal_a to_o the_o rectangle_n of_o the_o extreme_n and_o contrariwise_o 17._o p_o five_o i_o and_o 20._o p_o seven_o it_o be_v a_o corollary_n out_o of_o the_o 28._o e_fw-la x._o as_o in_o a_o e_fw-la e_o i_o i_o o._n 9_o if_o the_o base_a of_o a_o triangle_n do_v subtend_v a_o right_a angle_n the_o power_n of_o it_o be_v as_o much_o as_o of_o both_o the_o shank_n and_o contrariwise_o 47,48_o p_o i_o 10_o if_o the_o quadrate_n of_o a_o odd_a number_n give_v for_o the_o first_o shank_n be_v make_v less_o by_o a_o unity_n the_o half_a of_o the_o remainder_n shall_v be_v the_o other_o shank_n increase_v by_o a_o unity_n it_o shall_v be_v the_o base_a or_o thus_o if_o the_o square_a of_o a_o odd_a number_n give_v for_o the_o first_o foot_n have_v a_o unity_n take_v from_o it_o the_o half_a of_o the_o remainder_n shall_v be_v the_o other_o foot_n and_o the_o same_o half_n increase_v by_o a_o unity_n shall_v be_v the_o base_a h._n again_o the_o quadrate_n or_o square_n of_o 3._o the_o first_o shank_n be_v 9_o and_o 9_o 1_o be_v 8_o who_o half_a 4_o be_v the_o other_o shank_n and_o 9_o 1_o be_v 10._o who_o half_o 5._o be_v the_o base_a plato_n way_n be_v thus_o by_o a_o even_a number_n 11_o if_o the_o half_a of_o a_o even_a number_n give_v for_o the_o first_o shank_n be_v square_v the_o square_a number_n diminish_v by_o a_o unity_n shall_v be_v the_o other_o shank_n and_o increase_v by_o a_o unity_n it_o shall_v be_v the_o base_a again_o the_o quadrate_n or_o square_n of_o 3._o the_o half_a of_o 6_o the_o first_o shank_n be_v 9_o and_o 9_o 1_o be_v 8_o for_o the_o second_o shank_n and_o out_o of_o this_o rate_n of_o rational_a power_n as_o vitruvius_n in_o the_o 2._o chapter_n of_o his_o ix_o book_n say_v pythagoras_n teach_v how_o to_o make_v a_o most_o exact_a and_o true_a squire_n by_o join_v of_o three_o ruler_n together_o in_o the_o form_n of_o a_o triangle_n which_o be_v one_o unto_o another_o as_o 3,4_o and_o 5._o be_v one_o to_o another_o from_o hence_o architecture_n learn_v a_o arithmetical_a proportion_n in_o the_o part_n of_o ladder_n and_o stair_n for_o that_o rate_n or_o proportion_n as_o in_o many_o business_n and_o measure_n be_v very_o commodious_a so_o also_o in_o building_n and_o make_v of_o ladder_n or_o stair_n that_o they_o may_v have_v moderate_a rise_v of_o the_o step_n it_o be_v very_o speedy_a for_o 9_o 1_o be_v 10_o base_a 12._o the_o power_n of_o the_o diagony_n be_v twice_o asmuch_o as_o be_v the_o power_n of_o the_o side_n and_o it_o be_v unto_o it_o also_o incommensurable_a or_o thus_o the_o diagonall_a line_n be_v in_o power_n double_a to_o the_o side_n and_o be_v incommensurable_a unto_o it_o h._n this_o be_v the_o way_n of_o double_v of_o a_o square_n teach_v by_o plato_n as_o vitruvius_n tell_v we_o which_o notwithstanding_o may_v be_v also_o double_v treble_v or_o according_a to_o any_o reason_n assign_v increase_v by_o the_o 25_o e_fw-la iiij_o as_o there_o be_v foretell_v but_o that_o the_o diagony_n be_v incommensurable_a unto_o the_o side_n it_o be_v the_o 116_o p_o x._o the_o reason_n be_v because_o otherwise_o there_o may_v be_v give_v one_o quadrate_n number_n double_a to_o another_o quadrate_n number_n which_o as_o theon_n and_o campanus_n teach_v we_o be_v impossible_a to_o be_v find_v but_o that_o reason_n which_o aristotle_n bring_v be_v more_o clear_a which_o be_v this_o because_o otherwise_o a_o even_a number_n shall_v be_v odd_a for_o if_o the_o diagony_n be_v 4_o and_o the_o side_n 3_o the_o square_a of_o the_o diagony_n 16_o shall_v be_v double_a to_o the_o square_n of_o the_o side_n and_o so_o the_o square_a of_o the_o side_n shall_v be_v 8._o and_o the_o same_o square_n shall_v be_v 9_o to_o wit_n the_o square_a of_o 3._o and_o so_o even_o shall_v be_v odd_a which_o be_v most_o absurd_a hither_o may_v be_v add_v that_o at_o the_o 42_o p_o x._o that_o the_o segment_n of_o a_o right_a line_n diverse_o cut_v the_o more_o unequal_a they_o be_v the_o great_a be_v their_o power_n 13_o if_o the_o base_a of_o a_o right_a angle_a triangle_n be_v cut_v by_o a_o perpendicular_a from_o the_o right_a angle_n in_o a_o double_a reason_n the_o power_n of_o it_o shall_v be_v half_a as_o much_o more_o as_o be_v the_o power_n of_o the_o great_a shank_n but_o thrice_o so_o much_o as_o be_v the_o power_n of_o the_o lesser_a if_o in_o a_o quadrupled_a reason_n it_o shall_v be_v four_o time_n and_o one_o four_o so_o much_o as_o be_v the_o great_a but_o five_o time_n so_o much_o as_o be_v the_o lesser_a at_o the_o 13_o 15_o 16_o p_o x_o iij._o and_o by_o the_o same_o argument_n it_o shall_v be_v treble_a unto_o the_o quadrate_n or_o square_n of_o e_o i._n the_o other_o of_o the_o fourfold_a or_o quadruple_a section_n be_v manifest_a in_o the_o figure_n follow_v by_o the_o like_a argument_n 14_o if_o a_o right_a line_n be_v cut_v into_o how_o many_o part_n so_o ever_o the_o power_n of_o it_o be_v manifold_a unto_o the_o power_n of_o segment_n denominate_v of_o the_o square_n of_o the_o number_n of_o the_o section_n or_o thus_o if_o a_o right_a be_v cut_v into_o how_o many_o part_n so_o ever_o it_o be_v in_o power_n the_o multiplex_n of_o the_o segment_n the_o square_a of_o the_o number_n of_o the_o section_n be_v denominate_v thereof_o h._n 15._o if_o a_o right_a line_n be_v cut_v into_o two_o segment_n the_o quadrate_n of_o the_o whole_a be_v equal_a to_o the_o quadrat_n of_o the_o segment_n and_o a_o double_a

arise_v the_o four_o rate_n or_o comparison_n 7._o if_o a_o right_a line_n be_v cut_v into_o two_o equal_a part_n and_o otherwise_o the_o oblong_a of_o the_o unequal_a segment_n with_o the_o quadrate_n of_o the_o segment_n between_o they_o be_v equal_a to_o the_o quadrate_n of_o the_o bisegment_n 5_o p_o ij_o the_o three_o section_n do_v follow_v from_o whence_o the_o five_o reason_n arise_v 8._o if_o a_o right_a line_n be_v cut_v into_o equal_a part_n and_o continue_v the_o oblong_v make_v of_o the_o continue_a and_o the_o continuation_n with_o the_o quadrate_n of_o the_o bisegment_n or_o half_n be_v equal_a to_o the_o quadrate_n of_o the_o line_n compound_v of_o the_o bisegment_n and_o continuation_n 6_o p_o ij_o from_o hence_o arise_v the_o mesographus_n or_o mesolabus_n of_o heron_n the_o mechanic_n so_o name_v of_o the_o invention_n of_o two_o line_n continual_o proportional_a between_o two_o line_n give_v whereupon_o arise_v the_o deliacke_a problem_n which_o trouble_v apollo_n himself_o now_o the_o mesographus_n of_o hero_n be_v a_o infinite_a right_a line_n which_o be_v stay_v with_o a_o scrue-pinne_n which_o be_v to_o be_v move_v up_o and_o down_o in_o riglet_n and_o it_o be_v as_o pappus_n say_v in_o the_o begin_n of_o his_o 111_o book_n for_o architect_n most_o fit_a and_o more_o ready_a than_o the_o plato_n mesographu_n the_o mechanical_a handle_n of_o this_o mesographu_n be_v describe_v by_o eutocius_n at_o the_o 1_o theorem_a of_o the_o 11_o book_n of_o the_o sphere_n but_o it_o be_v somewhat_o more_o plain_o and_o easy_o thus_o lay_v down_o by_o we_o 9_o if_o the_o mesographus_n touch_v the_o angle_n opposite_a to_o the_o angle_n make_v of_o the_o two_o line_n give_v do_v cut_v the_o say_v two_o line_n give_v comprehend_v a_o right_a angle_a parallelogramme_n and_o infinite_o continue_v equal_o distant_a from_o the_o centre_n the_o intersegment_n shall_v be_v the_o mean_v continual_o proportional_o between_o and_o two_o line_n give_v or_o thus_o if_o a_o mesographus_n touch_v the_o angle_n opposite_a to_o the_o angle_n make_v of_o the_o line_n give_v do_v cut_v the_o equal_a distance_n from_o the_o centre_n the_o two_o right_a line_n give_v contain_v a_o right_a angle_a parallelogramme_n and_o continue_v out_o infinite_o the_o segment_n shall_v be_v mean_a in_o continual_a proportion_n with_o the_o line_n give_v h._n as_o let_v the_o two_o right-line_n give_v be_v a_o e_fw-es and_o a_o i_o and_o let_v they_o comprehend_v the_o rectangled_a parallelogramme_n a_o o_o and_o let_v the_o say_v right_a line_n give_v be_v continue_v infinite_o a_o e_z toward_z u._fw-mi and_o a_o i_o towards_o y._n now_o let_v the_o mesographus_n u._fw-mi y_fw-mi touch_fw-mi o_o the_o angle_n opposite_a to_o a_o and_o let_v it_o cut_v the_o say_v continue_v line_n equal_o distant_a from_o the_o centre_n the_o centre_n be_v find_v by_o the_o 8_o e_fw-la iiij_o to_o wit_n by_o the_o meeting_n of_o the_o diagony_n for_o the_o equidistance_n from_o the_o centre_n the_o mesographus_n be_v to_o be_v move_v up_o or_o down_o until_o by_o the_o compass_n it_o be_v find_v now_o suppose_v the_o point_n of_o equidistancy_n thus_o find_v to_o be_v u._fw-mi and_o y._n i_o say_v that_o the_o portion_n of_o the_o continue_a line_n thus_o be_v the_o mean_a proportional_o seek_v and_o as_o a_o e_z be_v to_z i_z y_z so_o be_v i_z y_z to_z e_z u._fw-mi so_o be_v e_o u._fw-mi to_o a_o i_o the_o fourteen_o book_n of_o p._n ramus_n geometry_n of_o a_o right_a line_n proportional_o cut_v and_o of_o other_o quadrangle_v and_o multangel_n thus_o far_o of_o the_o threefold_a section_n from_o whence_o we_o have_v the_o five_o rational_a rate_n of_o equality_n there_o follow_v of_o the_o three_o section_n another_o section_n into_o two_o segment_n proportional_a to_o the_o whole_a the_o section_n itself_o be_v first_o to_o be_v define_v 1._o a_o right_a line_n be_v cut_v according_a to_o a_o mean_a and_o