Selected quad for the lemma: reason_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
reason_n	angle_n	equal_a	line_n	4,117	5	11.1250	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A34005	The sector on a quadrant, or A treatise containing the description and use of four several quadrants two small ones and two great ones, each rendred many wayes, both general and particular. Each of them accomodated for dyalling; for the resolving of all proportions instrumentally; and for the ready finding the hour and azimuth universally in the equal limbe. Of great use to seamen and practitioners in the mathematicks. Written by John Collins accountant philomath. Also An appendix touching reflected dyalling from a glass placed at any reclination.	Collins, John, 1625-1683.; Lyon, John, professor of mathematics. Appendix touching reflective dialling.; Sutton, Henry, mathematical instrument maker.	1659 (1659)	Wing C5382; ESTC R32501	226,510	415

b_o to_o its_o opposite_a side_n m_o o_fw-fr or_o d_o s_o so_o be_v the_o radius_fw-la that_o be_v the_o angle_n at_o o_o or_o d_o to_o its_o opposite_a side_n s_o b_o or_o m_n b._n this_o equality_n be_v admit_v if_o unto_o l_o b_o we_o add_v b_o saint_n the_o sum_n be_v l_o s_o the_o altitude_n for_o summer_n declination_n if_o from_o b_o l_o we_o take_v b_o m_o equal_a to_z b_o s_o the_o remainder_n m_o l_o be_v the_o altitude_n for_o the_o like_a declination_n towards_o the_o depress_v pole_n be_v the_o winter_n altitude_n of_o that_o azimuth_n but_o for_o azimuth_n above_o 90_o d_o from_o the_o meridian_n it_o may_v be_v observe_v in_o the_o two_o opposite_a triangle_n e_o at_fw-fr n_o and_o e_z a_o i_o count_v the_o azimuth_n e_o a_o each_o way_n from_o the_o vertical_a its_o equinoctial_a altitude_n a_o n_o in_o the_o winter_n be_v equal_a to_o its_o equinoctial_a depression_n a_o i_o in_o the_o summer_n and_o be_v to_o be_v find_v by_o the_o 1_o proportion_n the_o second_o proportion_n vary_v not_o as_o the_o sine_fw-la of_o the_o latitude_n n_o f_o equal_a to_o ae_n z_o be_v to_o n_o i_o equal_a to_o z_o b_o the_o cousin_n of_o the_o equinoctial_a altitude_n or_o depression_n so_o be_v the_o sine_fw-la of_o the_o declination_n i_o g_o equal_a to_o d_o s_o to_o sine_fw-la i_o k_o from_o which_o take_v a_o i_o the_o equinoctial_a depression_n rest_v a_o k_o the_o altitude_n seek_v to_o calculate_v a_o table_n of_o the_o sun_n altitude_n for_o all_o azimuth_n and_o hour_n under_o the_o equinoctial_a this_o will_v be_v two_o case_n of_o a_o quadrantal_n sphoerical_a triangle_n 1._o for_o the_o altitude_n on_o all_o azimuth_n there_o will_v be_v give_v the_o side_n a_o b_o a_o quadrant_n the_o angle_n at_o b_o the_o azimuth_n from_o the_o meridian_n and_o the_o side_n a_o d_o the_o compliment_n of_o the_o sun_n declination_n if_o the_o side_n b_o d_o be_v continue_v to_o a_o quadrant_n the_o angle_n at_o c_o will_v be_v a_o right_n angle_v beside_o which_o in_o the_o triangle_n a_o d_o c_o the_fw-mi will_v be_v give_v a_o d_o as_o before_o the_o compliment_n of_o the_o sun_n declination_n and_o a_o c_o the_o measure_n of_o the_o angle_n at_o b_o to_o find_v d_o c_o the_o sun_n altitude_n be_v the_o compliment_n of_o b_o d_o and_o so_o have_v the_o hipotenusal_a and_o one_o of_o the_o leg_n of_o a_o right_n angle_v sphoerical_a triangle_n by_o the_o 7_o case_n we_o may_v find_v the_o other_o leg_n the_o proportion_n suitable_a to_o this_o question_n will_v be_v as_o the_o sine_fw-la of_o the_o azimuth_n from_o east_n or_o west_n be_v to_o the_o radius_fw-la so_o be_v the_o sine_fw-la of_o the_o declination_n to_o the_o cousin_n of_o the_o altitude_n seek_v 2._o for_o the_o altitude_n on_o all_o hour_n there_o will_v be_v give_v the_o side_n a_o b_o a_o quadrant_n a_o d_o the_o compliment_n of_o the_o sun_n declination_n with_o the_o contain_v angle_n b_o a_o d_o the_o hour_n from_o noon_n to_o find_v the_o side_n b_o d_o the_o compliment_n of_o the_o sun_n altitude_n here_o again_o if_o b_o d_o be_v continue_v to_o a_o quadrant_n the_o angle_n at_o c_o be_v a_o right_a angle_n the_o side_n a_o d_o remain_v common_a the_o angle_n d_o a_o c_o be_v the_o compliment_n of_o the_o angle_n b_o a_o d_o see_v page_n 57_o where_o it_o be_v deliver_v that_o if_o a_o sphoerical_a triangle_n have_v one_o right_a angle_n and_o one_o side_n a_o quadrant_n it_o have_v two_o right_a angle_n and_o two_o quadrantal_n side_n and_o therefore_o the_o angle_n b_o a_o c_o be_v a_o right_a angle_n this_o be_v coincident_a with_o the_o 8_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n thereof_o be_v as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o declination_n so_o be_v the_o sine_fw-la of_o the_o hour_n from_o six_o to_o the_o sine_fw-la of_o the_o altitude_n seek_v affection_n of_o sphoerical_a triangle_n because_o the_o last_o affection_n in_o page_n 57_o be_v not_o brace_v in_o the_o beginning_n and_o a_o mistake_n of_o lesser_a for_o great_a in_o the_o last_o brace_n but_o one_o i_o think_v fit_a to_o recite_v it_o at_o large_a any_o side_n of_o a_o sphoerical_a triangle_n be_v continue_v if_o the_o other_o side_n together_o be_v equal_a to_o a_o semicircle_n the_o outward_a angle_n on_o the_o side_n continue_v shall_v be_v equal_a to_o the_o inward_a angle_n on_o the_o say_a side_n opposite_a thereto_o if_o the_o side_n be_v less_o than_o a_o semicircle_n the_o outward_a angle_n will_v be_v great_a than_o the_o inward_a opposite_a angle_n but_o if_o the_o say_a side_n be_v together_o great_a than_o a_o semicircle_n the_o outward_a angle_n will_v be_v less_o than_o the_o inward_a opposite_a angle_n in_o the_o triangle_n annex_v if_o the_o side_n a_o b_o and_o a_o c_o together_o be_v equal_a to_o a_o semicircle_n then_o be_v the_o angle_n a_o c_o d_o equal_a to_o the_o angle_n a_o b_o c._n if_o less_o than_o a_o semicircle_n than_o be_v the_o say_a angle_n great_a than_o the_o angle_n at_o b._n but_o if_o they_o be_v great_a than_o be_v the_o say_a angle_n a_o c_o d_o less_o than_o the_o angle_n at_o b._n by_o reason_n of_o the_o first_o affection_n in_o page_n 58_o which_o want_v a_o brace_n in_o the_o first_o line_n after_o the_o word_n two_o side_n we_o require_v in_o the_o first_o second_o and_o other_o case_n of_o oblique_a angle_a sphoerical_a triangle_n the_o sum_n of_o the_o two_o side_n or_o angle_n give_v to_o be_v less_o than_o a_o semicircle_n before_o i_o finish_v the_o trigonometrical_a part_n i_o think_v it_o not_o amiss_o to_o give_v a_o determination_n of_o the_o certain_a case_n about_o opposite_a side_n and_o angle_n in_o sphoerical_a triangle_n have_v before_o show_v which_o be_v the_o doubtful_a and_o the_o rather_o because_o that_o this_o be_v never_o yet_o speak_v to_o two_o side_n with_o a_o angle_n opposite_a to_o one_o of_o they_o to_o determine_v the_o affection_n of_o the_o angle_n opposite_a to_o the_o other_o 1._o if_o the_o give_v angle_n be_v acute_a and_o the_o opposite_a side_n less_o than_o a_o quadrant_n and_o the_o adjacent_a side_n less_o than_o the_o former_a side_n the_o angle_n it_o subtend_v be_v acute_a because_o subtend_v by_o a_o lesser_a side_n for_o in_o all_o sphoerical_a triangle_n the_o lesser_a side_n subtend_v the_o lesser_a angle_n and_o the_o converse_n 2._o if_o the_o give_v angle_n be_v acute_a and_o the_o opposite_a side_n less_o than_o a_o quadrant_n and_o the_o other_o side_n great_a than_o the_o former_a side_n this_o be_v a_o doubtful_a case_n if_o it_o be_v less_o than_o a_o quadrant_n it_o may_v subtend_v either_o a_o acute_a or_o a_o obtuse_a angle_n and_o so_o it_o may_v also_o do_v if_o it_o be_v great_a than_o a_o quadrant_n yet_o we_o may_v determine_v that_o when_o the_o give_v angle_n be_v acute_a and_o the_o opposite_a side_n less_o than_o a_o quadrant_n but_o great_a than_o the_o compliment_n of_o the_o adjacent_a side_n to_o a_o semicircle_n which_o it_o can_v be_v unless_o the_o adjacent_a side_n be_v great_a than_o a_o quadrant_n the_o angle_n opposite_a thereto_o will_v be_v obtuse_a 3._o the_o give_a angle_n acute_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n and_o the_o other_o side_n great_a if_o two_o