Selected quad for the lemma: book_n

Word A Word B Word C Word D Occurrence Frequency Band MI MI Band Prominent
book_n definition_n proposition_n use_v 1,977 5 10.7773 5 false
View all documents for the selected quad

Text snippets containing the quad

ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A38722 The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...; Huict livres des Eléments d'Euclide rendus plus faciles. English Dechales, Claude-François Milliet, 1621-1678.; Euclid. Elements.; Williams, Reeve, fl. 1682-1703. 1685 (1685) Wing E3399; ESTC R10241 136,603 430

There are 9 snippets containing the selected quad. | View original text

the_o element_n of_o euclid_n explain_v and_o demonstrate_v in_o a_o new_a and_o most_o easy_a method_n with_o the_o use_v of_o each_o proposition_n in_o all_o the_o part_n of_o the_o mathematics_n by_o claude_n francois_n milliet_n d'chales_n a_o jesuit_n do_v out_o of_o french_z correct_v and_o augment_v and_o illustrate_v with_o nine_o copper_n plate_n and_o the_o effigy_n of_o euclid_n by_o reeve_n williams_n philomath_n london_n print_v for_o philip_n lea_n globemaker_n at_o the_o atlas_n and_o hercules_n in_o the_o poultry_n near_o cheapside_n 1685._o to_o the_o honourable_a samuel_n pepys_n esq_n principal_a officer_n of_o the_o navy_n secretary_z to_o the_o admiralty_n and_o precedent_n of_o the_o royal_a society_n honour_a sir_n the_o countenance_n and_o encouragement_n you_o have_v always_o give_v to_o mathematical_a learning_n especial_o as_o it_o have_v a_o tendency_n to_o promote_v the_o public_a good_a have_v embolden_v i_o humble_o to_o present_v your_o honour_n with_o this_o little_a piece_v which_o have_v the_o admirable_a euclid_n for_o its_o author_n and_o the_o learned_a d'chale_n for_o the_o commentator_n the_o excellency_n of_o the_o subject_a with_o the_o apt_a and_o profitable_a application_n thereof_o in_o its_o use_v do_v first_o induce_v i_o to_o translate_v it_o for_o my_o own_o use_n the_o benefit_n and_o quicken_n in_o those_o mathematical_a study_n that_o some_o profess_v to_o have_v receive_v do_v prevail_v with_o i_o to_o make_v it_o public_a and_o the_o great_a obligation_n i_o lie_v under_o from_o the_o many_o undeserved_a favour_n of_o your_o honour_n towards_o i_o i_o think_v do_v engage_v i_o on_o this_o occasion_n to_o make_v some_o public_a testimony_n and_o acknowledgement_n thereof_o i_o therefore_o humble_o beg_v your_o honour_n patronage_n of_o this_o little_a book_n and_o your_o pardon_n for_o this_o address_n entreat_v you_o will_v be_v please_v to_o look_v upon_o it_o with_o that_o benign_a aspect_n as_o you_o have_v be_v please_v always_o to_o vouchsafe_v to_o he_o who_o be_v your_o humble_a and_o most_o oblige_a servant_n reeve_n williams_z the_o author_n preface_n to_o the_o reader_n have_v long_o since_o observe_v that_o the_o great_a part_n of_o those_o that_o learn_v euclid_n element_n be_v very_o often_o dissatisfy_v therewith_o because_o they_o know_v not_o the_o use_n of_o proposition_n so_o inconsiderable_a in_o appearance_n and_o yet_o so_o difficult_a i_o think_v it_o may_v be_v to_o good_a purpose_n not_o only_o to_o make_v they_o as_o easy_a as_o possible_a but_o also_o to_o add_v some_o use_v after_o each_o proposition_n to_o show_v how_o they_o be_v applicable_a to_o practice_n and_o this_o have_v oblige_v i_o to_o change_v some_o of_o the_o demnostration_n which_o i_o look_v upon_o to_o be_v too_o troublesome_a and_o above_o the_o usual_a reach_n of_o beginner_n and_o to_o substitute_v other_o more_o intelligible_a for_o this_o cause_n i_o have_v demonstrate_v the_o five_o book_n after_o a_o method_n much_o more_o clear_a than_o that_o that_o make_v use_v of_o equimultiple_n i_o have_v not_o give_v all_o the_o use_v of_o the_o proposition_n for_o that_o will_v have_v make_v it_o necessary_a to_o recite_v all_o the_o mathematics_n and_o will_v have_v make_v the_o book_n too_o big_a and_o too_o hard_o wherefore_o i_o have_v only_o make_v choice_n of_o some_o of_o the_o plain_a and_o easy_a to_o conceive_v i_o will_v not_o have_v you_o to_o stand_v too_o much_o upon_o they_o nor_o make_v it_o your_o study_n to_o understand_v they_o perfect_o because_o they_o depend_v upon_o other_o principle_n beside_o for_o which_o reason_n i_o have_v distinguish_v they_o with_o the_o italic_a letter_n this_o be_v the_o design_n of_o this_o small_a treatise_n which_o i_o willing_o publish_v in_o a_o time_n when_o the_o mathematics_n be_v more_o than_o ever_o study_v milliet_n d'chales_n the_o translator_n preface_n the_o reader_n have_v peruse_v the_o author_n preface_n with_o this_o far_a intimation_n that_o he_o will_v find_v the_o subject_a and_o scope_n of_o this_o work_n succinct_o and_o pertinent_o present_v to_o he_o in_o the_o argument_n before_o each_o particular_a book_n may_v i_o presume_v expect_v the_o loss_n from_o i_o and_o i_o shall_v not_o at_o all_o endeavour_n to_o bespeak_v the_o reader_n acceptance_n of_o euclid_n element_n or_o persuade_v he_o to_o believe_v the_o necessary_a and_o excellent_a vsefulness_n thereof_o because_o every_o man_n experience_n so_o far_o as_o he_o understand_v they_o be_v a_o abundant_a testimony_n thereunto_o neither_o shall_v i_o need_v to_o commend_v the_o method_n with_o the_o use_n of_o our_o author_n d'chales_n who_o be_v well_o know_v to_o the_o learned_a of_o this_o age_n by_o his_o several_a excellent_a mathematical_a tract_n etc._n tract_n cursus_fw-la mathe_n his_o navigation_n his_o local_a motion_n etc._n etc._n for_o whosoever_o shall_v be_v a_o while_n conversant_a with_o this_o book_n may_v i_o presume_v fiind_v that_o instruction_n and_o incourgment_n in_o the_o learning_n of_o euclid_n eliment_n as_o he_o have_v not_o before_o meet_v with_o in_o our_o english_a tongue_n and_o this_o as_o it_o have_v be_v my_o own_o experience_n since_o i_o translate_v it_o from_o the_o french_a for_o the_o use_n of_o my_o english_a scholar_n so_o it_o be_v one_o great_a cause_n of_o its_o come_n abroad_o into_o the_o world_n for_o such_o as_o have_v learned_a by_o it_o find_v it_o difficult_a to_o attain_v unless_o transcribe_v which_o they_o think_v tedious_a be_v a_o subject_n so_o voluminous_a in_o manuscript_n and_o full_a of_o scheme_n i_o do_v therefore_o at_o their_o request_n and_o the_o importunity_n of_o some_o friend_n condescend_v to_o the_o print_n hereof_o though_o not_o without_o much_o averseness_n to_o my_o own_o mind_n be_v unwilling_a to_o expose_v myself_o in_o any_o public_a thing_n in_o this_o nice_a critical_a age_n but_o that_o difficulty_n be_v now_o overcome_v i_o shall_v only_o give_v the_o reader_n to_o understand_v that_o i_o have_v