extreme_a rate_n when_o as_o the_o whole_a shall_v be_v to_o the_o great_a segment_n so_o the_o great_a shall_v be_v unto_o the_o lesser_a 3._o d_o uj._o this_o line_n be_v cut_v so_o that_o the_o whole_a line_n itself_o with_o the_o two_o segment_n do_v make_v the_o three_o bound_n of_o the_o proportioned_a and_o the_o whole_a itself_o be_v first_o bind_v the_o great_a segment_n be_v the_o middle_n bind_v the_o lesser_a the_o three_o bind_v 2._o if_o a_o right_a line_n cut_v proportional_o be_v rational_a unto_o the_o measure_n give_v the_o segment_n be_v unto_o the_o same_o and_o between_o themselves_o irrational_a è_fw-fr 6_o p_o xiij_o a_o triangle_n and_o all_o triangulate_v that_o be_v figure_n make_v of_o triangle_n except_o a_o rightangled-parallelogramme_a be_v in_o geometry_n hold_v to_o be_v irrational_o this_o be_v therefore_o the_o definition_n of_o a_o proportional_a section_n the_o section_n itself_o follow_v which_o be_v by_o the_o rate_n of_o a_o oblong_a with_o a_o quadrate_n 3._o if_o a_o quadrate_n be_v make_v of_o a_o right_a line_n give_v the_o difference_n of_o the_o right_a line_n from_o the_o midst_n of_o the_o conterminall_a side_n of_o the_o say_a quadrate_n make_v above_o the_o same_o half_a shall_v be_v the_o great_a segment_n of_o the_o line_n give_v proportional_o cut_v 11_o p_o ij_o or_o thus_o if_o a_o square_a be_v make_v of_o a_o right_a line_n give_v the_o difference_n of_o a_o right_a line_n draw_v from_o the_o angle_n of_o the_o square_n make_v unto_o the_o midst_n of_o the_o next_o side_n above_o the_o half_a of_o the_o side_n shall_v be_v the_o great_a segment_n of_o the_o line_n give_v be_v proportional_o cut_v h._n for_o of_o y_o a_o let_v the_o quadrate_n a_fw-fr y_fw-fr s_o r_o be_v make_v and_o let_v s_o r_o be_v continue_v unto_o l._n now_o by_o the_o 8_o e_z xiij_o the_o oblong_n of_o o_fw-fr y_fw-fr and_o a_o y_z with_o the_o quadrate_n of_o you_o a_o be_v equal_a to_o the_o quadrate_n of_o u._fw-mi y_fw-mi that_o be_v by_o the_o construction_n of_o u._fw-mi e_fw-es and_o therefore_o by_o the_o 9_o e_fw-la xij_o it_o be_v equal_a to_o the_o quadrate_n e_o a_o and_o a_o u._fw-mi take_v away_o from_o each_o side_n the_o common_a oblong_v a_o l_o and_o the_o quadrate_n y_fw-fr r_o shall_v be_v equal_a to_o the_o oblong_n r_o i._n therefore_o the_o three_o right_a line_n e_z a_o a_o r_o and_o r_o e_o by_o the_o 8_o e_fw-la xij_o be_v continual_a proportional_a and_o the_o right_a line_n a_o e_o be_v cut_v proportional_o therefore_o 4_o if_o a_o right_a line_n cut_v proportional_o be_v continue_v with_o the_o great_a segment_n the_o whole_a shall_v be_v cut_v proportional_o and_o the_o great_a segment_n shall_v be_v the_o line_n give_v 5_o p_o xiij_o as_o in_o the_o same_o example_n the_o right_a line_n o_o y_fw-fr be_v continue_v with_o the_o great_a segment_n and_o the_o oblong_a of_o the_o whole_a and_o the_o lesser_a segment_n be_v equal_a to_o the_o quadrate_n of_o the_o great_a and_o thus_o one_o may_v by_o infinite_o proportional_o cut_v increase_n a_o right_a line_n and_o again_o decrease_v it_o the_o lesser_a segment_n of_o a_o right_a line_n proportional_o cut_v be_v the_o great_a segment_n of_o the_o great_a proportional_o cut_v and_o from_o hence_o a_o decrease_a may_v be_v make_v infinite_o 5_o the_o great_a segment_n continue_v to_o the_o half_a of_o the_o whole_a be_v of_o power_n quintuple_a unto_o the_o say_v half_a that_o be_v five_o time_n so_o great_a as_o it_o be_v and_o if_o the_o power_n of_o a_o right_a line_n be_v quintuple_a to_o his_o segment_n the_o remainder_n make_v the_o double_a of_o the_o former_a be_v cut_v proportional_o and_o the_o great_a segment_n be_v the_o same_o remainder_n 1._o and_o 2._o p_o x_o iij._o this_o be_v the_o fabric_n or_o manner_n of_o make_v a_o proportional_a section_n a_o threefold_a rate_n follow_v the_o first_o be_v of_o the_o great_a segment_n the_o converse_n be_v apparent_a in_o the_o same_o example_n for_o see_v that_o i_o o_o be_v of_o power_n five_o time_n so_o much_o as_o be_v a_o o_o the_o gnomon_n l_o m_z n_o shall_v be_v four_o time_n so_o much_o as_o be_v u._fw-mi a_o who_o quadruple_a also_o by_o the_o 14._o e_fw-la xij_o be_v a_o v_o therefore_o it_o be_v equal_a to_o the_o gnomon_n now_o a_o j_o be_v equal_a to_o a_o e_o therefore_o it_o be_v the_o double_a also_o of_o a_o o_o that_o be_v of_o a_o y_o and_o therefore_o by_o the_o 24._o e_fw-la x._o it_o be_v the_o double_a of_o a_o t_o and_o therefore_o it_o be_v equal_a to_o the_o compliment_n i_o y_fw-fr and_o y_z s_z therefore_o the_o other_o diagonall_a y_o r_o be_v equal_a to_o the_o other_o rectangle_n i_o v._n wherefore_o by_o the_o 8_o e_fw-la xij_o as_o e_z v_o that_o be_v a_o e_z be_v to_z y_z t_o that_o

be_v a_o i_o so_o be_v a_o i_o unto_z i_z e_z wherefore_o by_o the_o ●_o e_o a_o e_z be_v proportional_a cut_n and_o the_o great_a segment_n be_v a_o i_o the_o same_o remain_v the_o other_o propriety_n of_o the_o quintuple_a do_v follow_v 6_o the_o lesser_a segment_n continue_v to_o the_o half_a of_o the_o great_a be_v of_o power_n quintuple_a to_o the_o same_o half_a è_fw-mi 3_o p_o x_o iij._o the_o rate_n of_o the_o triple_a follow_v 7_o the_o whole_a line_n and_o the_o lesser_a segment_n be_v in_o power_n treble_a unto_o the_o great_a è_fw-it 4_o p_o xiij_o 8_o a_o obliquangled_a parallelogramme_n be_v either_o a_o rhombus_fw-la or_o a_o rhomboide_n 9_o a_o rhombus_fw-la be_v a_o obliquangled_a equilater_n parallelogramme_n 32_o dj_o it_o be_v otherwise_o of_o some_o call_v a_o diamond_n 10_o a_o rhomboide_n be_v a_o obliquangled_a parallelogram●e_n not_o equilater_n 33._o dj_o and_o a_o rhomboide_n be_v so_o oppose_v to_o a_o oblong_a as_o a_o rhombus_fw-la be_v to_o a_o quadrate_n and_o the_o rhomboide_n be_v so_o call_v as_o one_o will_v say_v rhombuslike_n although_o beside_o the_o inequality_n of_o the_o angle_v it_o have_v nothing_o like_o to_o a_o rhombus_fw-la a_o example_n of_o measure_v of_o a_o rhombus_fw-la be_v thus_o 11_o a_o trapezium_fw-la be_v a_o quadrangle_n not_o parallelogramme_n 34._o dj_o the_o example_n both_o of_o the_o figure_n and_o of_o the_o measure_n of_o the_o same_o let_v these_o be_v therefore_o triangulate_v quadrangle_v be_v of_o this_o sort_n 12_o a_o multangle_n be_v a_o figure_n that_o be_v comprehend_v of_o more_o than_o four_o right_a line_n 23._o dj_o by_o this_o general_a name_n all_o other_o sort_n of_o right_n line_v figure_n hereafter_o follow_v be_v by_o euclid_n comprehend_v as_o be_v the_o quinquangle_n sexangle_v septangle_n and_o such_o like_a innumerable_a take_v their_o name_n of_o the_o number_n of_o their_o angle_n in_o every_o kind_n of_o multangle_n there_o be_v one_o ordinate_a as_o we_o have_v in_o the_o former_a signify_v of_o which_o in_o this_o place_n we_o will_v say_v nothing_o but_o this_o one_o thing_n of_o the_o quinquangle_n the_o rest_n shall_v be_v reserve_v until_o we_o come_v to_o adscription_n 13_o multangle_v triangulate_v do_v take_v their_o measure_n also_o from_o their_o triangle_n 14_o if_o a_o equilater_n quinquangle_v have_v three_o side_n equal_a it_o be_v equiangle_v 7_o p_o 13._o this_o of_o some_o from_o the_o greek_a be_v call_v a_o pentagon_n of_o other_o a_o pentangle_v by_o a_o name_n partly_o greek_a partly_o latin_a the_o fifteen_o book_n of_o geometry_n of_o the_o line_n in_o a_o circle_n as_o yet_o we_o have_v have_v the_o geometry_n of_o rectilineal_n the_o geometry_n of_o curvilineal_n of_o which_o the_o circle_n be_v the_o chief_a do_v follow_v 1._o a_o circle_n be_v a_o round_a plain_n ●_o 15_o dj_o the_o mean_n to_o describe_v a_o circle_n be_v the_o same_o which_o be_v to_o make_v a_o periphery_a but_o with_o some_o difference_n for_o there_o be_v consider_v no_o more_o but_o the_o motion_n the_o point_n in_o the_o end_n of_o the_o ray_n describe_v the_o periphery_a here_o be_v consider_v the_o motion_n of_o the_o whole_a ray_n make_v the_o whole_a plot_n contain_v within_o the_o periphery_n a_o circle_n of_o all_o plain_n be_v the_o most_o ordinate_a figure_n as_o be_v before_o teach_v at_o the_o 10_o e_fw-la iiij_o 2_o circle_n be_v as_o the_o quadrate_n or_o square_n make_v of_o their_o diameter_n 2_o p._n x_o ij_o therefore_o 3._o the_o diameter_n be_v as_o their_o periphery_n pappus_n 5_o l_o x_o j_o and_o 26_o the_o 18._o as_o here_o thou_o see_v in_o a_o e_fw-la and_o i_z o._n 4._o circular_a geometry_n be_v either_o in_o line_n or_o in_o the_o segment_n of_o a_o circle_n this_o partition_n of_o the_o subject_a matter_n howsoever_o be_v take_v for_o the_o distinguish_n and_o sever_n with_o some_o light_n a_o matter_n somewhat_o confuse_v and_o indeed_o concern_v line_n the_o consideration_n of_o secant_v be_v here_o the_o foremost_a and_o first_o of_o inscript_n 5._o if_o a_o right_a line_n be_v bound_v by_o two_o point_n in_o the_o periphery_n it_o shall_v fall_v within_o the_o circle_n 2_o p_o iij._o from_o hence_o do_v follow_v the_o infinite_a section_n of_o which_o we_o speak_v at_o the_o 6_o e_fw-la i_o this_o proposition_n teach_v how_o a_o rightline_n be_v to_o be_v inscribe_v in_o a_o circle_n to_o wit_n by_o take_v of_o two_o point_n in_o the_o periphery_a 6._o if_o from_o the_o end_n of_o the_o diameter_n and_o with_o a_o ray_n of_o it_o equal_a to_o the_o right_a line_n give_v a_o periphery_n be_v describe_v