side_n be_v great_a than_o quadrant_n if_o one_o of_o they_o subtend_v a_o acute_a angle_n the_o other_o must_v subtend_v a_o obtuse_a angle_n by_o the_o one_a affection_n in_o pag_n 58._o 4._o the_o give_a angle_n acute_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n the_o other_o side_n can_v be_v lesser_a than_o the_o former_a by_o what_o be_v now_o speak_v 5._o if_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n less_o than_o quadrant_n the_o other_o side_n less_o subtend_v a_o acute_a angle_n 6._o if_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n less_o than_o a_o quadrant_n the_o other_o side_n great_a than_o the_o former_a side_n must_v of_o necessity_n be_v also_o great_a than_o a_o quadrant_n otherwise_o two_o side_n less_o than_o quadrant_n shall_v subtend_v two_o obtuse_a angle_n contrary_a to_o the_o first_o affection_n in_o p_o 58._o 7._o if_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n if_o the_o other_o side_n be_v great_a than_o the_o former_a it_o will_v subtend_v a_o more_o obtuse_a angle_n 8._o if_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n if_o the_o other_o side_n be_v less_o than_o the_o former_a whether_o it_o be_v lesser_a or_o great_a than_o a_o quadrant_n it_o may_v either_o subtend_v a_o acute_a or_o obtuse_a

divide_v but_o the_o author_n make_v his_o table_n in_o page_n 5._o without_o double_v to_o be_v graduate_v from_o a_o quadrant_n divide_v into_o 45_o equal_a part_n again_o if_o upon_o the_o centre_n c_o with_o a_o pair_n of_o compass_n each_o degree_n of_o the_o line_n of_o sines_n be_v transfer_v into_o the_o semicircle_n c_o g_o a_o it_o shall_v divide_v it_o into_o 90_o equal_a part_n the_o reason_n whereof_o be_v plain_a because_o the_o sine_fw-la of_o a_o arch_n be_v half_a the_o chord_n of_o twice_o that_o arch_n and_o therefore_o the_o sin_n be_v make_v to_o twice_o the_o radius_fw-la of_o this_o circle_n shall_v be_v transfer_v into_o it_o become_v chord_n of_o the_o like_a arch_n to_o divide_v a_o semicircle_n into_o 90_o equal_a part_n again_o upon_o the_o point_n a_o erect_v a_o line_n of_o tangent_n of_o the_o same_o radius_fw-la with_o the_o former_a sine_fw-la which_o we_o may_v suppose_v to_o be_v infinite_o continue_v here_o we_o use_v a_o portion_n of_o it_o a_o e._n if_o from_o the_o point_n c_o the_o other_o extremity_n of_o the_o diameter_n line_n be_v draw_v cut_v the_o low_a semicircle_n as_o a_o line_n draw_v from_o e_z intersect_v it_o at_o f_o through_o each_o degree_n of_o the_o say_v tangent_fw-la the_o say_v low_a semicircle_n shall_v be_v divide_v into_o 90_o equal_a part_n the_o reason_n be_v evident_a a_o line_n of_o tangent_n from_o the_o centre_n shall_v divide_v a_o quadrant_n into_o 90_o equal_a part_n and_o because_o a_o angle_n in_o the_o circumference_n be_v but_o half_a so_o much_o as_o it_o be_v in_o the_o centre_n be_v transfer_v thither_o a_o whole_a semicircle_n shall_v be_v fill_v with_o no_o more_o part_n the_o chief_a use_n of_o this_o circle_n be_v to_o operate_v proportion_n in_o tangent_n alone_o or_o in_o sines_n and_o tangent_n joint_o build_v upon_o this_o foundation_n that_o equiangle_v plain_a triangle_n have_v their_o side_n proportional_a in_o straight_a line_n it_o will_v be_v evident_a from_o the_o point_n d_o to_o e_z draw_v a_o straight_a line_n intersect_v the_o diameter_n at_o l_o and_o then_o it_o lie_v as_o c_o l_o to_o c_z d_o so_o be_v a_o l_o to_o a_o e_o it_o be_v also_o true_a in_o a_o circle_n provide_v it_o be_v evince_v that_o the_o point_n b_o l_o f_o fall_n in_o a_o straight_a line_n hereof_o i_o have_v a_o geometrical_a demonstration_n which_o will_v require_v more_o scheme_n which_o by_o reason_n of_o its_o length_n and_o difficulty_n i_o think_v fit_a at_o present_a not_o to_o insert_v possible_o a_o easy_a may_v be_v find_v hereafter_o as_o also_o a_o algebraick_n demonstration_n by_o the_o right_n honourable_a the_o lord_n brunkard_n whereby_o after_o many_o algebraick_n inference_n it_o be_v evince_v that_o as_o l_o k_n be_v to_o k_o b_o ∷_o so_o be_v l_o n_o to_o ●_o f_o whence_o it_o will_v follow_v that_o the_o point_n b_o l_o f_o be_v in_o a_o right_a line_n if_o a_o ruler_n be_v lay_v from_o 45_o of_o the_o semicircle_n to_o every_o degree_n of_o the_o quadrant_n of_o latitude_n it_o will_v constitute_v upon_o the_o diameter_n the_o graduation_n of_o the_o line_n sol_fw-mi whereby_o proportion_n in_o sin_n may_v be_v operate_v without_o the_o other_o supply_n from_o the_o same_o scheme_n also_o follow_v the_o construction_n of_o the_o straight_a line_n of_o latitude_n from_o the_o point_n g_o at_o 90_o of_o the_o quadrant_n of_o latitude_n draw_v a_o straight_a line_n to_o c_o and_o transfer_v each_o degree_n of_o the_o quadrant_n of_o latitude_n with_o compass_n one_o foot_n rest_v upon_o c_o into_o the_o say_v straight_a line_n and_o it_o shall_v be_v constitute_v to_o calculate_v it_o the_o line_n of_o latitude_n c_o g_o bear_v such_o proportion_n to_o c_o a_o as_o the_o chord_n of_o 90_o do_v to_o the_o diameter_n which_o be_v the_o same_o that_o the_o sine_fw-la of_o 45_o bear_n to_o the_o radius_fw-la or_o which_o be_v all_o one_o that_o the_o radius_fw-la bear_v to_o the_o secant_fw-la of_o 45_o d_o which_o secant_fw-la be_v equal_a to_o the_o chord_n of_o 90_o d_o from_o the_o diagram_n the_o nature_n of_o the_o line_n of_o latitude_n may_v be_v discover_v any_o two_o line_n be_v draw_v to_o make_v a_o right_a angle_n if_o any_o ark_n of_o the_o line_n of_o latitude_n be_v prick_v off_o in_o one_o of_o those_o line_n retain_v a_o constant_a hipotenusal_a a_o c_o called_z the_o line_n of_o hour_n equal_a to_o the_o diameter_n of_o that_o circle_n from_o whence_o the_o line_n of_o latitude_n be_v constitute_v if_o the_o say_v hipotenusal_a from_o the_o point_n former_o prick_v off_o be_v make_v the_o hipotenusal_a to_o the_o leg_n of_o the_o right_a angle_n former_o prick_v off_o the_o say_a leg_n or_o side_n include_v the_o right_a angle_n shall_v bear_v such_o proportion_n one_o to_o another_o as_o the_o radius_fw-la do_v to_o the_o sine_fw-la of_o the_o ark_n so_o prick_v off_o and_o this_o be_v evident_a from_o the_o schem_n for_o such_o proportion_n as_o a_o c_o bear_v to_o c_o d_o do_v a_o b_o bear_v to_o b_o c_o for_o the_o angle_n at_o a_o be_v common_a to_o both_o triangle_n and_o the_o angle_n at_o b_o in_o the_o circumference_n be_v a_o right_a angle_n and_o consequent_o the_o angle_n a_o c_o b_o will_n be_v equal_a to_o the_o angle_n a_o d_o c_o and_o the_o leg_n a_o c_o to_o c_o d_o bear_n such_o proportion_n by_o construction_n as_o the_o radius_fw-la do_v to_o the_o sine_fw-la of_o a_o ark_n and_o the_o same_o proportion_n do_v a_o b_o bear_v to_o b_o c_o in_o all_o case_n retain_v one_o and_o the_o same_o hypotenusal_a a_o c_o the_o proportion_n therefore_o lie_v evident_a as_o the_o radius_fw-la the_o sine_fw-la of_o the_o angle_n at_o b_o to_o its_o opposite_a side_n a_o c_o the_o secant_fw-la of_o 45d_o so_o be_v the_o sine_fw-la of_o the_o angle_n at_o a_o to_o its_o opposite_a side_n b_o c_o seek_v now_o the_o quantity_n of_o the_o angle_n at_o a_o be_v find_v by_o seek_v the_o natural_a sine_fw-la of_o the_o ark_n propose_v in_o the_o table_n of_o natural_a tangent_n and_o have_v find_v what_o ark_n answer_v thereto_o the_o sine_fw-la of_o the_o say_a ark_n be_v to_o become_v the_o three_o term_n in_o the_o proportion_n but_o the_o cannon_n prescribe_v in_o the_o description_n of_o the_o small_a quadrant_n be_v more_o expedite_a than_o this_o which_o mr_n sutton_n have_v from_o mr_n dary_n long_v since_o for_o who_o and_o by_o who_o direction_n he_o make_v a_o quadrant_n with_o the_o line_n sol_fw-mi and_o two_o parallel_n line_n of_o sines_n upon_o it_o as_o be_v here_o add_v to_o the_o backside_n of_o this_o quadrant_n of_o the_o line_n of_o hour_n alias_o the_o diameter_n or_o proportional_a tangent_fw-la this_o scale_n be_v no_o other_o than_o two_o line_n of_o natural_a tangent_n to_o 45_o d_o each_o set_v together_o at_o the_o centre_n and_o from_o thence_o beginning_n and_o continue_a to_o each_o end_n of_o the_o diameter_n