faithful_o render_v this_o piece_n into_o english_a according_a to_o the_o sense_n of_o the_o author_n but_o here_o and_o there_o omit_v some_o small_a matter_n which_o i_o judge_v not_o so_o proper_o relate_v to_o the_o subject_n of_o this_o work_n and_o therein_o will_v make_v no_o want_n to_o the_o reader_n nor_o i_o hope_v be_v any_o offence_n to_o the_o ingenious_a author_n himself_o i_o have_v only_o one_o thing_n more_o to_o add_v and_o that_o upon_o the_o account_n of_o a_o objection_n i_o have_v meet_v with_o that_o here_o be_v not_o all_o the_o book_n of_o euclid_n and_o it_o be_v true_a here_o be_v not_o all_o here_o be_v only_o the_o first_o six_o book_n and_o the_o eleven_o and_o twelve_o the_o other_o be_v purposely_o omit_v by_o our_o learned_a author_n who_o judge_v the_o understanding_n of_o these_o to_o be_v sufficient_a for_o all_o the_o part_n of_o the_o mathematics_n 304._o mathematics_n see_v argument_n before_o the_o eleven_o book_n page_n 304._o and_o i_o can_v also_o give_v instance_n of_o other_o excellent_a author_n that_o be_v of_o his_o opinion_n and_o have_v take_v the_o like_a course_n nay_o the_o truth_n be_v some_o very_a learned_a in_o the_o mathematics_n have_v reduce_v the_o proposition_n of_o these_o book_n to_o a_o much_o lesser_a number_n and_o yet_o have_v think_v they_o a_o complete_a foundation_n to_o all_o the_o science_n mathematical_a but_o i_o shall_v not_o trouble_v my_o reader_n far_o on_o this_o account_n not_o doubt_v but_o when_o he_o have_v peruse_v and_o well_o consider_v our_o euclid_n he_o will_v have_v a_o better_a opinion_n thereof_o than_o any_o thing_n i_o can_v now_o say_v may_v just_o hope_v to_o beget_v in_o he_o and_o so_o i_o shall_v submit_v my_o whole_a concernment_n herein_o to_o the_o impartial_a reader_n and_o remain_v ready_a to_o serve_v he_o reeve_n williams_n form_n my_o school_n at_o the_o virginia_n coff-house_n in_o st._n michael_n alley_n in_o cornhill_n advertisement_n whereas_o in_o the_o french_a all_o the_o definition_n which_o need_v all_o the_o proposition_n as_o also_o all_o those_o use_n our_o author_n think_v fit_a to_o illustrate_v by_o scheme_n be_v do_v in_o the_o book_n in_o wood_n here_o they_o be_v in_o copper_n plate_n to_o be_v place_v at_o the_o end_n of_o the_o book_n the_o definition_n and_o use_n be_v in_o the_o plate_n mark_v with_o arithmetical_a charactars_n or_o zipher_n but_o the_o propsition_n in_o alphabetical_a zipher_n at_o the_o begin_n of_o the_o book_n or_o at_o least_o when_o you_o come_v to_o the_o first_o definition_n you_o be_v refer_v to_o the_o number_n of_o the_o plate_n in_o which_o you_o shall_v find_v the_o scheme_n proper_a thereto_o as_o also_o the_o def_n prop._n and_o use_v belong_v unto_o the_o book_n unless_o the_o plate_n can_v not_o contain_v they_o and_o then_o you_o be_v refer_v to_o the_o next_o plate_n
whereon_o the_o head_n thereof_o you_o shall_v find_v the_o book_n it_o belong_v to_o and_o the_o proposition_n or_o use_n continue_v in_o their_o order_n eight_o book_n of_o euclid_n element_n with_o the_o use_v of_o each_o proposition_n the_o first_o book_n euclid_n design_n in_o this_o book_n be_v to_o give_v the_o first_o principle_n of_o geometry_n and_o to_o do_v the_o same_o methodical_o he_o begin_v with_o the_o definition_n and_o explication_n of_o the_o most_o ordinary_a term_n than_o he_o exhibit_v certain_a supposition_n and_o have_v propose_v those_o maxim_n which_o natural_a reason_n teach_v he_o pretend_v to_o put_v forward_o nothing_o without_o demonstration_n and_o to_o convince_v any_o one_o which_o will_v consent_v to_o nothing_o but_o what_o he_o shall_v be_v oblige_v to_o acknowledge_v in_o his_o first_o proposition_n he_o treat_v of_o line_n and_o of_o the_o several_a angle_n make_v by_o their_o intersect_v each_o other_o and_o have_v occasion_n to_o demonstrate_v their_o propriety_n and_o compare_v certain_a triangle_n he_o do_v the_o same_o in_o the_o first_o eight_o proposition_n then_o teach_v the_o practical_a way_n of_o divide_v a_o angle_n and_o a_o line_n into_o two_o equal_a part_n and_o to_o draw_v a_o perpendicular_a he_o pursue_v to_o the_o propriety_n of_o a_o triangle_n and_o have_v show_v those_o of_o parallel_n line_n he_o make_v a_o end_n of_o the_o explication_n of_o this_o first_o figure_n and_o pass_v forward_o to_o parallellogram_n give_v a_o way_n to_o reduce_v all_o sort_n of_o polygon_n into_o a_o more_o regular_a figure_n he_o end_v this_o book_n with_o that_o celebrate_a proposition_n of_o pythagoras_n and_o demonstrate_v that_o in_o a_o rectangular_a triangle_n the_o square_a of_o the_o base_a be_v equal_a to_o the_o sum_n of_o the_o square_n of_o the_o side_n include_v the_o right_a angle_n definition_n 1._o a_o point_n be_v that_o which_o have_v no_o part_n this_o definition_n be_v to_o be_v understand_v in_o this_o sense_n the_o quantity_n which_o we_o conceive_v without_o distinguish_v its_o part_n or_o without_o think_v that_o it_o have_v any_o be_v a_o mathematical_a point_n far_o differ_v from_o those_o of_o zeno_n which_o be_v altogether_o indivisible_a since_o one_o may_v doubt_v with_o a_o great_a deal_n of_o reason_n if_o those_o last_o be_v possible_a which_o yet_o we_o can_v of_o the_o first_o if_o we_o conceive_v they_o as_o we_o ought_v 2._o a_o line_n be_v a_o length_n without_o breadth_n the_o sense_n of_o this_o definition_n be_v the_o same_o with_o that_o of_o the_o forego_n the_o quantity_n which_o we_o consider_v have_v length_n without_o make_v any_o reflection_n on_o its_o breadth_n be_v that_o we_o understand_v by_o the_o word_n line_n although_o one_o can_v draw_v a_o real_a line_n which_o have_v not_o a_o determinate_a breadth_n it_o be_v general_o say_v that_o a_o line_n be_v produce_v by_o the_o motion_n of_o a_o point_n which_o we_o ought_v well_o to_o take_v notice_n of_o see_v that_o by_o a_o motion_n after_o that_o manner_n may_v be_v produce_v all_o sort_n of_o quantity_n imagine_v then_o that_o a_o point_n move_v and_o that_o it_o leave_v a_o trace_n in_o the_o middle_n of_o the_o way_n which_o it_o pass_v the_o trace_v be_v a_o line_n 3._o the_o two_o end_n of_o a_o line_n be_v point_n 4._o a_o straight_a line_n be_v that_o who_o point_n be_v place_v exact_o in_o the_o midst_n or_o if_o you_o will_v rather_o have_v it_o a_o straight_a line_n be_v the_o short_a of_o all_o the_o line_n which_o may_v be_v draw_v from_o one_o point_n to_o another_o 5._o a_o superficies_n be_v a_o quantity_n to_o which_o be_v give_v length_n and_o breadth_n without_o consider_v the_o thickness_n 6._o the_o extremity_n of_o a_o superficies_n be_v line_n 7._o a_o plain_a or_o straight_o superficies_n be_v that_o who_o line_n