a_o right_a line_n draw_v from_o the_o say_a end_n unto_o the_o meeting_n of_o the_o periphery_n shall_v be_v inscribe_v into_o the_o circle_n equal_a to_o the_o right_a line_n give_v 1_o p_o iiij_o and_o this_o proposition_n teach_v how_o a_o right_a line_n give_v be_v to_o be_v inscribe_v into_o a_o circle_n equal_a to_o a_o line_n give_v moreover_o of_o all_o inscript_n the_o diameter_n be_v the_o chief_a for_o it_o show_v the_o centre_n and_o also_o the_o reason_n or_o proportion_n of_o all_o other_o inscript_n therefore_o the_o invention_n and_o make_n of_o the_o diameter_n of_o a_o circle_n be_v first_o to_o be_v teach_v 7._o if_o a_o inscript_n do_v cut_v into_o two_o equal_a part_n another_o inscript_n perpendicular_o it_o be_v the_o diameter_n of_o the_o circle_n and_o the_o midst_n of_o it_o be_v the_o centre_n 1_o p_o iij._o the_o cause_n be_v the_o same_o which_o be_v of_o the_o 5_o e_z x_o i_o because_o the_o inscript_n cut_v into_o half_n if_o for_o the_o side_n of_o the_o inscribe_v rectangle_n and_o it_o do_v subtend_v the_o periphery_n cut_v also_o into_o two_o part_n by_o the_o which_o both_o the_o inscript_n and_o periphery_a also_o be_v in_o like_a manner_n cut_v into_o two_o equal_a part_n therefore_o the_o right_a line_n thus_o half_v in_o the_o diameter_n of_o the_o rectangle_n but_o that_o the_o middle_n of_o the_o circle_n be_v the_o centre_n be_v manife_a out_o of_o the_o 7_o e_o v_o and_o 29_o e_fw-la iiij_o euclid_n think_v better_a of_o impossibile_fw-it than_o he_o do_v of_o the_o cause_n and_o thus_o he_o force_v it_o for_o if_o y_o be_v not_o the_o centre_n but_o s_o the_o part_n must_v be_v equal_a to_o the_o whole_a for_o the_o triangle_n a_o o_o s_o shall_v be_v equilater_n to_o the_o triangle_n e_o o_fw-fr s._n for_o a_o o_o oe_o be_v equal_a by_o the_o grant_n item_n be_v a_o and_o s_z e_z be_v the_o ray_n of_o the_o circle_n and_o s_o o_o be_v common_a to_o both_o the_o triangle_n therefore_o by_o the_o 1_o e_fw-la seven_o the_o angle_v no_o each_o side_n at_o o_o be_v equal_a and_o by_o the_o 13_o e_o v_o they_o be_v both_o right_a angle_n therefore_o s_o o_o e_o be_v a_o right_a angle_n it_o be_v therefore_o equal_a by_o the_o grant_n to_o the_o right_a angle_n y_fw-fr o_fw-fr e_fw-es that_o be_v the_o part_n be_v equal_a to_o the_o whole_a which_o be_v impossible_a wherefore_o y_o be_v not_o the_o centre_n the_o same_o will_v fall_v out_o of_o any_o other_o point_n whatsoever_o ●ut_v of_o y._n therefore_o 8._o if_o two_o r●ght_a line_n do_v perpendicular_o half_a two_o inscript_n the_o meeting_n of_o these_o two_o bisecant_v shall_v be_v the_o centre_n of_o the_o circle_n è_fw-mi 25_o p_o iij._o and_o one_o may_n 9_o draw_v a_o periphery_a by_o three_o point_n which_o do_v not_o fall_v in_o a_o right_a line_n 10._o if_o a_o diameter_n do_v half_o a_o inscript_n that_o be_v n●t_v a_o diameter_n it_o do_v cut_v it_o perpendicular_o and_o contrariwise_o 3_o p_o iij._o 11._o if_o inscript_n which_o be_v not_o diameter_n do_v cut_v one_o another_o the_o segment_n shall_v be_v unequal_a 4_o p_o iij._o but_o rate_n have_v be_v hitherto_o in_o the_o part_n of_o inscript_n proportion_n in_o the_o same_o part_n follow_v 12_o if_o two_o inscript_n do_v cut_v one_o another_o the_o rectangle_n of_o the_o segment_n of_o the_o one_o be_v equal_a to_o the_o rectangle_n of_o the_o segment_n of_o the_o other_o 35_o p_o iij._o and_o this_o be_v the_o comparison_n of_o the_o part_n inscript_n the_o rate_n of_o whole_a inscript_n do_v follow_v the_o which_o whole_a one_o diameter_n do_v make_v 13_o inscript_n be_v equal_a distant_a from_o the_o centre_n unto_o which_o the_o perpendicular_o from_o the_o centre_n be_v equal_a 4_o d_o iij._o 14._o if_o inscript_n be_v equal_a they_o be_v equal_o distant_a from_o the_o centre_n and_o contrariwise_o 13_o p_o iij._o the_o diameter_n in_o the_o same_o circle_n by_o the_o 28_o e_o iiij●_n be_v equal_a and_o they_o be_v equal_o distant_a from_o the_o centre_n see_v they_o be_v by_o the_o centre_n or_o rather_o be_v no_o whit_n at_o all_o

be_v continual_a hitherto_o it_o have_v be_v prove_v that_o the_o quinquangle_v make_v be_v a_o equilater_n and_o plain_a it_o remain_v that_o it_o be_v prove_v to_o be_v equiangle_v let_v therefore_o the_o right_a line_n e_o p_o and_o e_o c_o be_v draw_v i_o say_v that_o the_o angle_n p_o b_o e_o and_o e_o z_o i_o be_v equal_a because_o they_o have_v by_o the_o construction_n the_o base_n of_o equal_a shank_n equal_a be_v to_o wit_n in_o value_n the_o quadruple_a of_o l_o e._n for_o the_o right_a line_n l_o f_o cut_v proportional_o and_o increase_v with_o the_o great_a segment_n d_o f_o that_o be_v f_o c_o be_v cut_v also_o proportional_o by_o the_o 4_o e_fw-la fourteen_o and_o by_o the_o 7_o e_fw-la fourteen_o the_o whole_a line_n proportional_o cut_v and_o the_o lesser_a segment_n that_o be_v c_o p_o be_v of_o treble_a value_n to_o the_o great_a f_o l_o that_o be_v of_o the_o say_v l_o e._n therefore_o e_o l_o and_o l_o c_o that_o be_v e_o c_o and_o c_o p_o that_o be_v e_o p_o be_v of_o quadruple_a power_n to_o e_o l_o and_o therefore_o by_o the_o 14_o e_fw-la xij_o it_o be_v the_o double_a of_o it_o and_o e_fw-it i_fw-it itself_o in_o like_a manner_n by_o the_o fabric_n or_o construction_n be_v the_o double_a of_o the_o same_o therefore_o the_o base_n be_v equal_a and_o after_o the_o same_o manner_n by_o draw_v the_o right_a line_n i_o d_o and_o i_z b_o the_o three_o angle_n b_o p_o i_o shall_v be_v conclude_v to_o be_v equal_a to_o the_o angle_n e_o z_o i._n therefore_o by_o the_o 13_o e_fw-la fourteen_o five_o angle_n be_v equal_a 23._o the_o diagony_n be_v irrational_a unto_o the_o side_n of_o the_o dodecahedrum_fw-la this_o be_v the_o five_o example_n of_o irrationality_n and_o incommensurability_n the_o first_o be_v of_o the_o diagony_n and_o side_n of_o a_o quadrate_n or_o square_n the_o second_o be_v of_o a_o line_n proportional_o cut_v and_o his_o segment_n the_o three_o be_v of_o the_o diameter_n of_o a_o circle_n and_o the_o side_n of_o a_o inscribe_v quinquangle_v the_o four_o be_v of_o the_o diagony_n and_o side_n of_o a_o icosahedrum_fw-la the_o five_o now_o be_v of_o the_o diagony_n and_o side_n of_o a_o dodecahedrum_fw-la 24_o if_o the_o side_n of_o a_o cube_fw-la be_v cut_v proportional_o the_o great_a segment_n shall_v be_v the_o side_n of_o a_o dodecahedrum_fw-la the_o semidiagony_n and_o ray_n of_o the_o circle_n thus_o find_v the_o altitude_n remain_v take_v out_o therefore_o the_o quadrate_n of_o the_o ray_n of_o the_o circle_n 16_o 4_o 225_o out_o of_o the_o quadrate_n of_o the_o semidiagony_n 47._o 12458_o 17161._o the_o side_n of_o the_o remainder_n 3●_n 2●14406_n 3861225_o be_v for_o the_o altitude_n or_o height_n who_o ⅓_n be_v 5_o 3._o the_o quinquangled_a base_n be_v almost_o 38._o which_o multiply_v by_o 5_o 3_o do_v make_v 63_o ⅓_n for_o the_o solidity_n of_o one_o pyramid_n which_o multiply_v by_o 12_o do_v make_v 760._o for_o the_o solidity_n of_o the_o whole_a dodetacedrum_fw-la 25_o there_o be_v but_o five_o ordinate_a solid_a plain_n this_o appear_v plain_o out_o of_o the_o nature_n of_o a_o solid_a angle_n by_o the_o kind_n of_o plain_a figure_n of_o two_o plain_a angle_n a_o solid_a angle_n can_v be_v comprehend_v of_o three_o angle_n of_o a_o ordinate_a triangle_n be_v the_o angle_n of_o a_o tetrahedrum_fw-la comprehend_v of_o four_o a_o octahedrum_fw-la of_o five_o a_o icosahedrum_fw-la of_o six_o none_o can_v be_v comprehend_v for_o six_o such_o like_a plain_a angle_n be_v equal_a to_o 12_o three_o of_o one_o right_a angle_n that_o be_v to_o four_o right_a angle_n but_o plain_a angle_n make_v a_o solid_a angle_n be_v lesser_a than_o four_o right_a angle_n by_o the_o 8_o e_fw-la xxij_o of_o seven_o therefore_o and_o of_o more_o it_o be_v much_o less_o possible_a of_o three_o quadrate_n angle_v the_o angle_n of_o a_o cube_fw-la be_v comprehend_v of_o 4._o such_o angle_v none_o may_v be_v comprehend_v for_o the_o same_o cause_n of_o three_o angle_n of_o a_o ordinate_a quinquangle_n be_v make_v the_o angle_n of_o a_o dodecahedrum_fw-la of_o 4._o none_o may_v possible_o be_v make_v for_o every_o such_o angle_n for_o every_o one_o of_o they_o several_o do_v countervail_v one_o right_a angle_n and_o 1_o 5_o of_o the_o same_o therefore_o they_o will_v be_v four_o and_o three_o fifth_n of_o more_o therefore_o much_o less_o may_v it_o be_v possible_a this_o demonstration_n do_v indeed_o very_o accurate_o and_o manifest_o appear_v although_o there_o may_v be_v a_o innumerable_a sort_n of_o ordinate_a plain_n yet_o of_o the_o kind_n of_o angle_n five_o only_a ordinate_a body_n may_v be_v make_v from_o whence_o the_o tetrahedrum_fw-la octahedrum_fw-la and_o icosahedrum_fw-la be_v make_v upon_o a_o triangular_a base_a the_o cube_n upon_o a_o quadrangular_a and_o the_o dodecahedrum_fw-la upon_o a_o quinquangular_a of_o geometry_n the_o twenty_o six_o book_n of_o a_o sphere_n 1_o a_o emboss_a solid_a be_v that_o which_o be_v comprehend_v of_o a_o emboss_a surface_n 2._o and_o it_o be_v either_o a_o sphere_n or_o a_o mingle_a form_n 3._o a_o sphere_n be_v a_o round_a imbossement_fw-fr therefore_o 4._o a_o sphere_n be_v make_v by_o the_o conversion_n of_o a_o semicircle_n the_o diameter_n stand_v still_o 14_o d_o xj_o as_o here_o thou_o see_v 5._o the_o great_a circle_n of_o