and_o from_o one_o end_n thereof_o number_v with_o 90_o d_o to_o the_o other_o end_n this_o line_n may_v fit_o be_v call_v a_o proportional_a tangent_fw-la for_o whersoever_o any_o ark_n be_v assume_v in_o it_o to_o be_v a_o tangent_fw-la the_o remain_a part_n of_o the_o diameter_n be_v the_o radius_fw-la to_o the_o say_v tangent_fw-la so_o in_o the_o former_a schem_n if_o c_o l_o be_v the_o tangent_fw-la of_o any_o ark_n the_o radius_fw-la thereto_o shall_v be_v a_o l._n in_o the_o schem_n annex_v let_v a_o b_o be_v the_o radius_fw-la of_o a_o line_n of_o tangent_n equal_a to_o c_o d_o and_o also_o parralel_v thereto_o and_o from_o the_o point_n b_o to_o c_o draw_v the_o line_n b_o c_o and_o let_v it_o be_v require_v to_o divide_v the_o same_o into_o a_o line_n of_o proportional_a tangent_n i_o say_v line_n draw_v from_o the_o point_n d_o to_o every_o degree_n of_o the_o tangent_fw-la a_o b_o shall_v divide_v one_o half_a of_o it_o as_o require_v from_o the_o similitude_n of_o two_o right_n angle_v equiangle_v plain_a triangle_n which_o will_v have_v their_o side_n proportional_a it_o will_v therefore_o hold_v as_o c_o f_o to_o c_o d_o so_o f_o b_o to_z b_o e_z and_o the_o converse_n as_o the_o second_o term_n c_o d_o to_z the_o fourth_z b_o e_z so_o be_v the_o first_o c_o f_o to_o the_o three_o f_o b_o and_o therefore_o c_o f_o bear_v such_o proportion_n to_o f_o b_o as_o c_z d_o do_v to_o b_o e_o which_o be_v the_o same_o that_o the_o radius_fw-la bear_v to_o the_o tangent_fw-la of_o the_o ark_n propose_v if_o it_o be_v doubt_v whether_o the_o diameter_n will_v be_v a_o double_a tangent_fw-la or_o the_o line_n here_o describe_v such_o a_o line_n a_o proportion_n shall_v be_v give_v to_o find_v by_o experience_n or_o calculation_n what_o line_n

this_o latter_a case_n resolve_v the_o opposite_a triangle_n example_n in_o the_o former_a triangle_n give_v ☉_o p_o comple_v declination_n 66_o 29′_n z_o p_o comple_v latitude_n 38_o 28_o angle_n ☉_o z_o p_o the_o azimuth_n 63_o 54_o to_o find_v the_o hour_n z_o p_o ☉_o 105_o the_o first_o operation_n will_v find_v the_o angle_n of_o position_n as_o before_o 37_o d_o 32′_n the_o second_o operation_n half_a difference_n of_o the_o give_v angle_n 13_o d_o 11_o m_o tangent_fw-la 936966_o half_a sum_n of_o the_o side_n 52_o 28′_n 30″_n sine_fw-la 989931_o 1926897_o half_a difference_n of_o the_o side_n 14_o d_o 00′_n 30″_n sine_fw-la 938393_o tangent_fw-la 37_o d_o 30′_n 988504_o comple_v be_v 52_o 30_o double_a make_n 105_o d_o the_o angle_n seek_v 7._o two_o angle_n with_o a_o side_n opposite_a to_o one_o of_o they_o be_v give_v to_o find_v the_o three_o angle_n the_o kind_n of_o the_o side_n opposite_a to_o the_o other_o angle_n be_v foreknow_v first_o find_v the_o side_n opposite_a to_o the_o other_o angle_n by_o 4_o case_n and_o then_o we_o have_v two_o angle_n and_o their_o opposite_a side_n to_o find_v the_o three_o angle_n by_o transpose_v the_o order_n of_o either_o of_o the_o proportion_n use_v in_o the_o first_o case_n the_o latter_a will_v be_v as_o the_o cousin_n of_o half_a the_o difference_n of_o the_o side_n to_o the_o tangent_fw-la of_o half_a the_o sum_n of_o the_o angle_n so_o the_o cousin_n of_o half_a the_o sum_n of_o the_o side_n to_o the_o cotangent_fw-la of_o half_a the_o contain_v angle_n example_n in_o the_o triangle_n z_o ☉_o p_o data_fw-la angle_v ☉_o 37_o d_o 32′_n angle_n p_o 105_o 00_o side_n ☉_o z_o 80_o 31_o to_o find_v the_o angle_n z_o 63_o 54_o the_o first_o operation_n will_v find_v z_o p_o 38_o 28_o the_o second_o operation_n half_a sum_n of_o the_o angle_n 71_o d_o 16′_n tangent_fw-la 1046963_o half_a sum_n of_o the_o side_n 59_o d_o 29′_n 30″_n sine_fw-la compl_n 970558_o 2017521_o half_a difference_n of_o the_o side_n 21_o d_o 1′_n 30″_n cousin_n 997007_o tangent_fw-la 58_o 3′_n 1020514_o compl_n 31_o 57′_n double_v be_v 63_o 54′_n the_o angle_n seek_v 8._o two_o angle_n with_o a_o side_n opposite_a to_o one_o of_o they_o be_v give_v to_o find_v the_o interjacent_a side_n the_o kind_n of_o the_o side_n opposite_a to_o the_o other_o angle_n be_v fore_o know_v first_o find_v the_o side_n opposite_a to_o the_o other_o angle_n by_o 4_o case_n and_o then_o you_o have_v two_o side_n and_o their_o opposite_a angle_n give_v to_o find_v the_o 3_o side_n by_o tranpose_v the_o order_n of_o either_o of_o the_o proportion_n use_v in_o the_o 2_o case_n the_o latter_a will_v be_v as_o the_o cousin_n of_o half_a the_o difference_n of_o the_o two_o angle_n to_o the_o tangent_fw-la of_o half_a the_o sum_n of_o the_o two_o side_n so_o the_o cousin_n of_o half_a the_o sum_n of_o the_o two_o give_v angle_n to_o the_o tangent_fw-la of_o half_a the_o three_o side_n example_n in_o the_o former_a triangle_n give_v the_o hour_n angle_n at_o p_o 105_o 00_o azimuth_n angle_n at_o z_o 63_o 54_o compl_a altitude_n side_n z_o ☉_o 80_o 31_o to_o find_v the_o compl._n of_o the_o latitude_n the_o side_n z_o p_o 38_o 28_o the_o first_o operation_n will_v find_v the_o side_n p_o ☉_o 66_o 29′_n second_o operation_n half_n the_o sum_n of_o the_o two_o side_n 73_o 30′_n tangent_fw-la 1052839_o half_n the_o sum_n of_o the_o two_o angle_n 84_o 27_o cousin_n 898549_o 1951388_o half_a the_o difference_n of_o the_o two_o angle_n 20_o 33′_n cousin_n 997144_o tangent_fw-la of_o 19_o 14′_n 954244_o doubled_n be_v 38_o 28_o the_o side_n seek_v these_o 6_o last_o precedent_n case_n may_v be_v call_v the_o doubtful_a case_n because_o that_o three_o give_v term_n be_v not_o sufficient_a data_fw-la to_o find_v one_o single_a answer_n without_o the_o quality_n of_o a_o four_o which_o be_v demonstrate_v by_o clavius_n in_o theodosium_n and_o see_v it_o pass_v without_o due_a caution_n in_o our_o english_a book_n i_o shall_v insert_v it_o from_o he_o let_v a_o d_o and_o a_o c_o be_v two_o equal_a side_n include_v the_o angle_n d_o ac_fw-la and_o both_o of_o they_o less_o or_o great_a than_o a_o quadrant_n draw_v through_o the_o point_n c_o and_o d_o the_o arch_n of_o a_o great_a circle_n c_o d_o continue_v it_o and_o draw_v thereunto_o another_o arch_n or_o side_n from_o a_o namely_o a_o b_o neither_o through_o the_o pole_n of_o the_o arch_n c_o d_o nor_o through_o the_o pole_n of_o the_o arch_n a_o d_o so_o that_o the_o angle_n b_o and_o b_o a_o d_o may_v not_o be_v right_a angle_n nor_o the_o angle_n a_o d_o b_o if_o then_o each_o of_o these_o side_n a_o d_o a_o c_o be_v less_o than_o a_o quadrant_n the_o two_o angle_n c_o and_o a_o d_o c_o will_n be_v acute_a and_o if_o these_o ark_n be_v great_a respective_o than_o a_o quadrant_n the_o two_o angle_n c_o and_o a_o d_o c_o will_v be_v obtuse_a whence_o it_o come_v to_o pass_v that_o the_o angle_n a_o d_o b_o be_v obtuse_a when_o the_o angle_n a_o d_o c_o be_v acute_a and_o the_o contrary_a now_o forasmuch_o as_o the_o side_n a_o c_o and_o a_o d_o be_v equal_a to_o each_o other_o the_o other_o data_fw-la viz._n the_o side_n a_o b_o and_o the_o angle_n at_o bare_a common_a to_o both_o for_o in_o each_o triangle_n a_o b_o d_o and_o a_o b_o c_o there_o be_v give_v two_o side_n with_o the_o angle_n at_o b_o opposite_a to_o one_o of_o they_o now_o this_o be_v not_o sufficient_a data_fw-la to_o find_v the_o angle_n opposite_a to_o the_o other_o side_n which_o may_v be_v either_o the_o acute_a angle_n at_o c_o or_o the_o obtuse_a angle_n adb_v the_o compliment_n thereof_o to_o a_o semicircle_n nor_o to_o find_v the_o three_o side_n which_o may_v be_v either_o b_o d_o or_o the_o whole_a side_n b_o c_o nor_o the_o angle_n include_v which_o may_v be_v either_o b_o a_o d_o or_o b_o a_o c_o therefore_o in_o these_o 3_o case_n we_o have_v require_v the_o quality_n of_o the_o angle_n opposite_a to_o the_o other_o give_v side_n a_o b_o and_o though_o it_o be_v not_o so_o much_o observe_v in_o the_o other_o trigonometry_n by_o perpendicular_o let_v fall_v without_o the_o knowledge_n of_o the_o say_v angle_v it_o can_v not_o be_v determine_v whether_o the_o perpendicular_a will_v fall_v with_o in_o or_o without_o the_o triangle_n nor_o whether_o the_o angle_n find_v in_o the_o first_o case_n be_v the_o thing_n seek_v or_o its_o compliment_n to_o 180_o nor_o