be_v place_v equal_o between_o the_o extremity_n or_o that_o to_o which_o a_o straight_a line_n may_v be_v apply_v any_o manner_n of_o way_n 1._o plate_n i._n fig._n 1._o i_o have_v already_o take_v notice_n that_o motion_n be_v capable_a of_o produce_v all_o sort_n of_o quantity_n whence_o we_o say_v that_o when_o a_o line_n pass_v over_o another_o it_o produce_v a_o superficies_n or_o a_o plain_n and_o that_o that_o motion_n have_v a_o likeness_n to_o arithmetical_a multiplication_n imagine_v that_o the_o line_n ab_fw-la move_v along_o the_o line_n bc_n keep_v the_o same_o situation_n without_o incline_v one_o way_n or_o the_o other_o the_o point_n a_o shall_v describe_v the_o line_n ad_fw-la the_o point_n b_o the_o line_n bc_n and_o the_o other_o point_n between_o other_o parallel_n line_n which_o shall_v compose_v the_o superficies_n abcd._n i_o add_v that_o this_o motion_n correspond_v with_o arithmetical_a multiplication_n for_o if_o i_o know_v the_o number_n of_o point_n which_o be_v in_o the_o line_n ab_fw-la bc_n multiply_v of_o they_o one_o by_o the_o other_o i_o shall_v have_v the_o number_n of_o point_n which_o compose_v the_o superficies_n abcd_v as_o if_o ab_fw-la contain_v four_o point_n and_o bc_n six_o say_v four_o time_n six_o be_v twenty_o four_o the_o superficies_n ab_fw-la cd_o shall_v be_v compose_v of_o twenty_o four_o point_n now_o i_o may_v take_v for_o a_o mathematical_a point_n any_o quantity_n whatsoever_o for_o example_n a_o foot_n provide_v i_o do_v not_o subdivide_v the_o same_o into_o part_n 8._o a_o plain_a angle_n be_v the_o open_n of_o two_o line_n which_o intersect_v each_o other_o and_o which_o compose_v not_o one_o single_a line_n 2._o fig._n 2._o as_o the_o opening_z d_o of_o the_o line_n ab_fw-la cb_n which_o be_v not_o part_n of_o the_o same_o line_n a_o right_a line_a angle_n be_v the_o open_n of_o two_o straight_a line_n it_o be_v principal_o of_o this_o sort_n of_o angle_n which_o i_o intend_v to_o treat_v of_o at_o present_a because_o experience_n do_v make_v i_o perceive_v that_o the_o most_o part_n of_o those_o who_o begin_v do_v mistake_v the_o measure_v the_o quantity_n of_o a_o angle_n by_o the_o length_n of_o the_o line_n which_o compose_v the_o same_o 4._o fig._n 3_o 4._o the_o most_o open_a angle_n be_v the_o great_a that_o be_v to_o say_v when_o the_o line_n include_v a_o angle_n be_v far_o asunder_o than_o those_o of_o another_o angle_n take_v they_o at_o the_o same_o distance_n from_o the_o point_n of_o intersection_n of_o their_o line_n the_o first_o be_v great_a than_o the_o second_o so_o the_o angle_n a_o be_v great_a than_o e_z because_o if_o we_o take_v the_o point_n b_o and_o d_o as_o far_o distant_a from_o the_o point_n a_o as_o the_o point_n g_o and_o l_o be_v from_o the_o point_n e_o the_o point_n b_o and_o d_o be_v far_o asunder_o than_o the_o point_n g_o and_o l_o from_o whence_o i_o conclude_v that_o if_o eglantine_n el_fw-es be_v continue_v the_o angle_n e_o will_v be_v of_o the_o same_o measure_n and_o less_o than_o the_o angle_n a._n we_o make_v use_v of_o three_o letter_n to_o express_v a_o angle_n and_o the_o second_o letter_n denote_v the_o angular_a point_n as_o the_o angle_n bad_a be_v the_o angle_n which_o the_o line_n ba_o ad_fw-la do_v form_n at_o the_o point_n a_o the_o angle_n bac_n be_v that_o which_o be_v form_v by_o the_o line_n ba_o ac_fw-la the_o angle_n god_n be_v comprehend_v under_o the_o line_n ca_n ad._n 3._o fig._n 3._o the_o arch_n of_o a_o circle_n be_v the_o measure_n of_o a_o angle_n thus_o design_v to_o measure_v the_o quantity_n of_o the_o angle_n bad_a i_o put_v one_o foot_n of_o the_o compass_n on_o the_o point_v a_o and_o with_o the_o other_o i_o describe_v a_o arch_n of_o a_o circle_n bcd_v the_o angle_n shall_v be_v the_o great_a by_o how_o much_o the_o arch_n bcd_v which_o be_v the_o measure_n thereof_o shall_v contain_v a_o great_a portion_n of_o a_o circle_n and_o because_o that_o common_o a_o arch_n of_o a_o circle_n be_v divide_v into_o three_o hundred_o and_o sixty_o equal_a part_n call_v degree_n it_o be_v say_v that_o a_o angle_n contain_v twenty_o thirty_o forty_o degree_n when_o the_o arch_n include_v betwixt_o its_o line_n contain_v twenty_o thirty_o forty_o degree_n so_o the_o angle_n be_v great_a which_o contain_v the_o great_a number_n of_o degree_n as_o the_o angle_n bad_a be_v great_a than_o gel_n the_o line_n ca_n divide_v the_o angle_n bad_a in_o the_o middle_n because_o the_o arch_n bc_n cd_o be_v equal_a and_o the_o angle_n bac_n be_v a_o part_n of_o bad_a because_o the_o arch_a bc_n be_v part_n of_o the_o arch_n bd._n 10._o when_o a_o line_n fall_v on_o another_o line_n make_v the_o angle_n on_o each_o side_n thereof_o equal_a those_o angles_n be_v right_a angle_n and_o the_o line_n so_o fall_v be_v a_o perpendicular_a 5._o fig._n 5._o as_o if_o the_o line_n ab_fw-la fall_v on_o cd_o
the_o double_a area_n of_o the_o triangle_n i_o make_v use_v thereof_o in_o several_a other_o proposition_n as_o in_o the_o seven_o proposition_n fourteen_o problem_n to_o describe_v a_o square_n equal_a to_o a_o right_n line_v figure_n give_v to_o make_v a_o square_a equal_a to_o a_o right_o line_v figure_n a_o make_v by_o the_o 45_o of_o the_o 1_o a_o rectangle_n bc_n de_fw-fr equal_a to_o the_o right_o line_v figure_n a_o if_o its_o side_n cd_o dc_o be_v equal_a we_o shall_v have_v already_o our_o desire_n if_o they_o be_v unequal_a continue_v the_o line_n bc_n until_o cf_n be_v equal_a to_o cd_o and_o divide_v the_o line_n bf_n in_o the_o middle_n in_o the_o point_n g_o describe_v the_o semicircle_n fhb_n then_o continue_v dc_o to_o h_n the_o square_a of_o the_o line_n ch_z be_v equal_a to_o the_o right_o line_v figure_n a_o draw_v the_o line_n gh_n demonstration_n the_o line_n bf_n be_v equal_o divide_v in_o g_o and_o unequal_o in_o c_z thence_o by_o the_o 5_o the_o rectangle_n comprehend_v under_o bc_n cf_n or_o cd_o that_o be_v to_o say_v the_o rectangle_n bd_o with_o the_o square_a cg_n be_v equal_a to_o the_o square_n of_o gb_n or_o to_o its_o equal_a gh_o now_o by_o the_o 47_o of_o the_o 1_o the_o square_a of_o gh_n be_v equal_a to_o the_o square_n of_o ch_z cg_n therefore_o the_o rectangle_n bd_o and_o the_o square_a of_o cg_n be_v equal_a to_o the_o square_n of_o cg_n and_o of_o ch_z and_o take_v away_o the_o square_n of_o cg_n which_o be_v common_a to_o both_o the_o rectangle_n bd_o or_o the_o right_o line_v figure_n a_o be_v equal_a to_o the_o square_n of_o ch._n use_v this_o proposition_n serve_v in_o the_o first_o place_n to_o reduce_v into_o a_o square_a any_o right_o line_v figure_n whatever_o and_o whereas_o a_o square_a be_v the_o first_o measure_n of_o all_o superficies_n because_o