a_o sphere_n be_v that_o which_o cut_v the_o sphere_n into_o two_o equal_a part_n therefore_o 6._o that_o circle_n which_o be_v near_a to_o the_o great_a be_v great_a than_o that_o which_o be_v far_o off_o and_o 7._o those_o which_o be_v equal_o distant_a from_o the_o great_a be_v equal_a as_o in_o the_o example_n above_o write_v 8._o the_o plain_a of_o the_o diameter_n and_o six_o part_n of_o the_o sphearicall_a be_v the_o solidity_n of_o the_o sphere_n therefore_o 9_o as_o 21_o be_v unto_o 11_o so_o be_v the_o cube_fw-la of_o the_o diameter_n unto_o the_o sphere_n as_o here_o the_o cube_n of_o 14_o be_v 2744._o for_o it_o be_v a_o easy_a matter_n for_o he_o that_o will_v compare_v the_o cube_fw-la 2744_o with_o the_o sphere_n to_o find_v that_o 2744_o to_o be_v to_o 1437_o ⅓_n in_o the_o least_o bound_n of_o the_o same_o reason_n as_o 21_o be_v unto_o 11._o thus_o much_o therefore_o of_o the_o geodesy_n of_o the_o sphere_n the_o geodesy_n of_o the_o setour_n and_o section_n of_o the_o sphere_n shall_v follow_v in_o the_o next_o place_n and_o 10._o the_o plain_a of_o the_o ray_n and_o of_o the_o six_o part_n of_o the_o sphearicall_a be_v the_o hemisphere_n but_o it_o be_v more_o accurate_a and_o preciser_n cause_n to_o take_v the_o half_a of_o the_o sphere_n 11._o sphere_n have_v a_o treble_a reason_n of_o their_o diameter_n so_o before_o it_o be_v tell_v you_o that_o circle_n be_v one_o to_o another_o as_o the_o square_n of_o their_o diameter_n be_v one_o to_o another_o because_o they_o be_v like_o plain_n and_o the_o diameter_n in_o circle_n be_v as_o now_o they_o be_v in_o sphere_n the_o homologall_a side_n therefore_o see_v that_o sphere_n be_v figure_n alike_o and_o of_o treble_a dimension_n they_o have_v a_o treble_a reason_n of_o their_o diameter_n 12_o the_o five_o ordinate_a body_n be_v inscribe_v into_o the_o same_o sphere_n by_o the_o conversion_n of_o a_o semicircle_n have_v for_o the_o diameter_n in_o a_o tetrahedrum_fw-la a_o right_a line_n of_o value_n sesquialter_fw-la unto_o the_o side_n of_o the_o say_v tetrahedrum_fw-la in_o the_o other_o four_o ordinate_a body_n the_o diagony_n of_o the_o same_o ordinate_a the_o adscription_n of_o ordinate_a plainebody_n be_v unto_o a_o sphere_n as_o before_o the_o adscription_n plain_a surface_n be_v into_o a_o circle_n of_o a_o triangle_n i_o mean_v and_o ordinate_a triangulate_a as_o quadrangle_n quinquangle_v sexangle_v decangle_v and_o quindecangle_v but_o indeed_o the_o geometer_n have_v both_o inscribe_v and_o circumscribe_v those_o plain_a figure_n within_o a_o circle_n but_o these_o five_o ordinate_a body_n and_o over_o and_o above_o the_o polyhedrum_fw-la the_o stereometer_n have_v only_o inscribe_v within_o the_o sphere_n the_o polyhedrum_fw-la we_o have_v pass_v over_o and_o we_o purpose_v only_o to_o touch_v the_o other_o ordinate_a body_n 13_o out_o of_o the_o reason_n of_o the_o axletree_n of_o the_o sphearicall_a the_o side_n of_o the_o tetraedrum_n cube_fw-la octahedrum_fw-la and_o dodecahedrum_fw-la be_v find_v out_o if_o the_o same_o axis_fw-la be_v cut_v into_o two_o half_n as_o in_o you_o and_o the_o perpendicular_a u._fw-mi y_fw-mi be_v erect_v and_o y_o and_o a_o be_v knit_v together_o the_o same_z y_z a_o thus_o knit_v they_o shall_v be_v the_o side_n of_o the_o octahedrum_fw-la as_o be_v manifest_a in_o like_a manner_n by_o the_o say_v 10_o e_fw-la viij_o and_o 25_o e_fw-la iiij_o the_o side_n of_o the_o icosahedrum_fw-la be_v have_v

as_o be_v manifest_a by_o division_n the_o example_n be_v thus_o and_o 26._o if_o four_o right_a line_n be_v proportional_a between_o themselves_o like_a figure_n like_o situate_a upon_o they_o shall_v be_v also_o proportional_a between_o themselves_o and_o contrariwise_o out_o of_o the_o 22._o puj._n and_o 37._o pxj._n the_o proportion_n may_v also_o here_o in_o part_n be_v express_v by_o number_n and_o yet_o a_o continual_a be_v not_o require_v as_o it_o be_v in_o the_o former_a in_o plain_n let_v the_o first_o example_n be_v as_o follow_v the_o cause_n of_o proportional_a figure_n for_o that_o twice_o two_o figure_n have_v the_o same_o reason_n double_v in_o solid_n let_v this_o be_v the_o second_o example_n and_o yet_o here_o the_o figure_n be_v not_o proportional_a unto_o the_o right_a line_n as_o before_o figure_n of_o equal_a height_n be_v unto_o their_o base_a but_o they_o themselves_o be_v proportional_a one_o to_o another_o and_o yet_o be_v they_o not_o proportional_a in_o the_o same_o kind_n of_o proportion_n the_o cause_n also_o be_v here_o the_o same_o that_o be_v before_o to_o wit_n because_o twice_o two_o figure_n have_v the_o same_o reason_n treble_v 27._o figure_n fill_v a_o place_n be_v those_o which_o be_v any_o way_n set_v about_o the_o same_o point_n do_v leave_v no_o void_a room_n this_o be_v the_o definition_n of_o the_o ancient_a geometer_n as_o appear_v out_o of_o simplicius_n in_o his_o commentary_n upon_o the_o 8._o chapter_n of_o aristotle_n iij._o book_n of_o heaven_n which_o kind_n of_o figure_n aristotle_n in_o the_o same_o place_n deem_v to_o be_v only_o ordinate_a and_o yet_o not_o all_o of_o that_o kind●_n but_o only_o three_o among_o the_o plain_n to_o wit_n a_o triangle_n a_o quadrate_n and_o a_o sexangle_n among_o solid_n two_o the_o pyramid_n and_o the_o cube_n but_o if_o the_o fill_n of_o a_o place_n be_v judge_v by_o right_a angle_n 4._o in_o a_o plain_a and_o 8._o in_o a_o solid_a the_o oblong_a of_o plain_n and_o the_o octahedrum_fw-la of_o solid_n shall_v as_o shall_v appear_v in_o their_o place_n fill_v a_o place_n and_o yet_o be_v not_o this_o geometry_n of_o aristotle_n accurate_a enough_o but_o right_a angle_n do_v determine_v this_o sentence_n and_o so_o do_v euclid_n out_o of_o the_o angle_n demonstrate_v that_o there_o be_v only_o five_o ordinate_a solid_n and_o so_o do_v potamon_n the_o geometer_n as_o simplicus_n testify_v demonstrate_v the_o question_n of_o figure_n fill_v a_o place_n last_o if_o figure_n by_o lay_v of_o their_o corner_n together_o do_v make_v in_o a_o plain_a 4._o right_a angle_n or_o in_o a_o solid_a 8._o they_o do_v fill_v a_o place_n of_o this_o problem_n the_o ancient_a geometer_n have_v write_v as_o we_o hear_v even_o now_o and_o of_o the_o latter_a writer_n regiomontanus_n be_v say_v to_o have_v write_v accurate_o and_o of_o this_o argument_n maucolycus_fw-la have_v promise_v a_o treatise_n neither_o of_o which_o as_o yet_o it_o have_v be_v our_o good_a hap_n to_o see_v neither_o of_o these_o be_v figure_n of_o this_o nature_n as_o in_o their_o due_a place_n shall_v be_v prove_v and_o demonstrate_v 28._o a_o round_a figure_n be_v that_o all_o who_o ray_n be_v equal_a rotundum_fw-la a_o roundle_n let_v it_o be_v here_o use_v for_o rotunda_n figura_fw-la a_o round_a figure_n and_o in_o deed_n thomas_n finkius_n or_o finche_n as_o we_o will_v call_v he_o a_o learned_a dane_n sequester_v this_o argument_n from_o the_o rest_n of_o the_o body_n of_o geometry_n have_v entitle_v that_o his_o work_n de_fw-fr geometria_n retundi_fw-la of_o the_o geometry_n of_o the_o round_a or_o roundle_n 29._o the_o diameter_n of_o a_o roundle_n be_v cut_v in_o two_o by_o equal_a ray_n the_o reason_n be_v because_o the_o half_n of_o the_o diameter_n be_v the_o ray_n or_o because_o the_o diameter_n be_v nothing_o else_o but_o a_o double_a ray_n therefore_o if_o thou_o shall_v cut_v off_o from_o the_o diameter_n so_o much_o as_o be_v the_o radius_fw-la or_o ray_n it_o follow_v that_o so_o much_o shall_v still_o remain_v as_o thou_o have_v cut_v of_o to_o wit_n one_o ray_n which_o be_v the_o other_o half_n of_o the_o diameter_n sn._n and_o here_o observe_v that_o bisecare_v do_v here_o and_o in_o other_o place_n follow_v signify_v to_o cut_v a_o thing_n into_o two_o equal_a part_n or_o portions●_n and_o so_o bisegmentum_fw-la to_o be_v one_o such_o portion●_n and_o bisectio_fw-la such_o a_o like_a cut_n or_o division_n 30._o round_n of_o equal_a diameter_n be_v equal_a out_o of_o the_o 1._o d._n i●●_n circle_n and_o sphere_n be_v equal_a which_o have_v equal_a diameter_n for_o the_o ray_n which_o do_v measure_v the_o space_n between_o the_o centre_n and_o perimeter_n be_v equal_a of_o which_o be_v double_v the_o diameter_n do_v consist_v sn._n the_o five_o book_n of_o ramus_n his_o geometry_n which_o be_v of_o line_n and_o angle_n in_o a_o plain_a surface_n 1._o a_o lineate_v be_v either_o a_o surface_n or_o a_o body_n lineatum_n or_o lineamentum_fw-la a_o magnitude_n make_v of_o line_n as_o be_v define_v at_o 1._o e._n iij._o be_v here_o divide_v into_o two_o kind_n which_o be_v easy_o conceive_v out_o of_o the_o say_a definition_n there_o in_o which_o a_o line_n be_v exclude_v and_o a_o surface_n &_o a_o body_n be_v comprehend_v and_o from_o hence_o arise_v the_o division_n of_o the_o art_n metriall_a into_o geometry_n of_o a_o surface_n and_o stereometry_n of_o a_o body_n after_o which_o manner_n plato_n in_o his_o seven_o book_n of_o his_o commonwealth_n and_o aristotle_n in_o the_o 7._o chapter_n of_o the_o first_o book_n of_o his_o posteriorum_n do_v distinguish_v between_o geometry_n and_o stereometry_n and_o yet_o the_o name_n of_o geometry_n be_v use_v to_o signify_v the_o whole_a art_n of_o measure_v in_o general_n 2._o a_o surface_n be_v a_o lineate_v only_o broad_a 5._o dj_o epiphania_fw-la the_o greek_a word_n which_o import_v