whether_o the_o angle_n or_o segment_v find_v by_o one_a and_o 2_o operation_n in_o the_o other_o case_n be_v to_o be_v add_v together_o or_o substract_v from_o each_o other_o to_o obtain_v the_o side_n or_o angle_v seek_v so_o also_o two_o angle_n with_o a_o side_n opposite_a to_o one_o of_o they_o be_v not_o sufficient_a data_fw-la to_o obtain_v a_o four_o thing_n in_o the_o say_a triangle_n without_o the_o affection_n of_o the_o side_n opposite_a to_o the_o other_o give_v angle_n let_v a_o b_o and_o a_o c_o be_v two_o unequal_a side_n contain_v the_o angle_n b_o a_o c_o both_o together_o equal_a to_o a_o semicircle_n one_o being_n great_a the_o other_o less_o than_o a_o quadrant_n draw_v through_o the_o point_n b_o and_o c_o the_o arch_n of_o a_o great_a circle_n b_o c_o continue_v it_o and_o draw_v thereto_o from_o a_o another_o side_n ad_fw-la but_o not_o through_o the_o pole_n of_o a_o c_o nor_o through_o the_o pole_n of_o b_o c_o so_fw-mi that_o the_o angle_n d_o and_o c_o a_o d_o may_v not_o be_v right_a angle_n nor_o the_o angle_n a_o c_o d_o a_o right_a angle_n for_o if_o it_o be_v a_o right_a angle_n the_o angle_n a_o b_o c_o whereto_o it_o be_v equal_a shall_v be_v also_o a_o right_a angle_n and_o so_o the_o two_o side_n a_o b_o and_o a_o c_o by_o reason_n of_o their_o right_a angle_n at_o b_o and_o c_o shall_v be_v equal_a and_o be_v quadrant_n contrary_a to_o the_o supposition_n now_o the_o angle_n a_o c_o d_o and_o a_o b_o c_o being_n equal_v which_o be_v thus_o prove_v suppose_v the_o two_o side_n a_o b_o and_o b_o d_o to_o be_v continue_v to_o a_o semicircle_n at_o e_o then_o will_v the_o say_a angle_n be_v equal_a to_o its_o opposite_a angle_n at_o b_o the_o side_n a_o c_o by_o supposition_n be_v equal_a to_o the_o side_n a_o e_o the_o compliment_n of_o the_o side_n a_o b_o to_o a_o semicircle_n but_o equal_a side_n subtend_v equal_a angle_n therefore_o the_o angle_n at_o c_o be_v equal_a to_o the_o angle_n at_o b_o or_o at_o e_o which_o be_v admit_v retain_v the_o side_n a_o d_o and_o angle_n at_o d_o we_o have_v another_o angle_n opposite_a thereto_o either_o c_z or_o b_o

angle_a triangle_n c_o g_o f_o right_a angle_v at_o g_o as_o the_o sine_z of_o the_o latitude_n the_o angle_n at_o f_o to_o its_o opposite_a side_n c_o g_o the_o sine_z of_o the_o declination_n so_o the_o radius_fw-la the_o angle_n at_o g_o to_z the_o secant_fw-la c_o z_o f._n again_o in_o summer_n as_o the_o cousin_n of_o the_o latitude_n the_o angle_n at_o ☉_o to_o its_o opposite_a side_n d_o z_o f_o the_o difference_n between_o the_o former_a secant_fw-la and_o the_o sine_fw-la of_o the_o altitude_n so_o be_v the_o sine_fw-la of_o the_o latitude_n the_o angle_n at_o f_o to_o its_o opposite_a side_n d_o ☉_o the_o sine_z of_o the_o azimuth_n from_o the_o vertical_a in_o the_o parralel_n of_o altitude_n in_o winter_n as_o cousin_n latitude_n angle_n at_o a_o to_z b_o e_z the_o sum_n of_o the_o former_a secant_fw-la equal_a to_o e_o m_o and_o of_o the_o sine_fw-la of_o the_o altitude_n m_o b_o so_o be_v the_o sine_fw-la of_o the_o latitude_n the_o angle_n b_o to_z a_o e_z equal_a to_o b_o d_o the_o sine_z of_o the_o azimuth_n in_o a_o parallel_n as_o before_o to_o be_v reduce_v to_o the_o common_a radius_fw-la from_o this_o schem_n may_v be_v observe_v the_o reason_n why_o the_o sun_n in_o those_o latitude_n upon_o some_o azimuth_n have_v two_o altitude_n because_o the_o parralel_n of_o his_o declination_n f_o r_o intersect_v and_o pass_v through_o the_o azimuth_n namely_o the_o prick_a ellipsis_n in_o the_o two_o point_n saint_n ☉_o i_o now_o proceed_v to_o the_o use_n in_o calculate_v a_o table_n of_o hour_n for_o those_o that_o have_v occasion_n to_o calculate_v a_o table_n of_o hour_n to_o any_o assign_a altitude_n and_o parralel_n of_o declination_n it_o will_v be_v the_o ready_a way_n to_o write_v down_o all_o the_o movable_a term_n first_o as_o the_o natural_a sin_n of_o the_o several_a altitude_n in_o a_o rule_v sheet_n of_o paper_n and_o then_o upon_o a_o piece_n of_o card_n to_o write_v down_o the_o natural_a sine_fw-la of_o the_o sun_n altitude_n at_o 6_o and_o remove_v to_o every_o altitude_n get_v the_o sum_n or_o difference_n according_o which_o be_v have_v seek_v the_o same_o in_o the_o natural_a sin_n and_o write_v down_o the_o log_n m_o that_o stand_v against_o it_o then_o upon_o the_o other_o end_n of_o the_o piece_n of_o card_n get_v the_o sum_n of_o the_o arithmetical_a compliment_n of_o the_o logarithmical_a cousin_n of_o the_o declination_n and_o of_o the_o logarithmica_fw-la cousin_n of_o the_o latitude_n and_o add_v this_o fix_a number_n to_o the_o logme_n before_o write_v down_o by_o remove_v the_o card_n to_o every_o one_o of_o they_o and_o the_o sum_n be_v the_o logme_n of_o the_o sine_fw-la of_o the_o hour_n from_o 6_o if_o the_o logme_n be_v well_o proportion_v out_o to_o the_o difference_n which_o may_v be_v sufficient_o do_v by_o guess_n example_n comp_n latitude_n 38_o 28′_n ar_n comp_n 0,2061683_o comp_n delinat_fw-la 66_o 29_o be_v comp_n 0,0376572_o fix_v number_n ,2438255_o let_v the_o altitude_n be_v 36_o 42′_n nat_n sine_fw-la 5976251_o natural_a sine_fw-la altitude_n at_o 6_o 3124174_o difference_n 2852077_o log_n against_o 9_o 4550441_o sine_fw-la of_o 30_o the_o hour_n from_o 6_o towards_o noon_n 9,6988696_o another_o example_n n_o s_o altitude_n at_o 6_o 3124174_o let_v the_o altitude_n be_v 13_o 46′_n n_o saint_n 2379684_o difference_n 744490_o logm_n 8,8715646_o the_o former_a fix_a number_n 0,2438255_o sine_fw-la of_o 7_o 30′_n the_o hour_n from_o 6_o towards_o midnight_n because_o the_o altitude_n be_v less_o than_o the_o altitude_n of_o 6._o 9,1153901_o this_o method_n of_o calculation_n will_v dispatch_v much_o fast_a than_o the_o common_a canon_n when_o three_o side_n be_v give_v to_o find_v a_o angle_n the_o azimuth_n may_v in_o like_a manner_n be_v calculate_v but_o will_v be_v more_o troublesome_a not_o have_v so_o many_o fix_a term_n in_o it_o and_o have_v get_v the_o hour_n the_o azimuth_n will_v be_v easy_o find_v in_o this_o case_n we_o have_v two_o side_n and_o a_o angle_n opposite_a to_o one_o of_o they_o give_v to_o find_v the_o angle_n opposite_a to_o the_o other_o and_o the_o proportion_n will_v hold_v as_o the_o cousin_n of_o the_o altitude_n to_o the_o sine_fw-la of_o the_o hour_n from_o the_o meridian_n so_o the_o cousin_n of_o the_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n from_o the_o meridian_n and_o in_o this_o case_n the_o three_o side_n be_v give_v we_o may_v determine_v the_o affection_n of_o any_o of_o the_o angle_n if_o the_o sun_n or_o star_n have_v declination_n towards_o the_o depress_a pole_n the_o azimuth_n be_v always_o obtuse_a and_o the_o hour_n and_o angle_v of_o position-acute_a if_o the_o sun_n etc._n etc._n have_v declination_n towards_o the_o elevate_v pole_n but_o less_o than_o the_o latitude_n of_o the_o place_n the_o angle_n of_o position_n be_v always_o acute_a the_o hour_n before_o 6_o obtuse_a the_o hour_n and_o azimuth_n between_o the_o altitude_n of_o 6_o and_o the_o vertical_a altitude_n both_o acute_a afterward_o the_o hour_n acute_a and_o the_o azimuth_n obtuse_a but_o when_o the_o sun_n or_o star_n come_v to_o the_o meridian_n between_o the_o zenith_n and_o the_o elevate_v pole_n as_o when_o their_o declination_n be_v great_a than_o the_o latitude_n of_o the_o place_n the_o azimuth_n be_v always_o acute_a the_o hour_n before_o 6_o obtuse_a afterward_o acute_a the_o angle_n of_o position_n from_o the_o time_n of_o rise_v to_o the_o remote_a azimuth_n from_o the_o meridian_n be_v acute_a afterward_o obtuse_a another_o general_a proportion_n for_o the_o hour_n as_o the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o latitude_n so_o the_o tangent_fw-la of_o the_o sun_n declination_n to_o the_o sine_fw-la of_o the_o hour_n of_o rise_v from_o six_o again_o to_o the_o rectangle_n of_o the_o cousin_n of_o the_o latitude_n and_o of_o the_o cousin_n of_o the_o declination_n be_v to_o the_o square_n of_o the_o radius_fw-la so_o be_v the_o sine_fw-la of_o the_o altitude_n to_o the_o difference_n of_o the_o verse_v sin_n of_o the_o