its_o length_n and_o breadth_n be_v equal_a we_o measure_v by_o this_o mean_v all_o right_n line_v figure_n in_o the_o second_o place_n this_o proposition_n teach_v we_o to_o find_v a_o mean_a proportion_n between_o two_o give_v line_n as_o we_o shall_v see_v in_o the_o thirteen_o proposition_n of_o the_o six_o book_n this_o proposition_n may_v also_o serve_v to_o square_v curve_v line_v figure_n and_o even_a circle_n themselves_o for_o any_o crooked_a or_o curve_v line_v figure_n may_v to_o sense_n be_v reduce_v to_o a_o right_o line_v figure_n as_o if_o we_o inscribe_v in_o a_o circle_n a_o polygon_n have_v a_o thousand_o side_n it_o shall_v not_o be_v sensible_o different_a from_o a_o circle_n and_o reduce_v the_o polygone_a into_o a_o square_a we_o square_v near_o the_o circle_n the_o three_o book_n of_o euclid_n element_n this_o three_o book_n explain_v the_o propriety_n of_o a_o circle_n and_o compare_v the_o divers_a line_n which_o may_v be_v draw_v within_o and_o without_o its_o circumference_n it_o far_o consider_v the_o circumstance_n of_o circle_n which_o cut_v each_o other_o or_o which_o touch_n a_o straight_a line_n and_o the_o different_a angle_n which_o be_v make_v as_o well_o those_o in_o their_o centre_n as_o in_o their_o circumference_n in_o fine_a it_o give_v the_o first_o principle_n for_o establish_v the_o practice_n of_o geometry_n by_o the_o which_o we_o make_v use_v and_o that_o very_o commodious_o of_o a_o circle_n in_o almost_o all_o treatise_n in_o the_o mathematics_n definition_n 1._o book_n def._n of_o the_o 3_o book_n those_o circle_n be_v equal_a who_o diameter_n or_o semidiameter_n be_v equal_a 2._o 1._o fig._n 1._o a_o line_n touch_v a_o circle_n when_o meeting_n with_o the_o circumference_n thereof_o it_o cut_v not_o the_o same_o as_o the_o line_n ab_fw-la 3._o 2._o fig._n 2._o circle_n touch_v each_o other_o when_o meeting_n they_o cut_v not_o each_o other_o as_o the_o circle_n ab_fw-la and_o c._n 4._o 3._o fig._n 3._o right_o line_n in_o a_o circle_n be_v equal_o distant_a from_o the_o centre_n when_o perpendicular_o draw_v from_o the_o centre_n to_o those_o line_n be_v equal_a as_o if_o the_o line_n of_o eglantine_n be_v perpendicular_o to_o the_o line_n ab_fw-la cd_o be_v equal_a ab_fw-la cd_o shall_v be_v equal_o distant_a from_o the_o centre_n because_o the_o distance_n ought_v always_o to_o be_v take_v or_o measure_v by_o perpendicular_a line_n 5._o 4._o fig._n 4._o a_o segment_n of_o a_o circle_n be_v a_o figure_n terminate_v on_o the_o one_o side_n by_o a_o straight_a line_n and_o on_o the_o other_o by_o the_o circumference_n of_o a_o circle_n as_o lon_n lmn_n 6._o the_o angle_n of_o a_o segment_n be_v a_o angle_n which_o the_o circumference_n make_v with_o a_o straight_a line_n as_o the_o angle_n oln_n lmn_n 7._o 5._o fig._n 5._o a_o angle_n be_v say_v to_o be_v in_o a_o segment_n of_o a_o circle_n when_o the_o line_n which_o form_n the_o same_o be_v therein_o as_o the_o angle_n fgh_o be_v in_o the_o segment_n fgh_o 8._o 6._o fig_n 6._o a_o angle_n be_v upon_o that_o arch_n to_o which_o it_o be_v opposite_a or_o to_o which_o it_o serve_v for_o a_o base_a as_o the_o angle_n fgh_o be_v upon_o the_o arch_n fih_o which_o may_v be_v say_v to_o be_v its_o base_a 9_o 6._o fig._n 6._o a_o sector_n be_v a_o figure_n comprehend_v under_o two_o semidiameter_n and_o under_o the_o arch_n which_o serve_v to_o they_o for_o a_o base_a as_o the_o figure_n sigh_o proposition_n i._o problem_n to_o find_v the_o centre_n of_o a_o circle_n if_o you_o will_v find_v the_o centre_n of_o the_o circle_n aebd_a draw_v the_o line_n ab_fw-la and_o divide_v the_o same_o in_o the_o middle_n in_o the_o point_n c_o at_o which_o point_n erect_v a_o perpendicular_a ed_z which_o you_o shall_v divide_v also_o equal_o in_o the_o point_n f._n this_o point_n f_o shall_v be_v the_o centre_n of_o the_o circle_n for_o if_o it_o be_v not_o imagine_v if_o you_o please_v that_o the_o point_n g_z be_v the_o centre_n draw_v the_o line_n give_v gb_n gc_fw-la demonstration_n if_o the_o point_n g_z be_v the_o centre_n the_o triangle_n gac_n gbc_n will_v have_v the_o side_n give_v gb_n equal_a by_o the_o definition_n of_o a_o circle_n ac_fw-la cb_n be_v equal_a to_o the_o line_n ab_fw-la have_v be_v divide_v in_o the_o middle_n in_o the_o point_n c._n and_o cg_n be_v common_a the_o angles_n gcb_n gca_n will_v then_o be_v equal_a by_o the_o 8_o of_o the_o 1_o and_o cg_n will_v be_v then_o a_o perpendicular_a and_o not_o cd_o which_o will_v be_v contrary_a to_o the_o hypothesis_n therefore_o the_o centre_n can_v be_v out_o of_o the_o line_n cd_o i_o further_o add_v that_o it_o must_v be_v in_o the_o point_n f_o which_o divide_v the_o same_o into_o two_o equal_a part_n otherwise_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n will_v not_o be_v equal_a corollary_n the_o centre_n of_o a_o circle_n be_v in_o a_o line_n which_o divide_v another_o line_n in_o the_o middle_n and_o that_o perpendicular_o use_v this_o first_o proposition_n be_v necessary_a to_o demonstrate_v those_o which_o follow_v proposition_n ii_o theorem_fw-la a_o straight_a line_n draw_v from_o one_o point_n of_o the_o circumference_n of_o a_o circle_n to_o another_o shall_v fall_v within_o the_o same_o let_v there_o be_v draw_v from_o the_o point_n b_o in_o the_o circumference_n a_o line_n to_o the_o point_n c._n i_o say_v that_o it_o shall_v fall_v whole_o within_o the_o circle_n to_o prove_v that_o it_o can_v fall_v without_o the_o circle_n as_o buc_n have_v find_v the_o centre_n thereof_o which_o be_v a_o draw_v the_o line_n ab_fw-la ac_fw-la au._n demonstration_n the_o sides_n ab_fw-la ac_fw-la of_o the_o triangle_n abc_n be_v equal_a whence_o by_o the_o 5_o of_o the_o 1_o the_o angel_n abc_n acb_n be_v equal_a and_o see_v the_o angle_n auc_n be_v exterior_a in_o respect_n of_o the_o triangle_n aub_n it_o be_v great_a than_o abc_n by_o the_o 16_o of_o the_o 1_o it_o shall_v be_v also_o great_a than_o the_o angle_n acb_n thence_o by_o the_o 19_o of_o the_o 1_o in_o the_o triangle_n acv_n the_o side_n ac_fw-la opposite_a to_o the_o great_a angle_n auc_n be_v great_a than_o ave_fw-la and_o by_o consequence_n hence_v not_o reach_v the_o circumference_n of_o the_o circle_n see_v it_o be_v short_a than_o ac_fw-la which_o do_v but_o reach_v the_o same_o wherefore_o the_o point_n v_n be_v within_o the_o circle_n the_o same_o may_v be_v prove_v of_o any_o point_n in_o the_o line_n ab_fw-la and_o therefore_o the_o whole_a line_n ab_fw-la fall_v within_o the_o circle_n use_v it_o be_v on_o this_o proposition_n that_o be_v ground_v those_o which_o demonstrate_v that_o a_o circle_n touch_v a_o straight_a line_n but_o only_o in_o one_o point_n for_o if_o the_o line_n shall_v touch_v two_o point_n of_o its_o circumference_n it_o will_v be_v then_o draw_v from_o one_o