only_o the_o outter_n appearance_n of_o a_o thing_n be_v here_o more_o significant_a because_o of_o a_o magnitude_n there_o be_v nothing_o visible_a or_o to_o be_v see_v but_o the_o surface_n 3._o the_o bind_v of_o a_o surface_n be_v a_o line_n 6._o dj_o the_o matter_n in_o plain_n be_v manifest_a for_o a_o three_o corner_a surface_n be_v bound_v with_o 3._o line_n a_o four_o corner_a surface_n with_o four_o li●es_n and_o so_o forth_o a_o circle_n be_v bound_v with_o one_o line_n but_o in_o a_o sphearicall_a surface_n the_o matter_n be_v not_o so_o plain_a for_o it_o be_v whole_a seem_v not_o to_o be_v bound_v with_o a_o line_n yet_o if_o the_o manner_n of_o make_v of_o a_o sphearicall_a surface_n by_o the_o conversion_n or_o turn_v about_o of_o a_o semiperiphery_a the_o beginning_n of_o it_o as_o also_o the_o end_n shall_v be_v a_o line_n to_o wit_v a_o semiperiphery_a and_o as_o a_o point_n do_v not_o only_a actu_fw-la or_o indeed_o bind_v and_o end_v a_o line_n but_o be_v potentia_fw-la or_o in_o power_n the_o midst_n of_o it_o so_o also_o a_o line_n bound_v a_o surface_n actu_fw-la and_o a_o innumerable_a company_n of_o line_n may_v be_v take_v or_o suppose_v to_o be_v throughout_o the_o whole_a surface_n a_o surface_n therefore_o be_v make_v by_o the_o motion_n of_o a_o line_n as_o a_o line_n be_v make_v by_o the_o motion_n of_o a_o point_n 4._o a_o surface_n be_v either_o plain_a or_o bow_v the_o difference_n of_o a_o surface_n do_v answer_n to_o the_o difference_n of_o a_o line●_n in_o straightness_n and_o obliquity_n or_o crookedness_n obliquum_fw-la oblique_a there_o signify_v crooked_a not_o righ●_n or_o straight_o here_o uneven_a or_o bow_v either_o upward_a or_o downward_o sn._n 5._o a_o plain_a surface_n be_v a_o surface_n which_o lie_v ●qually_o between_o his_o bound_n out_o of_o the_o 7._o dj_o planum_fw-la a_o plain_a be_v take_v and_o use_v for_o a_o plain_a surface_n as_o before_o rotundum_n a_o round_a be_v use_v for_o a_o round_a figure_n therefore_o 6._o from_o a_o point_n unto_o a_o point_n we_o may_v in_o a_o plain_a surface_n draw_v a_o right_a line_n 1_o and_o 2._o post_n i_o three_o thing_n be_v from_o the_o former_a ground_n beg_v the_o first_o be_v of_o a_o right_a line_n a_o right_a line_n and_o a_o periphery_n be_v in_o the_o ij_o book_n define_v but_o the_o fabric_n or_o make_v of_o they_o both_o be_v here_o say_v to_o be_v proper_o in_o a_o plain_a now_o the_o geometrical_a instrument_n for_o the_o draw_v of_o a_o right_a plain_n be_v call_v amussis_fw-la &_o by_o petolemey_n in_o the_o 2._o chapter_n of_o his_o first_o book_n of_o his_o music_n regula_n a_o rular_a such_o as_o here_o thou_o see_v and_o from_o a_o point_n unto_o a_o point_n be_v this_o just_o demand_v to_o be_v do_v not_o unto_o point_n for_o neither_o do_v all_o point_n fall_v in_o a_o

whole_a shall_v be_v the_o gnomon_n of_o the_o next_o great_a quadrate_n for_o the_o side_n be_v one_o of_o the_o compliment_n and_o be_v double_v it_o be_v the_o side_n of_o both_o together_o and_o a_o unity_n be_v the_o latter_a diagonall_a so_o the_o side_n of_o 148_o be_v 12_o 4_o 25._o the_o reason_n of_o this_o depend_v on_o the_o same_o proposition_n from_o whence_o also_o the_o whole_a side_n be_v find_v for_o see_v that_o the_o side_n of_o every_o quadrate_n lesser_a than_o the_o next_o follower_n differ_v only_o from_o the_o side_n of_o the_o quadrate_n next_o above_o great_a than_o it_o but_o by_o a_o 1._o the_o same_o unity_n both_o twice_o multiply_v by_o the_o side_n of_o the_o former_a quadrate_n and_o also_o once_o by_o itself_o do_v make_v the_o gnomon_n of_o the_o great_a to_o be_v add_v to_o the_o quadrate_n for_o it_o do_v make_v the_o quadrate_n 169._o whereby_o be_v understand_v that_o look_v how_o much_o the_o numerator_n 4._o be_v short_a of_o the_o denominatour_n 25._o so_o much_o be_v the_o quadrate_n 148._o short_a of_o the_o next_o great_a quadrate_n for_o it_o thou_o do_v add_v 21._o which_o be_v the_o difference_n whereby_o 4_o be_v short_a of_o 25._o thou_o shall_v make_v the_o quadrate_n 169._o who_o side_n be_v 13._o the_o second_o be_v by_o the_o reduction_n as_o i_o say_v of_o the_o number_n give_v unto_o part_n assign_v of_o some_o great_a denomination_n as_o 100_o or_o 1000_o or_o some_o small_a than_o those_o and_o those_o quadrate_n that_o their_o true_a and_o certain_a may_v be_v know_v now_o look_v how_o much_o the_o small_a they_o be_v so_o much_o near_o to_o the_o truth_n shall_v the_o side_n find_v be_v moreover_o in_o lesser_a part_n the_o second_o way_n beside_o the_o other_o do_v show_v the_o side_n to_o be_v somewhat_o great_a than_o the_o side_n by_o the_o first_o way_n find_v as_o in_o 7._o the_o side_n by_o the_o first_o way_n be_v 3_o 25._o but_o by_o the_o second_o way_n the_o side_n of_o 7._o reduce_v unto_o thousand_o quadrate_n that_o be_v unto_o 7000000_o 1000000_o that_o be_v 2645_o 1000_o and_o beside_o there_o do_v remain_v 3975._o but_o 645_o 1000_o be_v great_a than_o 3_o 5._o for_o â…—_n reduce_v unto_o 1000_o be_v but_o 600_o 1000_o therefore_o the_o second_o way_n in_o this_o example_n do_v exceed_v the_o first_o by_o 45_o 1000_o those_o remain_v 3975._o be_v also_o neglect_v therefore_o this_o be_v the_o analysis_n or_o manner_n of_o find_v the_o side_n of_o a_o quadrate_n by_o the_o first_o rate_n of_o a_o quadrate_n equal_a to_o a_o double_a rectangle_n and_o quadrate_n the_o geodesy_n or_o measure_n of_o a_o triangle_n there_o be_v one_o general_a geodesy_n or_o way_n of_o measure_v any_o manner_n of_o triangle_n whatsoever_o in_o hero_n by_o addition_n of_o the_o side_n halve_v of_o the_o sum_n subduction_n multiplication_n and_o invention_n of_o the_o quadrate_v side_n after_o this_o manner_n 18_o if_o from_o the_o half_a of_o the_o sum_n of_o the_o side_n the_o side_n be_v several_o subduct_v the_o side_n of_o the_o quadrate_n continual_o make_v of_o the_o half_a and_o the_o remain_v shall_v be_v the_o content_a of_o the_o triangle_n this_o general_a way_n of_o measure_v a_o triangle_n be_v most_o easy_a and_o speedy_a where_o the_o side_n be_v express_v by_o whole_a number_n the_o special_a geodesy_n of_o rectangle_n triangle_n be_v before_o teach_v at_o the_o 9_o e_z x_o i_o but_o of_o a_o oblique_a angle_n it_o shall_v hereafter_o be_v speak_v but_o the_o general_a way_n be_v far_o more_o excellent_a than_o the_o special_o for_o by_o the_o reduction_n of_o a_o obliquangle_v many_o fraud_n and_o error_n do_v fall_v out_o which_o cause_v the_o learned_a cardine_n merry_o to_o wish_v that_o he_o have_v but_o as_o much_o land_n as_o be_v lose_v by_o that_o false_a kind_n of_o measure_v 19_o if_o the_o base_a of_o a_o triangle_n do_v subtend_v a_o obtuse_a angle_n the_o power_n of_o it_o be_v more_o than_o the_o power_n of_o the_o shank_n by_o a_o double_a right_a angle_n of_o the_o one_o and_o of_o the_o continuation_n from_o the_o say_v obtusangle_v unto_o the_o perpendicular_a of_o the_o top_n 12._o p_o ij_o or_o thus_o if_o the_o base_a of_o a_o triangle_n do_v subtend_v a_o obtuse_a angle_n it_o be_v in_o power_n more_o than_o the_o foot_n by_o the_o right_a angle_a figure_n twice_o take_v which_o be_v contain_v under_o one_o of_o the_o foot_n and_o the_o line_n continue_v from_o the_o say_a foot_n unto_o the_o perpendicular_a draw_v from_o the_o top_n of_o the_o triangle_n h._n there_o be_v a_o comparison_n of_o a_o quadrate_n with_o two_o in_o like_a manner_n triangle_n and_o as_o many_o quadrate_n but_o of_o unequality_n for_o by_o 9_o e_fw-la the_o quadrate_n of_o a_o i_o be_v equal_a to_o the_o quadrate_n of_o a_o o_o and_o o_o ay_o that_o be_v to_o three_o quadrate_n of_o i_o o_o o_o e_o e_o a_o and_o the_o double_a rectangle_n aforesaid_a but_o the_o quadrate_n of_o the_o shank_n a_o e_o e_o i_o be_v equal_a to_o those_o three_o quadrate_n to_o wit_n of_o a_o i_o his_o own_o quadrate_n and_o of_o e_o i_o two_o the_o first_o i_o o_o the_o second_o o_o e_o by_o the_o 9_o e._n therefore_o the_o excess_n remain_v of_o a_o double_a rectangle_n of_o geometry_n the_o thirteen_o book_n of_o a_o oblong_a 1_o a_o oblong_o be_v a_o rectangle_n of_o inequal_a side_n 31._o d_o i_o this_o second_o kind_n of_o rectangle_n be_v of_o euclid_n in_o his_o element_n proper_o name_v for_o a_o definition_n sake_n only_o the_o rate_n of_o oblong_n be_v very_o copious_a out_o of_o a_o threefold_a section_n of_o a_o right_a line_n give_v sometime_o rational_a and_o expresable_a by_o a_o number_n the_o first_o section_n be_v as_o you_o please_v that_o be_v into_o two_o segment_n equal_a or_o unequal_a from_o whence_o a_o fivefold_a rate_n arise_v 2_o a_o oblong_o make_v of_o a_o whole_a line_n give_v and_o of_o one_o segment_n of_o the_o same_o be_v equal_a to_o a_o rectangle_n make_v of_o both_o the_o segment_n and_o the_o square_a of_o the_o say_a segment_n 3._o p_o ij_o it_o be_v a_o consectary_n out_o of_o the_o 7_o e_fw-la xj_o for_o the_o rectangle_n of_o the_o segment_n and_o the_o quadrate_n be_v make_v of_o one_o side_n and_o of_o the_o segment_n of_o the_o other_o now_o a_o rectangle_n be_v here_o therefore_o propose_v because_o it_o may_v be_v also_o a_o quadrate_n to_o wit_n if_o the_o line_n be_v cut_v into_o two_o equal_a part_n secondary_o 3_o oblong_v make_v of_o the_o whole_a line_n give_v and_o of_o the_o segment_n be_v equal_a to_o the_o