semidiurnal_a ark_n and_o of_o the_o hour_n seek_v have_v get_v the_o logarithm_n of_o this_o difference_n take_v the_o natural_a number_n out_o of_o the_o sin_n or_o tangent_n that_o stand_v against_o it_o according_o as_o the_o logme_n be_v seek_v and_o in_o winter_n add_v it_o to_o the_o natural_a sine_fw-la of_o the_o hour_n of_o rise_v from_o 6_o the_o sum_n be_v the_o natural_a sine_fw-la of_o the_o hour_n from_o 6_o towards_o noon_n in_o summer_n get_v the_o difference_n between_o this_o four_o and_o the_o sine_fw-la of_o rise_v from_o 6_o the_o say_a difference_n be_v the_o natural_a sine_fw-la of_o the_o hour_n from_o 6_o towards_o noon_n when_o the_o number_n find_v by_o the_o proportion_n be_v great_a than_o the_o sine_fw-la of_o rise_v towards_o midnight_n when_o less_o the_o canon_n be_v the_o same_o without_o variation_n as_o well_o for_o south_n declination_n as_o for_o north_n and_o therefore_o we_o may_v by_o help_n thereof_o find_v two_o hour_n to_o the_o same_o altitude_n example_n comp_n lat_n 38_o 28′_n ar_n comp_n sine_fw-la 2061683_o comp_n declin_n 77_o ar_n comp_n 0112761_o fix_a number_n 2174444_o let_v the_o altitude_n be_v 14_o 38′_n sine_fw-la 9,4024889_o natural_a sine_fw-la against_o it_o 41660_o 00_o 9,6199333_o nat_n sine_fw-la of_o rise_v from_o 6_o 29058_o 79_o sum_n 70718_o 79_o n_o sine_fw-la of_o 45_o the_o hour_n from_o six_o in_o winter_n difference_n 12601_o 21_o sine_fw-la of_o 7_o 14′_n the_o hour_n from_o 6_o in_o summer_n towards_o noon_n to_o the_o former_a altitude_n and_o like_o declination_n towards_o elevate_v pole_n another_o example_n for_o the_o same_o latitude_n and_o declination_n logme_n let_v the_o altitude_n be_v 20_o 25′_n sine_fw-la 9,5426321_o the_o former_a fix_a logme_n 2174444_o natural_a sine_fw-la against_o it_o 5754811_o sum_n 9,7600765_o natural_a sine_fw-la of_o rise_v 2905879_o sum_n 9,7600765_o sum_n 8660690_o sine_fw-la 60_o the_o ho_o from_o 6_o in_o wint_fw-mi difference_n 2848932_o sine_fw-la 16_o 33′_n the_o hour_n from_o 6_o in_o summer_n towards_o noon_n and_o thus_o may_v two_o hour_n be_v find_v at_o one_o operation_n for_o all_o altitude_n less_o than_o the_o winter_n meridian_n altitude_n to_o be_v convert_v into_o usual_a time_n by_o allow_v 15_o to_o a_o hour_n and_o 4_o to_o a_o degree_n to_o calculate_v a_o table_n of_o the_o sun_n altitude_n on_o all_o hour_n as_o the_o secant_fw-la of_o the_o latitude_n to_o the_o cousin_n of_o the_o declination_n or_o which_o be_v all_o one_o as_o the_o square_n of_o the_o radius_fw-la to_o the_o rectangle_n of_o the_o cosines_n both_o of_o the_o latitude_n and_o of_o the_o declination_n so_o be_v the_o sine_fw-la of_o the_o hour_n from_o 6._o to_o a_o four_o namely_o in_o summer_n the_o difference_n of_o the_o sin_n of_o the_o sun_n

angle_n but_o we_o may_v determine_v that_o when_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n if_o the_o compliment_n of_o the_o adjacent_a side_n to_o a_o semicircle_n be_v great_a than_o the_o say_v opposite_a side_n the_o angle_n subtend_v by_o the_o say_v adjacent_a side_n be_v acute_a two_o angle_n with_o a_o side_n opposite_a to_o one_o of_o they_o to_o determine_v the_o affection_n of_o the_o side_n opposite_a to_o the_o other_o 1._o if_o the_o give_v angle_n be_v acute_a and_o the_o opposite_a side_n less_o than_o a_o quadrant_n if_o the_o other_o angle_n be_v less_o the_o side_n opposite_a thereto_o shall_v be_v less_o than_o a_o quadrant_n because_o it_o subtend_v a_o lesser_a angle_n 2._o if_o the_o give_v angle_n be_v acute_a and_o the_o opposite_a side_n less_o than_o a_o quadrant_n if_o the_o other_o angle_n be_v great_a the_o case_n be_v ambiguous_a yet_o we_o may_v determine_v if_o the_o give_v angle_n be_v acute_a and_o the_o opposite_a side_n lesser_a than_o a_o quadrant_n if_o the_o compliment_n of_o the_o other_o angle_n to_o a_o semicircle_n be_v less_o than_o the_o acute_a angle_n which_o it_o can_v be_v but_o when_o the_o latter_a angle_n be_v obtuse_a the_o side_n subtend_v it_o shall_v be_v great_a than_o a_o quadrant_n 3._o if_o the_o give_v angle_n be_v acute_a and_o the_o opposite_a side_n grnate_a than_o a_o quadrant_n if_o the_o other_o angle_n be_v lesser_a then_o by_o 12_o of_o 4_o book_n of_o regiomontanus_n if_o two_o acute_a angle_n be_v unequal_a the_o side_n opposite_a to_o the_o lesser_a of_o they_o shall_v be_v less_o than_o a_o quadrant_n 4._o if_o the_o give_v angle_n be_v acute_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n if_o the_o other_o angle_n be_v great_a it_o must_v of_o necessity_n be_v obtuse_a because_o otherways_o two_o acute_a unequal_a angle_n the_o side_n opposite_a to_o the_o lesser_a of_o they_o shall_v not_o be_v lesser_a than_o a_o quadrant_n contraty_a the_o former_a place_n of_o regiomontanus_n 5._o if_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n lesser_a than_o a_o quadrant_n if_o the_o other_o angle_n be_v lesser_a it_o must_v of_o necessity_n be_v acute_a and_o the_o side_n subtend_v it_o less_o than_o a_o quadrant_n otherways_o two_o side_n less_o than_o quadrant_n shall_v subtend_v two_o obtuse_a angle_n contrary_a to_o one_a affection_n in_o page_n 58._o 6._o if_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n less_o than_o a_o quadrant_n if_o the_o other_o angle_n be_v great_a by_o 13_o prop_n of_o 4_o h_z of_o regiomontanus_n if_o a_o triangle_n have_v two_o obtuse_a unequal_a angle_n the_o side_n opposite_a to_o the_o great_a of_o they_o shall_v be_v great_a than_o a_o quadrant_n 7._o if_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n if_o the_o other_o angle_n be_v great_a or_o more_o obtuse_a than_o the_o former_a it_o be_v subtend_v by_o a_o great_a side_n 8._o if_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n if_o the_o other_o angle_n be_v less_o than_o the_o former_a the_o case_n be_v ambiguous_a yet_o we_o may_v determine_v that_o when_o the_o give_v angle_n be_v obtuse_a and_o the_o opposite_a side_n great_a than_o a_o quadrant_n if_o the_o other_o angle_n be_v less_o than_o the_o compliment_n of_o the_o say_v obtuse_a angle_n to_o a_o semicircle_n the_o side_n subtend_v it_o shall_v be_v less_o than_o a_o quadrant_n the_o former_a case_n that_o be_v still_o and_o always_o will_v be_v doubtful_a may_v be_v determine_v when_o three_o side_n be_v give_v a_o sphoerical_a triangle_n have_v two_o side_n less_o than_o quadrant_n and_o one_o great_a will_v always_o have_v one_o obtuse_a angle_n opposite_a to_o that_o great_a side_n and_o both_o the_o other_o angle_v acute_a a_o sphoerical_a triangle_n have_v three_o side_n less_o than_o quadrant_n can_v have_v but_o one_o obtuse_a angle_n and_o many_o time_n none_o and_o that_o obtuse_a angle_n shall_v be_v subtend_v by_o the_o great_a side_n but_o whether_o the_o great_a side_n subtend_v a_o acute_a or_o obtuse_a angle_n can_v be_v know_v unless_o give_v or_o find_v by_o calculation_n and_o that_o may_v be_v find_v several_a way_n first_o by_o help_n of_o the_o leg_n or_o side_n include_v the_o angle_n seek_v by_o 15_o case_n of_o right_n angle_v sphoerical_a triangle_n as_o the_o radius_fw-la to_o the_o cousin_n of_o one_o of_o those_o leg_n so_o be_v the_o cousin_n of_o the_o other_o leg_n to_o the_o sine_fw-la of_o a_o four_o arch._n if_o the_o three_o side_n be_v great_a than_o the_o compliment_n of_o the_o four_o arch_n to_o 90_o the_o angle_n include_v be_v obtuse_a if_o equal_a to_o it_o a_o right_a angle_n if_o less_o a_o acute_a angle_n second_o by_o help_n of_o one_o of_o the_o leg_n and_o the_o base_a or_o side_n subtend_v the_o angle_n seek_v by_o 7_o case_n of_o right_n angle_v sphoerical_a triangle_n as_o the_o cousin_n of_o the_o adjacent_a side_n be_v one_o of_o the_o lesser_a side_n be_v to_o the_o radius_fw-la so_o the_o cousin_n of_o the_o opposite_a side_n to_z the_o sine_z of_o a_o four_o arch._n if_o the_o three_o side_n be_v great_a than_o the_o compliment_n of_o the_o 4_o arch_n to_o 90_o the_o angle_n subtend_v be_v acute_a if_o equal_a to_o it_o a_o right_a angle_n if_o less_o a_o obtuse_a angle_n all_o other_o case_n need_v no_o determination_n if_o a_o triangle_n