inscribe_v by_o the_o little_a rectangle_v through_o which_o the_o circumference_n of_o the_o circle_n pass_v and_o all_o those_o rectangle_v take_v together_o be_v equal_a to_o the_o rectangle_n al._n imagine_v that_o the_o semicircle_n be_v make_v to_o roll_n about_o the_o diameter_n ebb_n the_o semicircle_n shall_v describe_v a_o hemi-sphere_n and_o the_o inscribe_v rectangle_v will_v describe_v inscribe_v cylinder_n in_o the_o semi-sphere_n and_o the_o circumscribe_v will_v describe_v other_o cylinder_n demonstration_n the_o circumscribe_v cylinder_n surpass_v more_o the_o inscribe_v than_o do_v the_o hemi-sphere_n surpass_v the_o same_o inscribe_v cylinder_n see_v that_o they_o be_v comprehend_v within_o the_o circumscribe_v cylinder_n now_o the_o circumscribe_v surpass_v the_o inscribe_v by_o so_o much_o as_o be_v the_o cylinder_n all_o therefore_o the_o hemi-sphere_n shall_v surpass_v by_o less_o the_o inscribe_v cylinder_n than_o do_v the_o cylinder_n make_v by_o the_o rectangle_n al._n the_o cylinder_n all_o be_v less_o than_o the_o cylinder_n mp_n for_o there_o be_v the_o same_o ratio_fw-la of_o a_o great_a circle_n of_o the_o sphere_n which_o serve_v for_o base_a to_o the_o cylinder_n all_o as_o of_o mp_n to_o r_n so_o then_o by_o the_o forego_n a_o cylinder_n which_o have_v for_o base_a a_o great_a circle_n of_o the_o sphere_n and_o the_o altitude_n r_n will_v be_v equal_a to_o the_o cylinder_n mp_n consequent_o the_o hemi-sphere_n which_o surpass_v the_o quantity_n d_o by_o the_o cylinder_n mp_n and_o the_o inscribe_v cylinder_n by_o a_o quantity_n less_o than_o all_o surpass_v the_o inscribe_v cylinder_n by_o less_o than_o the_o quantity_n d._n therefore_o the_o quantity_n d_o be_v less_o than_o the_o inscribe_v cylinder_n what_o i_o have_v say_v of_o a_o hemi_a sphere_n may_v be_v say_v of_o a_o whole_a sphere_n lemma_n ii_o like_a cylinder_n inscribe_v in_o two_o sphere_n be_v in_o triple_a ratio_fw-la of_o the_o diameter_n of_o the_o sphere_n ii_o lemma_n fig._n ii_o if_o the_o two_o like_a cylinder_n cd_o of_o be_v inscribe_v in_o the_o sphere_n a_o b_o they_o shall_v be_v in_o triple_a ratio_fw-la of_o the_o diameter_n lm_o no_o draw_v the_o line_n gd_v if_o demonstration_n the_o like_a right_n cylinder_v cd_o of_o be_v like_a so_o then_o there_o be_v the_o same_o ratio_fw-la of_o hd_a to_o dr_n as_o of_o qf_n to_o f_n as_o also_o the_o same_o ratio_fw-la of_o kd_a to_o kg_v as_o of_o pf_n to_o pi._n and_o consequent_o the_o triangle_n gdk_v ifp_n be_v like_o by_o the_o 6_o of_o the_o 6_o so_o there_o shall_v be_v the_o same_o ratio_fw-la of_o kd_v to_o pf_n as_o of_o gd_a to_o if_o or_o of_o lm_o to_o on_o now_o the_o like_a cylinder_n cd_o of_o be_v in_o triple_a ratio_fw-la of_o kd_a and_o pf_n the_o diameter_n of_o their_o base_n by_o the_o 12_o therefore_o the_o like_a cylinder_n cd_o of_o inscribe_v in_o the_o sphere_n a_o and_o b_o be_v in_o triple_a ratio_fw-la of_o the_o diameter_n of_o the_o sphere_n proposition_n xviii_o theorem_fw-la sphere_n be_v in_o triple_a ratio_fw-la of_o their_o diameter_n the_o sphere_n a_o and_o b_o be_v in_o triple_a ratio_fw-la of_o their_o diameter_n cd_o ef._n for_o if_o they_o be_v not_o in_o triple_a ratio_fw-la one_o of_o the_o sphere_n as_o a_o shall_v be_v in_o a_o great_a ratio_fw-la then_o triple_a of_o that_o of_o cd_o to_o of_o therefore_o a_o quantity_n g_z less_o than_o the_o sphere_n a_o shall_v be_v in_o triple_a ratio_fw-la of_o that_o of_o cd_o to_o of_o and_o so_o one_o might_n according_a to_o the_o first_o lemma_n inscribe_v in_o the_o sphere_n a_o cylinder_v of_o the_o same_o height_n great_a than_o the_o quantity_n g._n let_v there_o be_v inscribe_v in_o the_o sphere_n b_o as_o many_o like_a cylinder_n as_o those_o of_o the_o cylinder_n a._n demonstration_n the_o cylinder_n of_o the_o sphere_n a_o to_o those_o of_o the_o sphere_n b_o shall_v be_v in_o triple_a ratio_fw-la of_o that_o of_o cd_o to_o of_o by_o the_o precede_a now_o the_o quantity_n g_o in_o respect_n of_o the_o sphere_n b_o be_v in_o triple_a ratio_fw-la of_o cd_o to_o of_o there_o be_v then_o the_o same_o ratio_fw-la of_o the_o cylinder_n of_o the_o sphere_n a_o to_o the_o like_a cylinder_n of_o the_o sphere_n b._n so_o then_o the_o cylinder_n of_o a_o be_v great_a than_o the_o quantity_n g_o the_o cylinder_n b_o that_o be_v to_o say_v inscribe_v in_o the_o sphere_n b_o will_v be_v great_a than_o the_o sphere_n b_o which_o be_v impossible_a therefore_o the_o sphere_n a_o and_o b_o be_v in_o triple_a ratio_fw-la of_o that_o of_o their_o diameter_n coral_n sphere_n be_v in_o the_o same_o ratio_fw-la as_o be_v the_o cube_n of_o their_o diameter_n see_v that_o the_o cube_n be_v like_o solid_n be_v in_o triple_a ratio_fw-la of_o that_o of_o their_o side_n finis_fw-la errata_fw-la page_n 14._o line_n 21._o read_v afd_v p._n 23._o l._n 17._o for_o df_n r._n ef._n l._n 16._o r._n dfe_n l._n 27._o r._n fd._n p._n 24._o l._n 1_o and_o 9_o for_o fb_n r._n fd._n p._n 26._o l._n 8._o r._n he_o p._n 52._o l._n 14._o r._n acb_n p._n 55._o l._n 22._o r._n acf_n p._n 71._o l._n 3._o r._n acb_n p._n 82._o l._n 19_o r._n abc_n p._n 117._o l._n 13._o r._n gfe_n p._n 125._o l._n 9_o r._n ad._n p._n 178._o l._n 16._o r._n cfd_n p._n 194._o l._n ult_n r._n af._n p._n 197._o l._n 2._o r._n cda_n p._n 218._o l._n 7._o r._n ⅓_n and_o ⅓_n p._n 268._o l._n 24._o r._n ce._n p._n 281._o l._n 3._o r._n ecd_n p._n 289._o l._n 4._o r._n be_v simular_a p._n 294._o l._n 13._o r._n eight_o ¾_n p._n 309._o l._n 15._o r._n dbe_n advertisement_n of_o globe_n book_n map_n etc._n etc._n make_v and_o sell_v by_o philip_n lea_n at_o the_o sign_n of_o the_o atlas_n and_o hercules_n in_o the_o poultry_n near_o cheapside_n london_n 1._o a_o new_a size_n of_o globe_n about_o 15_o inch_n diameter_n make_v according_a to_o the_o more_o accurate_a observation_n and_o discovery_n of_o our_o modern_a astronomer_n and_o geographer_n and_o much_o different_a from_o all_o that_o ever_o be_v yet_o extant_a all_o the_o southern_a constellation_n according_a to_o mr._n hally_n observation_n in_o the_o island_n of_o st._n helena_n and_o many_o of_o the_o northern_a price_n four_o pound_n 2._o a_o size_n of_o globe_n of_o about_o ten_o inch_n diameter_n very_o much_o correct_v price_n 50_o s._n 3._o concave_a hemisphere_n three_o inch_n diameter_n which_o serve_v as_o a_o case_n for_o a_o terrestrial_a globe_n and_o may_v be_v carry_v in_o the_o pocket_n or_o fit_v up_o in_o frames_n price_n 15_o or_o 20_o s._n 4._o another_o globe_n of_o about_o four_o inch_n diameter_n fit_v to_o move_v in_o circular_a