quadrate_n of_o the_o whole_a 2_o p_o ij_o this_o be_v also_o a_o consectary_n out_o of_o the_o 4._o e_fw-la xj_o here_o the_o segment_n be_v more_o than_o two_o and_o yet_o notwithstanding_o from_o the_o first_o the_o rest_n may_v be_v take_v for_o one_o see_v that_o the_o particular_a rectangle_n in_o like_a manner_n be_v equal_a to_o they_o this_o proposition_n be_v use_v in_o the_o demonstration_n of_o the_o 9_o e_fw-la xviij_o three_o 4_o two_o oblong_v make_v of_o the_o whole_a line_n give_v and_o of_o the_o one_o segment_n with_o the_o three_o quadrate_n of_o the_o other_o segment_n be_v equal_a to_o the_o quadrate_n of_o the_o whole_a and_o of_o the_o say_a segment_n 7_o p_o ij_o 5_o the_o base_a of_o a_o acute_a triangle_n be_v of_o less_o power_n than_o the_o shank_n be_v by_o a_o double_a oblong_o make_v of_o one_o of_o the_o shank_n and_o the_o one_o segment_n of_o the_o same_o from_o the_o say_a angle_n unto_o the_o perpendicular_a of_o the_o top_n 13._o p.ij._n and_o from_o hence_o be_v have_v the_o segment_n of_o the_o shank_n towards_o the_o angle_n and_o by_o that_o the_o perpendicular_a in_o a_o triangle_n therefore_o 6._o if_o the_o square_a of_o the_o base_a of_o a_o acute_a angle_n be_v take_v out_o of_o the_o square_n of_o the_o shank_n the_o quotient_a of_o the_o half_a of_o the_o remain_n divide_v by_o the_o shank_n shall_v be_v the_o segment_n of_o the_o divide_v shank_n from_o the_o say_a angle_n unto_o the_o perpendicular_a of_o the_o top_n now_o again_o from_o 169_o the_o quadrate_n of_o the_o base_a 13_o take_v 25_o the_o quadrate_n of_o 5_o the_o say_a segment_n and_o the_o remain_n shall_v be_v 144_o for_o the_o quadrate_n of_o the_o perpendicular_a a_o o_o by_o the_o 9_o e_fw-la x_o ij_o here_o the_o perpendicular_a now_o find_v and_o the_o side_n cut_v be_v the_o side_n of_o the_o rectangle_n who_o half_a shall_v be_v the_o content_a of_o the_o triangle_n as_o here_o the_o rectangle_n of_o 21_o and_o 12_o be_v 252_o who_o half_o 126_o be_v the_o content_a of_o the_o triangle_n the_o second_o section_n follow_v from_o whence_o

unto_o the_o remainder_n which_o be_v thus_o find_v 5._o if_o a_o right_a line_n be_v cut_v proportional_o the_o base_a of_o that_o triangle_n who_o shank_n shall_v be_v equal_a to_o the_o whole_a line_n cut_v and_o the_o base_a to_o the_o great_a segment_n of_o the_o same_o shall_v have_v each_o of_o the_o angle_n at_o the_o base_a double_a to_o the_o remainder_n and_o the_o base_a shall_v be_v the_o side_n of_o the_o quinquangle_v inscribe_v with_o the_o triangle_n into_o a_o circle_n 10_o and_o 11._o p_o i_o i_o i_o i_o 6_o if_o two_o right_a line_n do_v subtend_v on_o each_o side_n two_o angle_n of_o a_o inscript_a quinquangle_n they_o be_v cut_v proportional_o and_o the_o great_a segment_n be_v the_o side_n of_o the_o say_a inscript_n è_fw-mi 8_o p_o x_o iij._o and_o from_o hence_o the_o fabric_n or_o construction_n of_o a_o ordinate_a quinquangle_n upon_o a_o right_a line_n give_v be_v manifest_a therefore_o 7_o if_o a_o right_a line_n give_v cut_v proportional_a be_v continue_v at_o each_o end_n with_o the_o great_a segment_n and_o six_o periphery_n at_o the_o distance_n of_o the_o line_n give_v shall_v meet_v two_o on_o each_o side_n from_o the_o end_n of_o the_o line_n give_v and_o the_o continue_a two_o other_o from_o their_o meeting_n right_a line_n draw_v from_o their_o meeting_n &_o the_o end_n of_o the_o assign_a shall_v make_v a_o ordinate_a quinquangle_n upon_o the_o assign_a 8_o if_o the_o diameter_n of_o a_o circle_n circumscribe_v about_o a_o quinquangle_n be_v rational_a it_o be_v irrational_a unto_o the_o side_n of_o the_o inscribe_v quinquangle_n è_fw-it 11._o p_o xiij_o so_o before_o the_o segment_n of_o a_o right_a line_n proportional_o cut_v be_v irrational_a the_o other_o triangulate_v hereafter_o multiply_v from_o the_o ternary_a quaternary_a or_o quinary_a of_o the_o side_n may_v be_v inscribe_v into_o a_o circle_n by_o a_o inscript_a triangle_n quadrate_n or_o quinquangle_v therefore_o by_o a_o triangle_n there_o may_v be_v inscribe_v a_o triangulate_a of_o 6._o 12,24,46_o angle_n by_o a_o quadrate_n a_o triangulate_a of_o 8._o 16,32,64_o angle_n by_o a_o quinquangle_n a_o triangulate_a of_o 10_o 20._o 40,80_o angle_n etc._n etc._n 9_o the_o ray_n of_o a_o circle_n be_v the_o side_n of_o the_o inscript_n sexangle_v è_fw-mi 15_o p_o iiij_o therefore_o 10_o three_o ordinate_a sexangle_n do_v fill_v up_o a_o place_n furthermore_o also_o no_o one_o figure_n among_o the_o plain_n do_v fill_v up_o a_o place_n a_o quinquangle_n do_v not_o for_o three_o angle_n a_o quinquangle_n may_v make_v only_o 3_o ●_o 5_o angle_n which_o be_v too_o little_a and_o four_o will_v make_v 4_o ●_o 5._o which_o be_v as_o much_o too_o great_a the_o angle_n of_o a_o septangle_n will_v make_v only_o two_o rightangle_v and_o 6_o 7_o of_o one_o three_o will_v make_v 3_o and_o 9_o 7_o that_o be_v in_o the_o whole_a 4._o 2_o 7_o which_o be_v too_o much_o etc._n etc._n to_o he_o that_o by_o induction_n shall_v thus_o make_v trial_n it_o will_v appear_v that_o a_o plain_a place_n may_v be_v fill_v up_o by_o three_o sort_n of_o ordinate_a plain_n only_o and_o 11_o if_o right_a line_n from_o one_o angle_n of_o a_o inscript_n sexangle_v unto_o the_o three_o angle_n on_o each_o side_n be_v knit_v together_o they_o shall_v inscribe_v a_o equilater_n triangle_n into_o the_o circle_n give_v 12_o the_o side_n of_o a_o inscribe_v equilater_n triangle_n have_v a_o treble_a power_n unto_o the_o ray_n of_o the_o circle_n 12._o p_o xiij_o 13_o if_o the_o side_n of_o a_o sexangle_n be_v cut_v proportional_o the_o great_a segment_n shall_v be_v the_o side_n of_o the_o decangle_n therefore_o 14_o if_o a_o decangle_n and_o a_o sexangle_v be_v inscribe_v in_o the_o same_o circle_n a_o right_a line_n continue_v and_o make_v of_o both_o side_n shall_v be_v cut_v proportional_o and_o the_o great_a segment_n shall_v be_v the_o side_n of_o a_o sexangle_n and_o if_o the_o great_a segment_n of_o a_o right_a line_n cut_v proportional_o be_v the_o side_n of_o a_o hexagon_n the_o rest_n shall_v be_v the_o side_n of_o a_o decagon_n 9_o p_o xiij_o the_o comparison_n of_o the_o decangle_n and_o sexangle_v with_o the_o quinangle_n follow_v 15_o if_o a_o decangle_n a_o sexangle_n and_o a_o pentangle_v be_v inscribe_v into_o the_o same_o circle_n the_o side_n of_o the_o pentangle_v shall_v in_o power_n countervail_v the_o side_n of_o the_o other_o and_o if_o a_o right_a line_n inscribe_v do_v countervail_v the_o side_n of_o the_o sexangle_n and_o decangle_v it_o be_v the_o side_n of_o the_o pentangle_v 10._o p_o fourteen_o let_v the_o proportion_n of_o this_o syllogism_n be_v demonstrate_v for_o this_o part_n only_o remain_v doubtful_a therefore_o two_o triangle_n a_o e_o i_o and_o y_fw-fr e_fw-it i_fw-it be_v equiangle_n have_v one_o common_a angle_n at_o e_o and_o also_o two_o equal_a one_o a_o e_o i_o and_o e_z i_z y_z the_o half_n to_o wit_n of_o the_o same_o e_o i_o s_o because_o that_o be_v by_o the_o 17_o e_fw-la uj_o one_o of_o the_o two_o equal_n unto_o the_o which_o e_o ay_o s_o the_o out_z angle_n be_v equal_a by_o the_o 15_o e._n uj._o and_o this_o do_v insist_v upon_o a_o half_a periphery_n for_o the_o half_a periphery_a a_o l_o s_o be_v equal_a to_o the_o half_a periphery_a a_o r_o s_o and_o also_o a_o l_o be_v equal_a to_o a_o r._n therefore_o the_o remnant_n l_o s_o be_v equal_a to_o the_o remnant_n r_o s_o and_o the_o whole_a r_o l_o be_v the_o double_a of_o the_o same_o r_o s_o and_o therefore_o e_o r_o be_v the_o double_a of_o e_o o_o and_o r_o s_o the_o double_a of_o o_o u._fw-mi for_o the_o bisegment_n be_v manifest_a by_o the_o 10_o e_z xv_o and_o the_o 11_o e_z xuj_o therefore_o the_o periphery_n e_o r_o s_o be_v the_o double_a of_o the_o periphery_n e_o o_fw-fr u._fw-mi and_o therefore_o the_o angle_n e_fw-it i_fw-it u._fw-mi be_v the_o half_a of_o the_o angle_n e_o i_o s_o by_z the_o 7_o e_z xuj_o therefore_o two_o angle_n of_o two_o triangle_n be_v equal_a wherefore_o the_o remainder_n by_o the_o 4_o e_fw-la seven_o be_v equal_a to_o the_o remainder_n wherefore_o by_o the_o 12_o e_z seven_o as_o the_o side_n a_o e_o be_v to_z e_o i_o so_o be_v e_z i_z to_z e_o y._n therefore_o by_o the_o 8_o e_fw-la xij_o the_o oblong_a of_o the_o extreme_n be_v equal_a to_o the_o quadrate_n of_o the_o mean_a now_o let_v o_o y_fw-es be_v knit_v together_o with_o a_o straight_o here_o again_o the_o two_o triangle_n a_o o_o e_o and_o a_o o_o y_fw-fr be_v equiangle_n have_v one_o common_a angle_n at_o a_o and_o a_o o_o y_fw-fr and_o o_z e_z a_o therefore_o also_o equal_a because_o both_o be_v equal_a to_o the_o angle_n at_o a_o that_o by_o the_o 17_o e_fw-la uj_o this_o by_o the_o 2_o e_z seven_o because_o the_o perpendicular_a half_v the_o side_n of_o the_o decangle_n do_v make_v two_o triangle_n equicrural_a and_o equal_a by_o the_o right_a angle_n of_o their_o shank_n and_o therefore_o they_o be_v equiangle_n therefore_o as_o e_z a_o be_v to_o a_o o_o so_o be_v e_z a_o to_o a_o y._n wherefore_o by_o the_o 8_o e_z xij_o the_o oblong_a of_o the_o two_o extreme_n be_v equal_a to_o the_o quadrate_n of_o the_o mean_a and_o the_o proposition_n of_o the_o syllogism_n which_o be_v to_o be_v demonstrate_v the_o converse_n from_o hence_o as_o manifest_v euclid_n do_v use_v at_o the_o 16_o p_o xiij_o 16._o if_o a_o triangle_n and_o a_o quinquangle_v be_v inscribe_v into_o the_o same_o circle_n at_o the_o same_o point_n the_o right_a line_n inscribe_v between_o the_o base_n of_o the_o both_o opposite_a to_o the_o say_a point_n shall_v be_v the_o side_n of_o the_o inscribe_v quindecangle_n 16._o p._n iiij_o therefore_o 17._o if_o a_o quinquangle_n and_o a_o sexangle_v be_v inscribe_v into_o the_o same_o circle_n at_o the_o same_o point_n the_o periphery_a intercept_v between_o both_o their_o side_n shall_v be_v the_o thirty_o part_n of_o the_o whole_a periphery_n of_o geometry_n the_o ninteenth_fw-mi book_n of_o the_o measure_v of_o ordinate_a multangle_n and_o of_o a_o circle_n out_o of_o the_o adscription_n of_o a_o circle_n and_o a_o rectilineall_a be_v draw_v the_o geodesy_n of_o ordinate_a multangle_v and_o first_o of_o the_o circle_n itself_o for_o the_o meeting_n of_o two_o right_a line_n equal_o divide_v two_o angle_n be_v the_o centre_n of_o the_o circumscribe_v circle_n from_o the_o centre_n unto_o the_o angle_n be_v the_o ray_n and_o then_o if_o the_o quadrate_n of_o half_a the_o side_n be_v take_v out_o of_o the_o quadrate_n of_o the_o ray_n the_o side_n of_o the_o remainder_n shall_v be_v the_o perpendicular_a by_o the_o 9_o e_fw-la xij_o therefore_o a_o special_a theorem_a be_v here_o thus_o make_v 1._o a_o plain_a make_v of_o the_o