have_v two_o side_n give_v big_a than_o quadrant_n make_v recourse_n to_o the_o opposite_a triangle_n and_o it_o will_v agree_v to_o these_o case_n if_o three_o angle_n be_v give_v to_o determine_v the_o affection_n of_o the_o side_n if_o they_o be_v all_o acute_a the_o three_o side_n subtend_v they_o will_v be_v all_o less_o than_o quadrant_n but_o observe_v that_o though_o a_o triangle_n that_o have_v but_o one_o side_n great_a than_o a_o quadrant_n can_v and_o shall_v always_o have_v but_o one_o obtuse_a angle_n yet_o a_o triangle_n that_o have_v but_o one_o obtuse_a angle_n may_v frequent_o have_v two_o side_n great_a than_o quadrant_n in_o this_o and_o all_o other_o case_n let_v the_o angle_n be_v change_v into_o side_n and_o the_o former_a rule_n will_v serve_v i_o shall_v have_v add_v a_o brief_a application_n of_o all_o the_o axiom_n that_o be_v necessary_a to_o be_v remember_v and_o have_v reduce_v the_o oblique_a case_n to_o settle_a proportion_n with_o the_o cadence_n of_o perpendicular_o only_o to_o show_v how_o they_o arise_v whereby_o they_o will_v be_v render_v very_o facile_a as_o also_o the_o demonstration_n of_o the_o affection_n which_o may_v be_v hereafter_o add_v to_o some_o other_o treatise_n to_o be_v bind_v with_o this_o book_n of_o work_a proportion_n by_o the_o line_n on_o the_o quadrant_n before_o i_o come_v to_o show_v how_o all_o proportion_n may_v in_o some_o measure_n be_v perform_v upon_o the_o line_n of_o this_o quadrant_n it_o be_v to_o be_v intimate_v in_o general_a that_o the_o work_n of_o a_o proportion_n upon_o a_o single_a natural_a line_n be_v the_o useful_a invention_n of_o the_o late_a learned_a mathematician_n mr_n samuel_n foster_n and_o publish_v after_o his_o decease_n as_o he_o in_o the_o use_n of_o his_o scale_n a_o book_n call_v posthuma_n fosteri_n as_o also_o by_o mr_n stirrup_n in_o a_o treatise_n of_o dyall_v in_o which_o book_n though_o it_o be_v there_o prescribe_v and_o from_o thence_o may_v be_v learned_a yet_o i_o acknowledge_v i_o receive_v some_o light_n concern_v it_o from_o some_o manuscript_n lend_v i_o by_o mr_n foster_n in_o his_o life_n time_n to_o transcribe_v for_o his_o and_o my_o own_o use_n touch_v instrumental_a application_n yet_o withal_o be_v it_o here_o intimate_v that_o there_o be_v no_o way_n use_v upon_o this_o quadrant_n for_o the_o obtain_v the_o hour_n and_o azimuth_n with_o compass_n and_o the_o converse_n of_o the_o 4_o axiom_n but_o what_o be_v whole_o my_o own_o and_o altogether_o novel_a though_o not_o worth_a the_o own_n for_o instrumental_a conclusion_n not_o be_v so_o exact_a as_o the_o table_n be_v of_o small_a esteem_n with_o the_o learned_a as_o in_o mr_n wingates_n preface_n to_o the_o posthuma_n beside_o the_o take_n off_o of_o any_o line_n from_o the_o limb_n to_o any_o radius_fw-la the_o explanation_n of_o the_o reason_n of_o proportion_n so_o wrought_v the_o supply_n of_o many_o defect_n and_o the_o inscribe_v of_o line_n in_o the_o limb_n i_o have_v not_o see_v any_o thing_n of_o mr_n foster_n or_o of_o any_o other_o man_n tend_v thereto_o of_o the_o line_n of_o equal_a part_n this_o line_n issue_v from_o the_o centre_n of_o the_o quadrant_n on_o the_o right_a edge_n of_o the_o foreside_n and_o will_v

sine_fw-la and_o enter_v the_o former_a extent_n between_o the_o scale_n and_o the_o thread_n and_o the_o foot_n of_o the_o compass_n will_v on_o the_o line_n of_o equal_a part_n show_v the_o four_o proportional_a the_o proportion_n for_o find_v the_o altitude_n of_o a_o tower_n at_o one_o station_n by_o the_o measure_a distance_n may_v also_o be_v wrought_v in_o in_o equal_a part_n and_o sin_n for_o as_o the_o cousin_n of_o the_o ark_n at_o first_o station_n to_o the_o measure_a distance_n thereof_o from_o the_o tower_n so_o be_v the_o sine_fw-la of_o the_o say_a ark_n to_o the_o altitude_n of_o the_o tower_n in_o that_o former_a scheme_n the_o measure_a distance_n b_o h_n be_v 85_o and_o the_o angle_n observe_v at_o h_n 48_o d_o 29′_n wherefore_o i_o lie_v the_o thread_n to_o the_o sine_fw-la of_o the_o say_a ark_n in_o the_o limb_n count_v from_o the_o right_a edge_n and_o from_o the_o measure_a distance_n in_o the_o equal_a part_n take_v the_o near_a extent_n to_o the_o thread_n then_o lay_v the_o thread_n to_o the_o cousin_n of_o the_o say_a ark_n in_o the_o limb_n and_o enter_v the_o former_a extent_n between_o the_o thread_n and_o the_o scale_n i_o shall_v find_v the_o foot_n of_o the_o compass_n to_o fall_v upon_o 96_o the_o altitude_n seek_v so_o also_o in_o the_o triangle_n a_o c_o b_o if_o there_o be_v give_v the_o side_n a_o c_o 194_o the_o measure_a distance_n between_o two_o station_n on_o the_o wall_n of_o a_o town_n besiege_v and_o the_o observe_v angle_n at_o a_o 25_o d_o 22′_n at_o c_o 113_o d_o 22′_n if_o b_o be_v a_o battery_n we_o may_v by_o this_o work_n find_v the_o distance_n of_o it_o from_o either_o a_o or_o c_o for_o have_v two_o angle_n give_v all_o the_o three_o be_v give_v it_o therefore_o hold_v as_o the_o sine_fw-la of_o the_o angle_n ot_fw-mi b_o 41d_o 16′_n to_o its_o opposite_a side_n a_o c_o 194_o so_o the_o sine_fw-la of_o the_o angle_n at_o c_o 66d_o 38′_n the_o compliment_n to_o its_o opposite_a side_n b_o a_o 270_o the_o distance_n of_o the_o battery_n from_o a_o such_o proportion_n as_o have_v the_o radius_fw-la in_o they_o will_v be_v more_o easy_o wrought_v we_o shall_v give_v some_o few_o example_n in_o use_n in_o navigation_n 1._o to_o find_v how_o many_o mile_n or_o league_n in_o each_o parralel_n of_o latitude_n answer_v to_o one_o degree_n of_o longitude_n as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o latitude_n so_o the_o number_n of_o miles_n in_o a_o degree_n in_o the_o equinoctial_a to_o the_o number_n of_o miles_n in_o the_o parralel_n so_o in_o 51_o d_o 32′_n of_o latitude_n if_o 60_o mile_n answer_v to_o a_o degree_n in_o the_o equinoctial_a 37_o ‑_o 3_o miles_n shall_v answer_v to_o one_o degree_n in_o this_o parralel_n this_o be_v wrought_v by_o lay_v the_o thread_n to_o 51_o d_o 32′_n in_o the_o limb_n from_o the_o left_a edge_n towards_o the_o right_n then_o take_v the_o near_a distance_n to_o it_o from_o 60_o in_o the_o equal_a part_n which_o measure_v from_o the_o centre_n will_v be_v find_v to_o reach_v to_o 37_o ‑_o 3_o as_o before_o the_o reason_n of_o this_o facile_a operation_n be_v because_o the_o near_a distance_n from_o the_o end_n of_o the_o line_n of_o equal_a part_n to_o the_o thread_n be_v equal_a to_o the_o cousin_n of_o the_o latitude_n the_o scale_n itself_o be_v equal_a to_o the_o radius_fw-la and_o therefore_o need_v not_o be_v take_v out_o of_o a_o scale_n of_o sines_n and_o enter_v upon_o the_o first_o term_v the_o radius_fw-la as_o in_o other_o proportion_n in_o sin_n of_o of_o the_o great_a to_o the_o less_o when_o wrought_v upon_o a_o single_a line_n only_o issue_v from_o the_o centre_n where_o the_o second_o term_n must_v be_v take_v out_o of_o a_o scale_n and_o enter_v upon_o the_o first_o term_n 2._o the_o course_n and_o distance_n give_v to_o find_v the_o difference_n of_o latitude_n in_o league_n or_o mile_n as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o rumb_n from_o the_o meridian_n so_o the_o distance_n sail_v to_o the_o difference_n of_o latitude_n in_o like_a part_n example_n a_o ship_n sail_v s_o w_n by_o w_n that_o be_v on_o a_o rumb_n 56_o d_o 15′_n from_o the_o meridian_n 60_o miles_n the_o difference_n of_o latitude_n in_o miles_n will_v be_v find_v to_o be_v 33_o ‑_o 3_o the_o operation_n be_v all_o one_o with_o the_o former_a lay_v the_o thread_n to_o the_o rumb_n in_o the_o limb_n and_o from_o 60_o take_v the_o near_a distance_n to_o it_o which_o measure_v in_o the_o scale_n of_o equal_a part_n will_v be_v find_v as_o before_o 3._o the_o course_n and_o distance_n give_v to_o find_v the_o departure_n from_o the_o meridian_n alias_o the_o variation_n as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o rumb_n from_o the_o meridian_n so_o the_o distance_n sail_v to_o the_o departure_n from_o the_o meridian_n in_o the_o former_a example_n to_o find_v the_o departure_n from_o the_o meridian_n lay_v the_o