line_n of_o brass_n for_o demonstrate_v the_o reason_n of_o dyall_v or_o be_v erect_v upon_o a_o small_a pedestal_n and_o fetch_v north_n and_o south_n will_v show_v the_o hour_n of_o the_o day_n by_o its_o own_o shadow_n or_o by_o the_o help_n of_o a_o move_a meridian_n will_v show_v the_o hour_n of_o the_o day_n in_o all_o part_n of_o the_o world_n etc._n etc._n 14._o there_o be_v in_o the_o press_n a_o particular_a description_n of_o the_o general_a use_n of_o quadrant_n for_o the_o easy_a resolve_v astronomical_a geometrical_a and_o gnomonical_a problem_n and_o find_v the_o hour_n and_o azimuth_n universal_o etc._n etc._n whereunto_o be_v add_v the_o use_n of_o the_o nocturnal_a and_o equinoctial_a dial._n 15._o a_o new_a map_n of_o england_n scotland_n and_o ireland_n with_o the_o road_n and_o a_o delineation_n of_o the_o genealogy_n of_o the_o king_n thereof_o from_o william_n the_o conqueror_n to_o this_o present_a time_n with_o a_o alphabetical_a table_n for_o the_o ready_a find_n of_o the_o place_n price_n 18_o d._n 16._o the_o element_n of_o euclid_n explain_v and_o demonstrate_v after_o a_o new_a and_o most_o easy_a method_n with_o the_o use_n of_o each_o proposition_n in_o all_o the_o part_n of_o the_o mathematics_n by_o claude_n francois_n millet_n dechales_n a_o jesuit_n 17._o new_a map_n of_o the_o world_n four_o quarter_n and_o of_o all_o the_o country_n and_o of_o all_o size_n make_v according_a to_o the_o late_a discovery_n extant_a may_v be_v have_v past_v upon_o cloth_n and_o colour_a also_o sea_n plait_n mathematical_a projection_n book_n and_o instrument_n whatsoever_o be_v make_v and_o sell_v by_o philip_n lea._n finis_fw-la euclid_n element_n with_o the_o use_v euclid_n london_n print_v for_o philip_n lea_n globe_n maker_n at_o the_o atlas_n and_o hercules_n in_o cheapside_n near_o friday_n street_n there_o all_o sort_n of_o globe_n sphere_n map_n sea-plat_n mathematical_a book_n and_o instrument_n be_v make_v and_o sell_v plate_n 2._o proposition_n and_o use_v of_o the_o first_o book_n see_v plate_n 3_o plate_n 1._o definition_n of_o the_o first_o book_n proposition_n and_o use_v of_o the_o first_o book_n see_v plate_n 2._o plate_n 3._o proposition_n &_o use_v of_o the_o first_o book_n definition_n proposition_n &_o use_v of_o the_o second_o book_n plate_n 4_o definition_n of_o the_o three_o book_n proposition_n proposition_n and_o use_v plate_n 5._o definition_n propsition_n of_o the_o fouth_z book_n plate_n 6._o definition_n of_o the_o six_o book_n proposition_n &_o use_v plate_n 7._o definition_n of_o the_o eleven_o book_n proposition_n plate_n 8._o proposi_fw-la of_o the_o twelve_o book_n
under_o ab_fw-la and_o ac_fw-la shall_v be_v three_o time_n 8_o or_o 24_o the_o square_a of_o ac_fw-la 3_o be_v 9_o the_o rectangle_n comprehend_v under_o ac_fw-la 3_o and_o cb_n 5_o be_v 3_o time_n 5_o or_o 15._o it_o be_v evident_a that_o 15_o and_o 9_o be_v 24._o use_v at_fw-fr  _fw-fr 43_o c_z 40._o 3_o b_o  _fw-fr 3_o 120._o  _fw-fr 9_o 129_o  _fw-fr  _fw-fr this_o proposition_n serve_v likewise_o to_o demonstrate_v the_o ordinary_a practice_n of_o multiplication_n for_o example_n if_o one_o will_v multiply_v the_o number_n 43_o by_o 3_o have_v separate_v the_o number_n of_o 43_o into_o two_o part_n in_o 40_o and_o 3_o three_o time_n 43_o shall_v be_v as_o much_o as_o three_o time_n 3_o which_o be_v nine_o the_o square_a of_o three_o and_o three_o time_n forty_o which_o be_v 120_o for_o 129_o be_v three_o time_n 43._o those_o which_o be_v young_a beginner_n ought_v not_o to_o be_v discourage_v if_o they_o do_v not_o conceive_v immediate_o these_o proposition_n for_o they_o be_v not_o difficult_a but_o because_o they_o do_v imagine_v they_o contain_v some_o great_a mystery_n proposition_n iu_o theorem_fw-la if_o a_o line_n be_v divide_v into_o two_o part_n the_o square_a of_o the_o whole_a line_n shall_v be_v equal_a to_o the_o two_o square_n make_v of_o its_o part_n and_o to_o two_o rectangle_v comprehend_v under_o the_o same_o part_n let_v the_o line_n ab_fw-la be_v divide_v in_o c_o and_o let_v the_o square_a thereof_o abde_n be_v make_v let_v the_o diagonal_a ebb_n be_v draw_v and_o the_o perpendicular_a cf_n cut_v the_o same_o and_o through_o that_o point_n let_v there_o be_v draw_v gl_n parallel_n to_o ab_fw-la it_o be_v evident_a that_o the_o square_a abde_n be_v equal_a to_o the_o four_o rectangle_v gf_n cl_n cg_n lf_n the_o two_o first_o be_v the_o square_a of_o ac_fw-la and_o of_o cb_n the_o two_o compliment_n be_v comprehend_v under_o ac_fw-la cb._n demonstration_n the_o side_n ae_n ab_fw-la be_v equal_a thence_o the_o angle_n aeb_fw-mi abe_n be_v half_a right_n and_o because_o of_o the_o parallel_n gl_n ab_fw-la the_o angle_n of_o the_o triangle_n of_o the_o square_a ge_z by_o the_o 29_o shall_v be_v equal_a as_o also_o the_o side_n by_o the_o 6_o of_o the_o 1._o thence_o gf_n be_v the_o square_a of_o ac_fw-la in_o like_a manner_n the_o rectangle_n cl_n be_v the_o square_a of_o cb_n the_o rectangle_n gc_n be_v comprehend_v under_o ac_fw-la and_o agnostus_n equal_a to_o bl_v or_o bc_n the_o rectangle_n lf_o be_v comprehend_v under_o ld_n equal_a to_o ac_fw-la and_o under_o fd_n equal_a to_o bc._n coral_n if_o a_o diagonal_a be_v draw_v in_o a_o square_a the_o rectangle_v through_o which_o it_o pass_v be_v square_n use_v a_o 144_o b_o 22_o c_o 12_o this_o proposition_n give_v we_o the_o practical_a way_n of_o find_v or_o extract_v the_o square_a root_n of_o a_o number_n propound_v let_v the_o same_o be_v the_o number_n a_o 144_o represent_v by_o the_o square_a ad_fw-la and_o its_o root_n by_o the_o line_n ab_fw-la moreover_o i_o know_v that_o the_o line_n require_v ab_fw-la must_v have_v two_o figure_n i_o therefore_o imagine_v that_o the_o line_n ab_fw-la be_v divide_v in_o c_o and_o that_o ac_fw-la represent_v the_o first_o figure_n and_o bc_n the_o second_o i_o seek_v the_o root_n of_o the_o first_o figure_n of_o the_o number_n 144_o which_o be_v 100_o and_o i_o find_v that_o it_o be_v 10_o and_o make_v its_o square_a 100_o represent_v by_o the_o square_a gf_n i_o subtract_v the_o same_o from_o 144_o and_o there_o remain_v 44_o for_o the_o rectangle_v gc_a fl_fw-mi and_o the_o square_a cl._n but_o because_o this_o gnomonicall_a figure_n be_v not_o proper_a i_o transport_v the_o rectangle_n fl_fw-mi in_o kg_v and_o so_o i_o have_v the_o rectangle_n kl_n contain_v 44._o i_o know_v also_o almost_o all_o the_o length_n of_o the_o side_n kb_n for_o ac_fw-la be_v 10_o therefore_o kc_n be_v 20_o i_o must_v then_o divide_v 44_o by_o 20_o that_o be_v to_o say_v to_o find_v the_o divisor_n i_o double_v the_o root_n find_v and_o i_o say_v how_o many_o time_n 20_o in_o 44_o i_o find_v