great_a than_o the_o base_a i_o u._n therefore_o by_o the_o 5_o e_fw-la seven_o the_o angle_n o_o e_o i_o be_v great_a than_o the_o angle_n i_o e_o u._fw-mi therefore_o two_o angle_n a_o e_o o_o and_o o_o e_fw-it i_fw-it be_v great_a than_o a_o e_o i._n 10_o a_o plain_a solid_a be_v a_o pyramid_n or_o a_o pyramidate_n 11_o a_o pyramid_n be_v a_o plain_a solid_a from_o a_o rectilineall_a base_a equal_o decrease_v as_o here_o thou_o conceive_v from_o the_o triangular_a base_a a_o e_o i_o unto_o the_o top_n o_o the_o triangle_n a_o o_o e_o a_o o_o ay_o and_o e_z o_o ay_o to_o be_v rear_v up_o therefore_o 12_o the_o side_n of_o a_o pyramid_n be_v one_o more_o than_o be_v the_o base_a the_o side_n be_v here_o name_v hedrae_fw-la and_o 13_o a_o pyramid_n be_v the_o first_o figure_n of_o solid_n for_o a_o pyramid_n in_o solid_n be_v as_o a_o triangle_n be_v in_o plain_n for_o a_o pyramid_n may_v be_v resolve_v into_o other_o solid_a figure_n but_o it_o can_v be_v resolve_v into_o any_o one_o more_o simple_a than_o itself_o and_o which_o consist_v of_o few_o side_n than_o it_o do_v therefore_o 14_o pyramid_n of_o equal_a height_n be_v as_o their_o base_n be_v 5_o e_z and_o 6._o p_o xij_o and_o 15_o those_o which_o be_v reciprocal_a in_o base_a and_o height_n be_v equal_a 9_o p_o xij_o these_o consectary_n be_v draw_v out_o of_o the_o 16_o 18_o e._n iiij_o 16_o a_o tetraedrum_n be_v a_o ordinate_a pyramid_n comprehend_v of_o four_o triangle_n 26._o d_o xj_o therefore_o 17_o the_o edge_n of_o a_o tetraedrum_n be_v six_o the_o plain_a angle_n twelve_o the_o solid_a angle_n four_o for_o a_o tetraedrum_n be_v comprehend_v of_o four_o triangle_n each_o of_o they_o have_v three_o side_n and_o three_o corner_n a_o piece_n and_o every_o side_n be_v twice_o take_v therefore_o the_o number_n of_o edge_n be_v but_o half_a so_o many_o and_o 18_o twelve_o tetraedra_n doe_n fill_v up_o a_o solid_a place_n because_o 8._o solid_a right_a angle_n fill_v a_o place_n and_o 12._o angle_n of_o the_o tetraedrum_n be_v equal_a between_o themselves_o see_v that_o both_o of_o they_o be_v comprehend_v of_o 24._o plain_a rightangle_n for_o a_o solid_a right_a angle_n be_v comprehend_v of_o three_o plain_a right_a angle_n and_o therefore_o 8._o be_v comprehend_v of_o 24._o in_o like_a manner_n the_o angle_n of_o a_o tetraedrum_n be_v comprehend_v of_o three_o plain_a equilater_n that_o be_v of_o six_o three_o of_o one_o right_a angle_n and_o therefore_o of_o two_o right_a angle_n therefore_o 12_o be_v comprehend_v of_o 24._o and_o 19_o if_o four_o ordinate_a and_o equal_a triangle_n be_v join_v together_o in_o solid_a angle_n they_o shall_v comprehend_v a_o tetraedrum_n 20._o if_o a_o right_a line_n who_o power_n be_v sesquialter_fw-la unto_o the_o side_n of_o a_o equilater_n triangle_n be_v cut_v after_o a_o double_a reason_n the_o double_a segment_n perpendicular_a to_o the_o centre_n of_o the_o triangle_n knit_v together_o with_o the_o angle_n thereof_o shall_v comprehend_v a_o tetraedrum_n 13_o p_o xiij_o for_o a_o solid_a to_o be_v comprehend_v of_o right_a line_n understand_v plain_n comprehend_v of_o right_a line_n as_o in_o other_o place_n follow_v the_o twenty_o three_o book_n of_o geometry_n of_o a_o prisma_fw-la 1_o a_o pyramidate_n be_v a_o plain_a solid_a comprehend_v of_o pyramid_n 2._o a_o pyramidate_n be_v a_o prisma_fw-la or_o a_o mingle_a polyedrum_fw-la 3._o a_o prisma_fw-la be_v a_o pyramidate_n who_o opposite_a plain_n be_v equal_a alike_o and_o parallel_v the_o rest_n parallelogramme_n 13_o dxj_o therefore_o 4._o the_o flatte_n of_o a_o prisma_fw-la be_v two_o more_o than_o be_v the_o angle_n in_o the_o base_a and_o indeed_o as_o the_o augmentation_n of_o a_o pyramid_n from_o a_o quaternary_a be_v infinite_a so_o be_v it_o of_o a_o prisma_fw-la from_o a_o quinary_a as_o if_o it_o be_v from_o a_o triangular_a quadrangular_a or_o quinquangular_a base_a you_o shall_v have_v a_o pentaedrum_fw-la hexaedrum_n heptaedrum_fw-la and_o so_o in_o infinite_a 5._o the_o plain_a of_o the_o base_a and_o height_n be_v the_o solidity_n of_o a_o right_a prisma_fw-la 6._o a_o prisma_fw-la be_v the_o triple_a of_o a_o pyramid_n of_o equal_a base_a and_o height_n è_fw-it 7_o p._n x_o i_o i_o if_o the_o base_a be_v triangular_a the_o prisma_fw-la may_v be_v resolve_v into_o prisma_n of_o triangular_a base_n and_o the_o theorem_a shall_v be_v conclude_v as_o afore_o therefore_o 7._o the_o plain_a make_v of_o the_o base_a and_o the_o three_o part_n of_o the_o height_n be_v the_o solidity_n of_o a_o pyramid_n of_o equal_a base_a and_o height_n so_o in_o the_o example_n follow_v let_v 36_o the_o quadrate_n of_o 6_o the_o ray_n be_v take_v out_o of_o 292_o 9_o 1156_o the_o quadrate_n of_o the_o side_n 17_o 3_o 34_o the_o side_n 16_o 3_o 34_o of_o 256_o 9_o 1156_o the_o remainder_n shall_v be_v the_o height_n who_o three_o part_n be_v 5_o 37_o 102_o the_o plain_a of_o which_o by_o the_o base_a 72_o ¼_n shall_v be_v 387_o 11_o 24_o for_o the_o solidity_n of_o the_o pyramid_n give_v after_o this_o manner_n you_o may_v measure_v a_o imperfect_a prisma_fw-la 8._o homogeneal_a prisma_n of_o equal_a height_n be_v one_o to_o another_o as_o their_o base_n be_v one_o to_o another_o 29_o 30,31_o 32_o p_o xj_o this_o element_n be_v a_o consectary_n out_o of_o the_o 16_o e_fw-la iiij_o and_o 9_o if_o they_o be_v reciprocal_a in_o base_a and_o height_n they_o be_v equal_a this_o be_v a_o consectary_n out_o the_o 18_o e_fw-la iiij_o and_o 10._o if_o a_o prisma_fw-la be_v cut_v by_o a_o plain_a parallel_n to_o his_o opposite_a flatte_n the_o segment_n be_v as_o the_o base_n be_v 25_o p._n xj_o 11._o a_o prisma_fw-la be_v either_o a_o pentaedrum_fw-la or_o compound_v of_o pentaedra_n here_o the_o resolution_n show_v the_o composition_n 12_o if_o of_o two_o pentaedra_n the_o one_o of_o a_o triangular_a base_a the_o other_o of_o a_o parallelogramme_n base_a double_a unto_o the_o triangular_a be_v of_o equal_a height_n they_o be_v equal_a 40._o p_o xj_o the_o cause_n be_v manifest_a and_o brief_a because_o they_o be_v the_o half_n of_o the_o same_o prisma_fw-la as_o here_o thou_o may_v perccive_v in_o a_o prisma_fw-la cut_v into_o two_o half_n by_o the_o diagoni'_v of_o the_o opposite_a side_n euclid_n do_v demonstrate_v it_o thus_o let_v the_o pentaedra_n a_o e_o i_o o_o u._fw-mi and_o y_z s_o r_o l_o m_o be_v of_o equal_a height_n the_o first_o of_o a_o triangular_a base_a e_o i_o o_o the_o second_o of_o a_o parallelogramme_n base_a s_o l_o double_a unto_o the_o triangular_a now_o let_v both_o of_o they_o be_v double_a and_o make_v up_o so_o that_o first_o be_v n●_n the_o second_o y_o s_o r_o l_o v_o f._n now_o again_o by_o the_o grant_n the_o base_a s_o l_o be_v the_o double_a of_o the_o base_a e_o i_o o_o who_o double_a be_v th●_n base_a e_o o_o by_o the_o 12_o e_fw-la x._o therefore_o the_o base_n s_o l_o and_o e_o o_o be_v equal_a and_o therefore_o see_v the_o prisma_n by_o the_o grant_n here_o be_v of_o equal_a height_n as_o the_o base_n by_o the_o conclusion_n be_v equal_a the_o prisma_n be_v equal_a and_o therefore_o also_o their_o half_n a_o e_o i_o o_o u._fw-mi and_o y_z s_z n_o l_o r_o be_v equal_a the_o measure_n of_o a_o pentaedrall_n prisma_fw-la be_v even_o now_o general_o teach_v the_o matter_n in_o special_a may_v be_v conceive_v in_o these_o two_o example_n follow_v the_o plain_a of_o 18._o the_o perimeter_n of_o the_o triangular_a base_a and_o 12_o the_o height_n be_v 216._o this_o add_v to_o the_o triangular_a base_a 15_o 18_o 3●_n or_o 15_o ⅗_n almost_o twice_o take_v that_o be_v 31_o ⅕_n do_v make_v 247_o ⅕_n for_o the_o sum_n of_o the_o whole_a surface_n but_o the_o plain_a of_o the_o same_o base_a 15_o ⅖_n and_o the_o height_n 12._o be_v 187_o ⅕_n for_o the_o whole_a solidity_n so_o in_o the_o pentaedrum_fw-la the_o second_o prisma_fw-la which_o be_v call_v cuneus_n a_o wedge_n of_o the_o sharpness_n and_o which_o also_o more_o proper_o of_o cut_v be_v call_v a_o prisma_fw-la the_o whole_a surface_n be_v 150_o and_o the_o solidity_n 90._o 13_o a_o prisma_fw-la compound_v of_o penta●dra's_n be_v either_o a_o hexaedrum_n or_o polyedrum_fw-la and_o the_o hexaedrum_n be_v either_o a_o parallelepipedum_fw-la or_o a_o trapezium_fw-la 14_o a_o parallelepipedum_fw-la be_v that_o who_o opposite_a plain_n be_v parallelogramme_n ê_a 24._o p_o xj_o therefore_o a_o parallelepipedum_fw-la in_o solid_n answer_v to_o a_o parallelogramme_n in_o plain_n for_o here_o the_o opposite_a hedrae_fw-la or_o flatte_n be_v parallel_v there_o the_o opposite_a side_n be_v parallel_v therefore_o 15_o it_o be_v cut_v into_o two_o half_n with_o a_o plain_a by_o the_o diagony_n of_o the_o opposite_a