thread_n to_o the_o rumb_n count_v from_o the_o right_a edge_n towards_o the_o left_a that_o be_v to_o 56_o 15′_n so_o count_v and_o from_o 60_o in_o the_o equal_a part_n be_v the_o miles_n sail_v take_v the_o near_a distance_n to_o it_o this_o extent_n measure_v in_o the_o say_a scale_n will_v be_v find_v to_o be_v 49_o ‑_o 9_o miles_n and_o so_o if_o the_o converse_n of_o this_o be_v to_o be_v wrought_v it_o be_v evident_a that_o the_o mile_n of_o departure_n must_v be_v take_v out_o of_o the_o scale_n of_o equal_a part_n and_o enter_v parralelly_n between_o the_o scale_n and_o the_o thread_n lie_v over_o the_o rumb_n many_o more_o example_n and_o proposition_n may_v be_v illustrate_v but_o these_o be_v sufficient_a those_o that_o will_v use_v a_o quadrant_n for_o this_o purpose_n may_v have_v the_o rumb_n trace_v out_o or_o prick_v upon_o the_o limb_n now_o we_o repair_v to_o the_o backside_n of_o the_o quadrant_n of_o the_o line_n of_o on_o the_o right_a edge_n of_o the_o backside_n the_o use_v of_o this_o line_n be_v manifold_a in_o dyall_v in_o draw_v projection_n in_o work_a proportion_n etc._n etc._n 1._o to_o take_v of_o a_o proportional_a sine_fw-la to_o any_o lesser_a radius_fw-la than_o the_o side_n of_o the_o quadrant_n or_o which_o be_v all_o one_o to_o divide_v any_o line_n short_a in_o length_n then_o the_o whole_a line_n of_o sines_n in_o such_o manner_n as_o the_o same_o be_v divide_v enter_v the_o length_n of_o the_o line_n propose_v at_o 90_o d_o the_o end_n of_o the_o scale_n of_o sines_n and_o to_o the_o other_o foot_n lay_v the_o thread_n according_a to_o near_a distance_n or_o measure_v the_o length_n of_o the_o line_n propose_v on_o the_o line_n of_o sines_n from_o the_o centre_n and_o observe_v to_o what_o sine_fw-la it_o be_v equal_a then_o lay_v the_o thread_n over_o the_o like_a arch_n in_o the_o limb_n and_o the_o near_a distance_n to_o it_o from_o each_o degree_n of_o the_o line_n of_o sines_n shall_v be_v the_o proportional_a part_n seek_v and_o if_o the_o thread_n be_v lay_v over_o 30_o d_o of_o the_o limb_n the_o near_a distance_n to_o it_o will_v be_v sin_n to_o half_a the_o common_a radius_fw-la 2._o from_o a_o line_n of_o sines_n to_o take_v off_o a_o tangent_fw-la the_o proportion_n to_o do_v it_o be_v as_o the_o cousin_n of_o a_o arch_n to_o the_o radius_fw-la of_o the_o line_n propose_v so_o the_o sine_fw-la of_o the_o say_a arch_n to_o the_o tangent_fw-la of_o the_o say_v arch._n enter_v the_o radius_fw-la of_o the_o tangent_fw-la propose_v at_o the_o cousin_n of_o the_o give_v arch_n and_o to_o the_o other_o foot_n lay_v the_o thread_n then_o from_o the_o sine_fw-la of_o that_o arch_n take_v the_o near_a distance_n to_o the_o thread_n this_o extent_n be_v the_o length_n of_o the_o tangent_fw-la seek_v thus_o to_o get_v the_o tangent_fw-la of_o 20_o d_o enter_v the_o radius_fw-la propose_v at_o the_o sine_fw-la of_o 70_o d_o then_o take_v the_o near_a distance_n to_o the_o thread_n from_o the_o sine_fw-la of_o 20_o d_o this_o extent_n be_v the_o tangent_fw-la of_o the_o say_a arch_n in_o reference_n to_o the_o limit_a radius_fw-la otherways_o by_o the_o limb._n lay_v the_o thread_n to_o the_o sine_fw-la of_o that_o arch_n count_v from_o the_o right_a edge_n whereto_o you_o will_v take_v out_o a_o tangent_fw-la and_o enter_v the_o radius_fw-la propose_v down_o the_o line_n of_o sines_n from_o the_o centre_n and_o take_v the_o near_a distance_n to_o the_o thread_n than_o lie_v the_o thread_n to_o the_o like_a arch_n from_o the_o left_a edge_n and_o enter_v the_o extent_n between_o the_o scale_n and_o the_o thread_n the_o distance_n of_o the_o foot_n of_o the_o compass_n from_o the_o centre_n shall_v be_v the_o length_n of_o the_o

altitude_n be_v well_o give_v these_o scale_n in_o their_o use_n presuppose_v the_o hour_n and_o azimuth_n of_o the_o sun_n to_o be_v near_o the_o noon_n meridian_n then_o 60d._o operation_n to_o find_v the_o hour_n take_v the_o distance_n between_o the_o altitude_n and_o the_o declination_n proper_a to_o the_o season_n of_o the_o year_n out_o of_o the_o hour_n scale_n and_o enter_v one_o foot_n of_o this_o extent_n at_o the_o cousin_n of_o the_o declination_n in_o the_o line_n of_o sin_n and_o lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n it_o show_v the_o hour_n from_o noon_n in_o the_o verse_v sin_n quadruple_v example_n when_o the_o sun_n have_v 23_o 31′_n of_o north_n declination_n and_o 60_o of_o altitude_n the_o hour_n from_o noon_n will_v be_v 13_o 58′_n to_o be_v convert_v into_o time_n when_o the_o hour_n be_v find_v to_o be_v less_o than_o 40_o from_o noon_n the_o former_a extent_n may_v be_v double_v and_o enter_v as_o before_o and_o it_o show_v the_o hour_n in_o the_o verse_v sin_n octuple_v and_o when_o the_o hour_n be_v less_o than_o 30_o from_o noon_n the_o former_a extent_n may_v be_v tripled_a and_o enter_v as_o before_o and_o after_o this_o manner_n it_o be_v possible_a to_o make_v the_o whole_a limb_n give_v the_o hour_n next_o noon_n the_o verse_v sine_fw-la duodecuple_v lie_v on_o the_o other_o side_n of_o the_o quadrant_n and_o in_o this_o case_n a_o ark_n must_v first_o be_v find_v in_o the_o limb_n and_o the_o thread_n lay_v over_o the_o say_a ark_n count_v from_o the_o other_o edge_n will_v intersect_v the_o say_v verse_v sine_fw-la at_o the_o ark_n seek_v to_o find_v the_o sun_n azimuth_n take_v the_o distance_n in_o the_o azimuth_n scale_n between_o the_o altitude_n and_o the_o declination_n proper_a to_o the_o season_n of_o the_o year_n and_o enter_v it_o at_o the_o cousin_n of_o the_o altitude_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n it_o will_v show_v the_o azimuth_n in_o the_o verse_v sin_n quadruple_v or_o when_o the_o azimuth_n be_v near_a noon_n according_a to_o the_o former_a restriction_n for_o the_o hour_n the_o extent_n may_v be_v double_v or_o tripled_a and_o the_o answer_v find_v in_o the_o verse_v sin_n octuple_v or_o duodecuple_v as_o be_v do_v for_o the_o hour_n example_n so_o when_o the_o sun_n have_v 23_o 31′_n of_o north_n declination_n his_o altitude_n be_v 60d._o the_o azimuth_n will_v be_v find_v to_o be_v 26_o 21′_n from_o the_o south_n by_o the_o like-reason_n when_o we_o find_v the_o hour_n and_o azimuth_n in_o the_o equal_a limb_n by_o the_o diagonal_a scale_n if_o those_o extent_n have_v be_v double_v the_o hour_n and_o azimuth_n near_o six_o or_o the_o vertical_a may_v have_v be_v find_v in_o a_o line_n of_o sin_n of_o 30_o put_v through_o the_o whole_a limb_n but_o that_o we_o think_v needless_a finis_fw-la the_o descrpition_n and_o use_v of_o a_o great_a universal_a quadrant_n with_o a_o quarter_n of_o stofter_n particular_a projection_n upon_o it_o invert_v contrive_v and_o write_v by_o john_n collins_n accountant_a and_o student_n in_o the_o mathematics_n london_n print_v in_o the_o year_n 1658._o the_o description_n of_o the_o great_a quadrant_n it_o have_v be_v hint_v before_o that_o though_o the_o former_a contrivance_n may_v serve_v for_o a_o small_a quadrant_n yet_o there_o may_v be_v a_o better_a for_o a_o great_a one_o the_o description_n of_o the_o fore-side_n on_o the_o right_a edge_n from_o the_o centre_n be_v place_v a_o line_n of_o sines_n on_o the_o left_a edge_n from_o the_o centre_n a_o line_n of_o verse_v sin_n to_o 180d._o the_o limb_n the_o same_o as_o in_o the_o small_a quadrant_n between_o the_o limb_n and_o the_o centre_n be_v place_v in_o circle_n a_o line_n of_o verse_v sin_n to_o 180_o another_o through_o the_o whole_a limb_n to_o 90d._o the_o line_n of_o lesser_a sin_n and_o secant_v the_o line_n of_o tangent_n the_o quadrat_fw-la and_o shadow_n above_o they_o the_o projection_n with_o the_o declination_n day_n of_o the_o month_n and_o almanac_n on_o the_o left_a edge_n be_v place_v the_o fit_a hour_n and_o azimuth_n scale_n within_o the_o projection_n abut_a against_o the_o sin_n be_v place_v a_o little_a scale_n call_v the_o scale_n of_o entrance_n be_v graduate_v to_o 62_o and_o be_v no_o other_o but_o a_o small_a line_n of_o sines_n number_v by_o the_o compliment_n at_o the_o end_n of_o the_o secant_fw-la be_v put_v on_o the_o verse_v sin_n