it_o 2_o time_n for_o the_o side_n bl_n but_o because_o 20_o be_v not_o the_o whole_a side_n kb_n but_o only_a kc_n this_o 2_o which_o come_v in_o the_o quotient_n be_v to_o be_v add_v to_o the_o divisor_n which_o then_o will_v be_v 22._o so_o i_o find_v the_o same_o 2_o time_n precise_o in_o 44_o the_o square_a root_n then_o shall_v be_v 12._o you_o see_v that_o the_o square_a of_o 144_o be_v equal_a to_o the_o square_n of_o 10_o to_o the_o square_n of_o 2_o which_o be_v 4_o and_o to_o twice_o 20_o which_o be_v two_o rectangle_v comprehend_v under_o 2_o and_o under_o 10._o proposition_n v._o theorem_fw-la if_o a_o right_a line_n be_v cut_v into_o equal_a part_n and_o into_o unequal_a part_n the_o rectangle_n comprehend_v under_o the_o unequal_a part_n together_o with_o the_o square_n which_o be_v of_o the_o middle_a part_n or_o difference_n of_o the_o part_n be_v equal_a to_o the_o square_n of_o half_a the_o line_n if_o the_o line_n ab_fw-la be_v divide_v equal_o in_o c_z and_o unequal_o in_o d_o the_o rectangle_n ah_o comprehend_v under_o the_o segment_n ad_fw-la db_fw-la together_o with_o the_o square_n of_o cd_o shall_v be_v equal_a to_o the_o square_a cf_n that_o be_v of_o half_a of_o ab_fw-la viz._n cb._n make_v a_o end_n of_o the_o figure_n as_o you_o see_v it_o the_o rectangle_v lg_n diego_n shall_v be_v square_n by_o the_o coral_n of_o the_o 4_o i_o prove_v that_o the_o rectangle_n ah_o comprehend_v under_o ad_fw-la and_o dh_n equal_a to_o db_v with_o the_o square_a lg_n be_v equal_a to_o the_o square_a cf._n demonstration_n the_o rectangle_n all_o be_v equal_a to_o the_o rectangle_n df_n the_o one_o and_o the_o other_o be_v comprehend_v under_o half_a the_o line_n ab_fw-la and_o under_o bd_o or_o dh_n equal_a thereto_o add_v to_o both_o the_o rectangle_n ch_z the_o rectangle_n ah_o shall_v be_v equal_a to_o the_o gnomon_n lbg_n again_o to_o both_o add_v the_o square_a lg_n the_o rectangle_n ah_o with_o the_o square_a lg_n shall_v be_v equal_a to_o the_o square_a cf._n arithmetical_o let_v ab_fw-la be_v 10_o ac_fw-la be_v 5_o as_o also_o cb._n let_v cd_o be_v 2_o and_o db_n 3_o the_o rectangle_n comprehend_v under_o ad_fw-la 7_o and_o db_n 3_o that_o be_v to_o say_v 21_o with_o the_o square_n of_o cd_o 2_o which_o be_v 4_o shall_v be_v equal_a to_o the_o square_n of_o cb_n 5_o which_o be_v 25._o use_v this_o proposition_n be_v very_o useful_a in_o the_o three_o book_n we_o make_v use_v thereof_o in_o algebra_n to_o demonstrate_v the_o way_n of_o find_v the_o root_n of_o a_o affect_a square_a or_o equation_n proposition_n vi_o theorem_fw-la if_o one_o add_v a_o line_n to_o another_o which_o be_v divide_v into_o two_o equal_a part_n the_o rectangle_n comprehend_v under_o the_o line_n compound_v of_o both_o and_o under_o the_o line_n add_v together_o with_o the_o square_n of_o half_a the_o divide_a line_n be_v equal_a to_o the_o square_n of_o a_o line_n compound_v of_o half_a the_o divide_a line_n and_o the_o line_n add_v if_o one_o add_v the_o line_n bd_o to_o the_o line_n ab_fw-la which_o be_v equal_o divide_v in_o c_o the_o rectangle_n a_fw-la comprehend_v under_o ad_fw-la and_o under_o dn_n or_o db_n with_o the_o square_n of_o cb_n be_v equal_a to_o the_o square_n of_o cd_o make_v the_o square_n of_o cd_o and_o have_v draw_v the_o diagonal_a fd_n draw_v bg_n parallel_v to_o fc_n which_o cut_v fd_v in_o the_o point_n h_n through_o which_o pass_v hn_n parallel_v to_o ab_fw-la kg_n shall_v be_v the_o square_n of_o bc_n and_o bn_n that_o of_o bd._n demonstration_n the_o rectangle_v ak_v ch_z on_o equal_a base_n ac_fw-la bc_n be_v equal_a by_o the_o 38_o of_o the_o 1_o the_o compliment_n ch_z he_o be_v equal_a by_o the_o 43_o of_o the_o 1_o therefore_o the_o rectangle_v ak_v he_o be_v equal_a add_v to_o both_o the_o rectangle_n cn_fw-la and_o the_o square_a kg_n the_o rectangle_v ak_v cn_fw-la that_o be_v to_o say_v the_o rectangle_n a_fw-la with_o the_o square_a kg_n shall_v be_v equal_a to_o the_o rectangle_v cn_fw-la he_o and_o to_o the_o square_a kg_n that_o be_v to_o say_v to_o the_o square_a ce._n arithmetical_o or_o by_o number_n let_v ab_fw-la be_v 8_o ac_fw-la 4_o cb_n 4_o bd_o 3_o than_o ad_fw-la shall_v be_v 11._o it_o be_v evident_a that_o the_o rectangle_n a_fw-la three_o time_n 11_o that_o be_v to_o say_v 33_o with_o the_o square_n of_o kg_v 16_o which_o together_o be_v 49_o be_v equal_a to_o the_o square_n of_o cd_o 7_o which_o be_v 49_o for_o 7_o time_n 7_o be_v 49._o use_v 6._o fig._n 6._o maurolycus_n measure_v the_o whole_a earth_n by_o one_o single_a
superficies_n 18._o a_o cone_n be_v a_o figure_n make_v when_o one_o side_n of_o a_o right_a angle_a triangle_n viz._n one_o of_o those_o that_o contain_v the_o right_a angle_n remain_v fix_v the_o triangle_n be_v turn_v round_o about_o till_o it_o return_v to_o the_o place_n from_o whence_o it_o first_o move_v and_o if_o the_o fix_a right_a line_n be_v equal_a to_o the_o other_o which_o contain_v the_o right_a angle_n than_o the_o cone_n be_v a_o rectangled_a cone_n but_o if_o it_o be_v less_o it_o be_v a_o obtuse_a angle_a cone_n if_o great_a a_o acute_a angle_a cone_n 19_o the_o axis_n of_o a_o cone_n be_v that_o fix_a line_n about_o which_o the_o triangle_n be_v move_v 20._o a_o cylinder_n be_v a_o figure_n make_v by_o the_o move_a round_n of_o a_o right_a angle_a parallelogram_n one_o of_o the_o side_n thereof_o namely_o which_o contain_v the_o right_a angle_n abide_v fix_v till_o the_o parallelogram_n be_v turn_v about_o to_o the_o same_o place_n whence_o it_o begin_v to_o move_v 21._o like_a cones_n and_o cylinder_n be_v those_o who_o axe_n and_o diameter_n of_o their_o base_n be_v proportional_a cones_n be_v right_a when_o the_o axis_n be_v perpendicular_a to_o the_o plain_a of_o the_o base_a and_o they_o be_v say_v to_o be_v scalene_n when_o the_o axis_n be_v incline_v to_o the_o base_a and_o the_o diameter_n of_o their_o base_n be_v in_o the_o same_o ratio_fw-la we_o add_v that_o incline_v cones_n to_o be_v like_o their_o axe_n must_v have_v the_o same_o inclination_n to_o the_o plane_n of_o their_o base_n proposition_n i._o theorem_fw-la i._n plate_n vii_o prop._n i._n a_o straight_a line_n can_v have_v one_o of_o its_o part_n in_o a_o plane_n and_o the_o other_o without_o it_o if_o the_o line_n ab_fw-la be_v in_o the_o plane_n ad_fw-la it_o be_v continue_v shall_v not_o go_v without_o but_o all_o its_o part_n shall_v be_v in_o the_o same_o plane_n for_o if_o it_o can_v be_v that_o bc_n be_v a_o part_n of_o ab_fw-la continue_v draw_v in_o the_o plane_n cd_o the_o line_n bd_o perpendicular_a to_o ab_fw-la draw_v also_o in_o the_o same_o plane_n be_v perpendicular_a to_o bd._n demonstration_n the_o angle_n abdella_n bde_n