side_n 28_o p_o xj_o it_o answer_v to_o the_o 34._o pj._n and_o 16_o if_o it_o be_v half_v by_o two_o plain_n half_v the_o opposite_a side_n the_o common_a bisection_n and_o diagony_n do_v half_a one_o another_o 39_o p_o xj_o 17_o if_o three_o line_n be_v proportional_a the_o parallelepipedum_fw-la of_o mean_a shall_v be_v equal_a to_o the_o equiangle_v p●rallelepipedum_n of_o all_o they_o è_fw-it 36._o p_o x_o i_o it_o be_v a_o consectary_n out_o of_o the_o 8_o e._n 18_o eight_o rectangle_v parallelepiped_n do_v fill_v a_o solid_a place_n 19_o the_o figurate_a of_o a_o rectangle_v parallelepipedum_fw-la be_v call_v a_o solid_a make_v of_o three_o number_n 17._o d_o seven_o as_o if_o thou_o shall_v multiply_v 1,2,3_o continual_o thou_o shall_v make_v the_o solid_a 6._o item_n if_o thou_o shall_v in_o like_a manner_n multiply_v 2,3,4_o thou_o shall_v make_v the_o solid_a 24._o and_o the_o side_n of_o that_o solid_a 6_o solid_a shall_v be_v 1,2,3_o of_o 24_o they_o shall_v be_v 2,3,4_o therefore_o 20_o if_o two_o solid_n be_v alike_o they_o have_v their_o side_n proportional_o and_o two_o mean_a proportional_o 21_o d_o seven_o 19_o 21._o p_o viij_o it_o be_v a_o consectary_n out_o of_o the_o 5_o e_fw-la xxij_o but_o the_o mean_a proportional_o be_v make_v of_o the_o side_n of_o the_o like_a solid_n to_o wit_n of_o the_o second_o three_o and_o four_o item_n of_o the_o three_o four_o and_o five_o as_o here_o tho●_n see_v of_o geometry_n the_o twenty_o four_o book_n of_o a_o cube_n 1_o a_o rightangle_v parallelepipedum_fw-la be_v either_o a_o cube_n or_o a_o oblong_a 2_o a_o cube_n be_v a_o right_n angle_v parallelepipedum_fw-la of_o equal_a flatte_n 25._o d_o xj_o as_o here_o thou_o see_v in_o these_o two_o figure_n therefore_o 3_o the_o side_n of_o a_o cube_fw-la be_v 12._o the_o plain_a angle_n 24._o the_o solid_a 8._o therefore_o 4_o if_o six_o equal_a quadrate_n be_v join_v with_o solid_a angle_n they_o shall_v comprehend_v a_o cube_fw-la as_o here_o in_o these_o two_o example_n therefore_o 5_o if_o from_o the_o angle_n of_o a_o quadrate_n perpend●culars_n equal_a to_o the_o side_n be_v tie_v together_o aloft_o they_o shall_v comprehend_v a_o cube_n è_fw-it 15_o p_o xj_o it_o be_v a_o consectary_n follow_v upon_o the_o former_a consectary_n for_o then_o shall_v six_o equal_a quadrate_n be_v knit_v together_o 6_o the_o diagony_n of_o a_o cube_n be_v of_o treble_a power_n unto_o the_o side_n for_o the_o diagony_n of_o a_o quadrate_n be_v of_o double_a power_n to_o the_o side_n by_z the_o 12_o e_z xij_o and_o the_o diagony_n of_o a_o cube_n be_v of_o as_o much_o power_n as_o the_o side_n the_o diagony_n of_o the_o quadrate_n by_o the_o same_o e._n therefore_o it_o be_v of_o treble_a power_n to_o the_o side_n 7_o if_o of_o four_o right_a line_n continual_o proportional_o the_o first_o be_v the_o half_a of_o the_o four_o the_o cube_fw-la of_o the_o first_o shall_v be_v the_o half_a of_o the_o cube_n of_o the_o second_o è_fw-la 33_o p_o xj_o it_o be_v a_o consectary_n out_o of_o the_o 25_o e_fw-la iiij_o from_o hence_o hypocrates_n first_o find_v how_o to_o answer_v apollo_n problem_n 8_o the_o solid_a plain_n of_o a_o cube_fw-la be_v call_v a_o cube_n to_o wit_n a_o solid_a of_o equal_a side_n 19_o d_o seven_o therefore_o 9_o it_o be_v make_v of_o a_o number_n multiply_v into_o his_o own_o quadrate_n so_o be_v a_o cube_n make_v by_o multiply_v a_o number_n by_o itself_o and_o the_o product_n again_o by_o the_o first_o such_o be_v these_o nine_o first_o cube_n make_v of_o the_o nine_o first_o arithmetical_a figure_n this_o be_v the_o general_a invention_n of_o a_o cube_n both_o geometrical_a and_o arithmetical_a 10_o if_o a_o right_a line_n be_v cut_v into_o two_o segment_n the_o cube_n of_o the_o whole_a shall_v be_v equal_a to_o the_o cube_n of_o the_o segment_n and_o a_o double_a solid_a thrice_o comprehend_v of_o the_o quadrate_n of_o his_o own_o segment_n and_o the_o other_o segment_n as_o for_o example_n the_o side_n 12_o let_v it_o be_v cut_v into_o two_o segment_n 10_o and_o 2._o the_o cube_fw-la of_o 12._o the_o whole_a which_o be_v 1728_o shall_v be_v equal_a to_o two_o cube_n 1000_o and_o 8_o make_v of_o the_o segment_v 10._o and_o 2._o and_o a_o double_a solid_a of_o which_o the_o first_o 600._o be_v thrice_o comprehend_v of_o 100_o the_o quadrate_n of_o his_o segment_n 10._o and_o of_o 2._o the_o other_o segment_n the_o second_o 120._o be_v thrice_o comprehend_v of_o 4_o the_o quadrate_n of_o his_o own_o segment_n and_o of_o 10._o the_o other_o segment_n now_o 1000_o 600_o 120._o 8_o be_v equal_a to_o 1728_o and_o therefore_o a_o right_a etc._n etc._n therefore_o 11._o the_o side_n of_o the_o first_o several_a cube_fw-la be_v the_o other_o side_n of_o the_o second_o solid_a and_o the_o quadrate_n of_o the_o same_o side_n be_v the_o other_o side_n of_o the_o first_o solid_a who_o other_o side_n be_v the_o side_n of_o the_o second_o cube_fw-la and_o the_o quadrate_n of_o the_o same_o other_o side_n be_v the_o other_o side_n of_o the_o second_o solid_a in_o that_o equation_n therefore_o of_o four_o solid_n with_o one_o solid_a thou_o shall_v consider_v a_o peculiar_a make_n and_o composition_n first_o that_o the_o last_o cube_fw-la be_v make_v of_o the_o last_o segment_n 2d_o then_o that_o the_o second_o solid_a of_o 4_o the_o quadrate_n of_o his_o own_o segment_n and_o of_o 10_o the_o other_o segment_n be_v thrice_o comprehend_v last_o that_o the_o first_o solid_a of_o 100_o the_o square_a of_o his_o own_o segment_n 10_o and_o the_o other_o segment_n 2_o be_v also_o thrice_o comprehend_v last_o that_o the_o cube_n 1000_o be_v make_v of_o the_o great_a segment_n 10._o out_o of_o this_o make_n etc._n etc._n the_o plain_a of_o the_o perimeter_n of_o the_o base_a 20_o and_o the_o altitude_n 5_o be_v 100_o this_o add_v to_o 25_o and_o 25_o both_o the_o base_n that_o be_v to_o 50_o make_v 150_o for_o the_o whole_a surface_n now_o the_o plain_a of_o 25_o the_o base_a and_o the_o height_n 5_o be_v 125_o for_o the_o whole_a solidity_n so_o in_o the_o oblong_a the_o plain_a of_o the_o base_n perimeter_n 20_o and_o the_o height_n 11_o be_v 220_o which_o add_v to_o the_o base_n 24_o and_o 24_o that_o be_v 48_o make_v 268_o for_o the_o whole_a surface_n but_o the_o plain_a of_o the_o base_a 24_o and_o the_o height_n 11_o be_v 264_o for_o the_o solidity_n thus_o be_v such_o kind_n of_o wall_n whether_o of_o mud_n brick_n or_o stone_n of_o most_o large_a house_n to_o be_v measure_v the_o same_o manner_n of_o geodesy_n be_v also_o to_o be_v use_v in_o the_o measure_n of_o a_o rhombe_n rhomboide_n trapezium_fw-la or_o mensall_n and_o any_o kind_n of_o multangled_a body_n the_o base_a be_v first_o to_o be_v measure_v as_o in_o the_o former_a then_o out_o of_o that_o and_o the_o height_n the_o solidity_n shall_v be_v manifest_v as_o in_o the_o rhombe_n the_o base_a be_v 24_o the_o height_n 4._o therefore_o the_o solidity_n be_v 96._o in_o the_o rhomboide_n the_o base_a be_v 64_o 3●_n 12●_n the_o hey_z 11._o therefore_o the_o solidity_n be_v 1028_o 44_o 1●9_n the_o same_o be_v the_o geodesy_n of_o a_o trapezium_n as_o in_o these_o example_n the_o surface_n of_o the_o first_o be_v 198_o the_o solidity_n 192_o ½_o the_o surface_n of_o the_o second_o be_v 158_o 3_o 49_o the_o solidity_n be_v 91_o 29_o 4●_n and_o from_o hence_o also_o may_v the_o capacity_n or_o content_n of_o vessel_n or_o measure_n make_v after_o any_o manner_n of_o plain_a solid_a be_v esteem_v and_o judge_v of_o as_o here_o thou_o see_v for_o here_o the_o plain_a of_o the_o sexangular_a base_a be_v 41_o 1_o 7_o for_o the_o ray_n by_o the_o 9_o e_fw-la xviij_o be_v the_o fide_fw-la and_o the_o height_n 5_o shall_v be_v 205_o 5_o 7._o therefore_o if_o a_o cubicall_a foot_n do_v contain_v 4_o quarter_n as_o we_o common_o call_v they_o then_o shall_v the_o vessel_n contain_v 822_o 6_o 7_o quarte_n that_o be_v almost_o 823_o quarte_n of_o geometry_n the_o twenty_o five_o book_n of_o mingle_a ordinate_a polyedra's_n 1._o a_o mingle_a ordinate_a polyedrum_fw-la be_v a_o pyramidate_n compound_v of_o pyramid_n with_o their_o top_n meet_v in_o the_o centre_n and_o their_o base_n only_o outward_o appear_v see_v therefore_o a_o mingle_a ordinate_a pyramidate_n be_v thus_o make_v or_o compound_v of_o pyramid_n the_o geodesy_n of_o it_o shall_v be_v have_v from_o the_o geodesy_n of_o the_o pyramid_n compound_v it_o and_o one_o base_a multiply_v by_o the_o number_n of_o all_o the_o base_n shall_v make_v the_o surface_n of_o the_o body_n and_o one_o pyramid_n by_o the_o number_n of_o all_o the_o pyramid_n shall_v make_v the_o solidity_n 2_o the_o height_n of_o the_o