be_v both_o right_a angle_n thence_o by_o the_o 14_o of_o the_o first_o ab_fw-la be_v do_v make_v but_o one_o line_n and_o consequent_o bc_n be_v not_o a_o part_n of_o the_o line_n ab_fw-la continue_v otherwise_o two_o strait_a line_n cb_n ebb_n will_v have_v the_o same_o part_n ab_fw-la that_o be_v ab_fw-la will_v be_v part_n of_o both_o which_o we_o have_v reject_v as_o false_a in_o the_o thirteen_o maxim_n of_o the_o first_o book_n use_v we_o establish_v on_o this_o proposition_n a_o principle_n in_o gnomonic_n to_o prove_v that_o the_o shadow_n of_o the_o stile_n fall_v not_o without_o the_o plane_n of_o a_o great_a circle_n in_o which_o the_o sun_n be_v see_v that_o the_o end_n or_o top_n of_o the_o stile_n be_v take_v for_o the_o centre_n of_o the_o heaven_n and_o consequent_o for_o the_o centre_n of_o all_o the_o great_a circle_n the_o shadow_n be_v always_o in_o a_o straight_a line_n with_o the_o ray_n draw_v from_o the_o sun_n to_o the_o opaque_fw-fr body_n this_o ray_n be_v in_o any_o great_a circle_n the_o shadow_n must_v also_o be_v therein_o proposition_n ii_o theorem_fw-la line_n which_o cut_v one_o another_o be_v in_o the_o same_o plane_n as_o well_o as_o all_o the_o part_n of_o a_o triangle_n if_o the_o two_o line_n be_v cd_o cut_v one_o another_o in_o the_o point_n a_o and_o if_o there_o be_v make_v a_o triangle_n by_o draw_v the_o base_a bc_n i_o say_v that_o all_o the_o part_n of_o the_o triangle_n abc_n be_v in_o the_o same_o plane_n and_o that_o the_o line_n be_v cd_o be_v likewise_o therein_o demonstration_n it_o can_v be_v say_v that_o any_o one_o part_n of_o the_o triangle_n abc_n be_v in_o a_o plane_n and_o that_o the_o other_o part_n be_v without_o without_o say_v that_o one_o part_n of_o a_o line_n be_v in_o one_o plane_n and_o that_o the_o other_o part_n of_o the_o same_o line_n be_v not_o therein_o which_o be_v contrary_a to_o the_o first_o proposition_n and_o see_v that_o the_o side_n of_o the_o triangle_n be_v in_o the_o same_o plane_n wherein_o the_o triangle_n be_v the_o line_n be_v cd_o shall_v be_v in_o the_o same_o plane_n use_v this_o proposition_n do_v sufficient_o determine_v a_o plane_n by_o two_o straight_a line_n mutual_o intersect_v each_o other_o or_o by_o a_o triangle_n i_o have_v make_v use_n thereof_o in_o optic_n to_o prove_v that_o the_o objective_a parallel_n line_n which_o fall_v on_o the_o tablet_n aught_o to_o be_v represent_v by_o line_n which_o concur_v in_o a_o point_n proposition_n iii_o theorem_fw-la the_o common_a section_n of_o two_o place_n be_v a_o straight_a line_n if_o two_o planes_n ab_fw-la cd_o cut_v one_o another_o their_o common_a section_n of_o shall_v be_v a_o straight_a line_n for_o if_o it_o be_v not_o take_v two_o point_n common_a to_o both_o plane_n which_o let_v be_v e_o and_o f_o and_o draw_v a_o strait_a line_n from_o the_o point_n e_o to_o the_o point_n f_o in_o the_o plane_n ab_fw-la which_o let_v be_v ehf_n draw_v also_o in_o the_o plane_n cd_o a_o straight_a line_n from_o e_o to_o f_o if_o it_o be_v not_o the_o same_o with_o the_o former_a let_v it_o be_v egf_n demonstration_n those_o line_n draw_v in_o the_o two_o plane_n be_v two_o different_a line_n and_o they_o comprehend_v a_o space_n which_o be_v contrary_a to_o the_o twelve_o maxim_n thence_o they_o be_v but_o one_o line_n which_o be_v in_o both_o plane_n shall_v be_v their_o common_a section_n use_v this_o proposition_n be_v fundamental_a we_o do_v suppose_v it_o in_o gnomonic_n when_o we_o represent_v in_o a_o dial_n the_o circle_n of_o the_o hour_n mark_v only_o the_o common_a section_n of_o their_o plane_n and_o that_o of_o the_o wall_n proposition_n iu_o theorem_fw-la if_o a_o line_n be_v perpendicular_a to_o two_o other_o line_n which_o cut_v one_o another_o it_o shall_v be_v also_o perpendicular_a to_o the_o plane_n of_o those_o line_n if_o the_o line_n ab_fw-la be_v perpendicular_a to_o the_o line_n cd_o of_o which_o cut_v one_o another_o in_o the_o point_n b_o in_o such_o manner_n that_o the_o angel_n abc_n abdella_n abe_n abf_n be_v right_a which_o a_o flat_a figure_n can_v represent_v it_o shall_v be_v perpendicular_a to_o the_o plane_n cd_o of_o that_o be_v to_o say_v that_o it_o shall_v be_v perpendicular_a to_o all_o the_o line_n that_o be_v draw_v in_o that_o plane_n through_o the_o point_n b_o as_o to_o the_o line_n gbh_n let_v equal_a line_n be_v cut_v bc_n bd_o be_v bf_n and_o let_v be_v draw_v the_o line_n aec_fw-la df_n ac_fw-la ad_fw-la ae_n of_o agnostus_n and_z ah_o demonstration_n the_o four_o triangle_n abc_n abdella_n abe_n abf_n have_v their_o angle_n right_o in_o the_o point_n b_o and_o the_o side_n bc_n bd_o be_v bf_n equal_a with_o the_o side_n ab_fw-la common_a to_o they_o all_o therefore_o their_o base_n ac_fw-la ad_fw-la ae_n of_o be_v equal_a by_o the_o four_o of_o the_o one_a 2._o the_o triangle_n ebc_n dbf_n shall_v be_v equal_a in_o every_o respect_n have_v the_o side_n bc_n bd_o be_v bf_n equal_a and_o the_o angel_n cbe_n dbf_n opposite_a at_o the_o vertex_fw-la be_v equal_a so_o than_o the_o angle_n be_v bdf_n bec_n bfd_n shall_v be_v equal_a by_o the_o four_o of_o the_o first_o and_o their_o base_n aec_fw-la df_n equal_a 3._o the_o triangle_n gbc_n dbh_n have_v their_o opposite_a angle_n cbg_n dbh_n equal_a as_o also_o the_o angle_n bdh_fw-mi bcg_n and_o the_o side_n bc_n bd_o they_o shall_v then_o have_v by_o the_o 26_o of_o the_o one_a their_o side_n bg_n bh_n cg_n dh_n equal_a 4._o the_o triangle_n ace_n afd_v have_v their_o side_n ac_fw-la ad_fw-la ae_n of_o equal_a and_o the_o base_n aec_fw-la df_n equal_a they_o shall_v have_v by_o the_o 8_o of_o the_o one_a the_o angles_n adf_n ace_n equal_a 5._o the_o triangle_n acg_n adh_n have_v the_o side_n ac_fw-la ad_fw-la cg_n dh_n equal_a with_o the_o angles_n adh_n agc_n thence_o they_o shall_v have_v their_o base_n agnostus_n ah_o equal_a last_o the_o triangle_n abh_n abg_n have_v all_o their_o side_n equal_a thence_o by_o the_o 27_o of_o the_o one_a the_o angles_n abg_n abh_n shall_v be_v equal_a and_o the_o line_n ab_fw-la perpendicular_a to_o gh_v so_o then_o the_o line_n ab_fw-la shall_v be_v perpendicular_a to_o any_o line_n which_o may_v be_v draw_v through_o the_o point_n b_o in_o the_o plane_n of_o the_o line_n cd_o of_o which_o i_o call_v perpendicular_a to_o the_o plane_n use_v this_o proposition_n come_v often_o in_o use_n in_o the_o first_o book_n of_o theodosius_n for_o example_n to_o demonstrate_v that_o the_o axis_n of_o the_o world_n be_v
square_n of_o the_o line_n add_v be_v double_a to_o the_o square_n of_o half_a the_o line_n and_o to_o the_o square_n which_o be_v compose_v of_o the_o half_a line_n and_o the_o line_n add_v if_o one_o suppose_v ab_fw-la to_o be_v