double_v that_o be_v to_o twice_o the_o radius_fw-la of_o the_o quadrant_n and_o at_o the_o end_n of_o the_o tangent_n tripled_a to_o some_o few_o degree_n to_o give_v the_o hour_n and_o azimuth_n near_o noon_n more_o exact_o the_o description_n of_o the_o backside_n on_o the_o right_a edge_n from_o the_o centre_n be_v place_v a_o line_n of_o equal_a part_n be_v 10_o inch_n precise_a decimal_o subdivide_v on_o the_o outside_n next_o the_o edge_n be_v place_v a_o large_a chord_n to_o 60_o equal_a in_o length_n to_o the_o radius_fw-la of_o the_o line_n of_o sines_n on_o the_o left_a edge_n be_v place_v a_o line_n of_o tangent_n issue_v from_o the_o centre_n continue_v to_o 63_o 26′_n and_o again_o continue_v apart_o from_o 60_o to_o 75_o the_o equal_a limb._n within_o it_o a_o quadrant_n of_o ascension_n divide_v into_o 24_o equal_a hour_n and_o its_o part_n with_o star_n affix_v and_o letter_n grave_v to_o refer_v to_o their_o name_n between_o it_o and_o the_o centre_n be_v place_v a_o circle_n whereof_o there_o be_v but_o three_o quadrant_n graduate_v the_o diameter_n of_o this_o circle_n be_v no_o other_o than_o the_o dyall_a scale_n of_o 6_o hour_n or_o double_a tangent_n divide_v into_o 90d._o two_o quadrant_n or_o the_o half_a of_o this_o circle_n beneath_o the_o diameter_n be_v divide_v into_o 90_o equal_a part_n or_o degree_n the_o upper_a divide_a quadrant_n be_v call_v the_o quadrant_n of_o latitude_n from_o the_o extremity_n of_o the_o say_a quadrant_n and_o perpendicular_a to_o the_o diameter_n be_v graduate_v a_o line_n of_o proportional_a sin_n m_n foster_n call_v it_o the_o line_n sol._n diagonal-wise_a from_o one_o extremity_n of_o the_o quadrant_n of_o latitude_n to_o the_o other_o be_v graduate_v a_o line_n of_o sines_n that_o end_n number_v with_o ●0_n d_o that_o be_v next_o the_o diameter_n be_v of_o the_o same_o radius_fw-la with_o the_o tangent_n opposite_a and_o parallel_a thereto_o from_o 45_o of_o the_o semicircle_n to_o the_o other_o extremity_n of_o the_o diameter_n be_v place_v a_o line_n of_o sines_n equal_a to_o the_o former_a diagonal-wise_a from_o the_o begin_n of_o the_o line_n sol_fw-mi to_o the_o end_n of_o the_o diameter_n be_v graduate_v a_o line_n of_o 60_o chord_n from_o the_o beginning_n of_o the_o diameter_n but_o below_o it_o towards_o 45_o of_o the_o semicircle_n be_v graduate_v the_o projection_n tangent_fw-la alius_fw-la a_o semi-tangent_a to_o 90_o be_v of_o the_o same_o radius_fw-la with_o the_o tangent_n the_o other_o quadrant_n of_o this_o circle_n be_v only_o a_o void_a line_n there_o pass_v through_o it_o from_o the_o centre_n a_o tangent_fw-la of_o 45_o for_o dyall_v divide_v into_o 3_o hour_n with_o its_o quarter_n and_o minute_n below_o the_o diameter_n be_v void_a space_n leave_v to_o graduate_v any_o table_n at_o pleasure_n and_o a_o line_n of_o chord_n may_v be_v there_o place_v most_o of_o these_o line_n and_o the_o projection_n have_v be_v already_o treat_v upon_o in_o the_o use_n of_o the_o small_a quadrant_n those_o that_o be_v add_v shall_v here_o be_v speak_v to_o of_o the_o line_n of_o verse_v sin_n on_o the_o left_a edge_n issue_v from_o the_o centre_n this_o line_n and_o the_o use_n of_o it_o be_v invent_v by_o the_o learned_a mathematician_n m._n samuel_n foster_n of_o gresham_n college_n decease_a from_o who_o i_o receive_v the_o use_n of_o it_o apply_v to_o a_o sector_n i_o shall_v and_o have_v add_v the_o proportion_n to_o be_v wrought_v upon_o it_o and_o in_o that_o and_o other_o respect_n diversify_v from_o what_o i_o receive_v wherein_o i_o shall_v not_o be_v tedious_a because_o there_o be_v other_o way_n to_o follow_v since_o find_v out_o by_o myself_o the_o chief_a use_n of_o it_o be_v to_o resolve_v the_o two_o case_n of_o the_o four_o axiom_n of_o spherical_a trigonometry_n as_o when_o three_o side_n be_v give_v to_o find_v a_o angle_n or_o two_o side_n with_o the_o angle_n comprehend_v to_o find_v the_o three_o side_n which_o be_v the_o case_n that_o find_v the_o hour_n and_o azimuth_n general_o and_o the_o sun_n altitude_n on_o all_o hour_n for_o the_o hour_n the_o learned_a author_n think_v meet_v to_o add_v a_o zodiaque_fw-la of_o the_o suns-place_n annex_v to_o it_o both_o in_o the_o use_n of_o his_o sector_n as_o also_o

doubtful_a whether_o the_o angle_n oppose_v to_o the_o great_a side_n be_v acute_a or_o obtuse_a yet_o a_o true_a sine_fw-la of_o the_o 4_o ●h_a proportional_a the_o sine_fw-la find_z will_v give_v the_o angle_n opposite_a to_o the_o other_o give_v side_n if_o it_o be_v acute_a and_o it_o will_v always_o be_v acute_a when_o the_o give_v angle_n be_v obtuse_a but_o if_o it_o be_v fore_o know_v to_o be_v obtuse_a the_o arch_n of_o the_o sine_fw-la find_v subtract_v from_o a_o semicircle_n and_o there_o will_v remain_v the_o angle_n seek_v in_o this_o case_n the_o quality_n or_o affection_n of_o the_o angle_n seek_v must_v be_v give_v and_o fore-known_a for_o otherwise_o it_o be_v impossible_a to_o give_v any_o other_o than_o a_o double_a answer_n the_o acute_a angle_n find_v or_o its_o compliment_n to_o 180_o yet_o some_o in_o this_o case_n have_v prescribe_v rule_n to_o know_v it_o suppose_v the_o three_o side_n give_v which_o be_v not_o if_o it_o be_v then_o in_o any_o plain_a triangle_n it_o will_v hold_v that_o if_o the_o square_a of_o any_o side_n be_v equal_a to_o the_o sum_n of_o the_o square_n of_o the_o other_o two_o side_n the_o angle_n it_o subtend_v be_v a_o right_a angle_n if_o less_o acute_a if_o great_a obtuse_a but_o if_o three_o side_n be_v give_v we_o may_v with_o as_o little_a trouble_n by_o follow_v proportion_n come_v by_o the_o quantity_n of_o any_o angle_n as_o by_o this_o rule_n to_o know_v the_o affection_n of_o it_o two_o side_n with_o a_o angle_n opposite_a to_o one_o of_o they_o to_o find_v the_o three_o side_n in_o this_o case_n as_o in_o the_o former_a the_o affection_n of_o the_o angle_n opposite_a to_o the_o other_o give_v side_n must_v be_v fore-known_a or_o else_o the_o answer_n may_v be_v double_a or_o doubtful_a in_o the_o triangle_n annex_v there_o be_v give_v the_o side_n a_o b_o and_o a_o c_o with_o the_o angle_n a_o b_o c_o to_n find_v the_o angle_n opposite_a to_o the_o other_o give_v side_n which_o may_v be_v either_o the_o angle_n at_o c_o or_o d_o and_o the_o three_o side_n which_o may_v be_v either_o b_o c_o or_o b_o d_o the_o reason_n hereof_o be_v because_o two_o of_o the_o give_v term_n the_o side_n a_o b_o and_o the_o angle_n at_o b_o remain_v the_o same_o in_o both_o triangle_n and_o the_o other_o give_v side_n may_v be_v either_o a_o c_o or_o a_o d_o equal_a to_o it_o the_o one_o fall_v as_o much_o without_o as_o the_o other_o do_v within_o the_o perpendicular_a a_o e._n the_o quality_n of_o the_o angle_n opposite_a to_o the_o other_o give_v side_n be_v know_v by_o the_o former_a case_n get_v the_o quantity_n and_o then_o have_v two_o angle_n the_o compliment_n of_o their_o sum_n to_o 180_o be_v equal_a to_o the_o three_o angle_n the_o case_n will_v be_v to_o find_v the_o say_a side_n by_o have_v choice_n of_o the_o other_o side_n and_o their_o opposite_a angle_n as_o follow_v 2._o two_o angle_n with_o a_o side_n opposite_a to_o one_o of_o they_o give_v to_o find_v the_o side_n opposite_a to_o the_o other_o as_o the_o sine_fw-la of_o the_o angle_n oppose_v to_o the_o give_v side_n be_v to_o the_o give_v side_n so_o be_v the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o side_n seek_v to_o the_o side_n seek_v ricciolus_n in_o the_o trigonomerical_a part_n of_o his_o late_a almagestum_n novum_n suggest_v in_o the_o resolution_n of_o this_o case_n that_o if_o the_o side_n opposite_a to_o the_o obtuse_a angle_n be_v seek_v it_o can_v be_v find_v under_o three_o operation_n as_o first_o to_o get_v the_o quantity_n of_o the_o perpendicular_a fall_n from_o the_o obtuse_a angle_n on_o its_o opposite_a side_n and_o then_o the_o quantity_n of_o the_o two_o segment_n of_o the_o side_n on_o which_o it_o fall_v and_o so_o add_v they_o together_o to_o obtain_v the_o side_n seek_v but_o this_o be_v a_o mistake_n and_o if_o it_o be_v true_a by_o the_o like_a reason_n a_o side_n opposite_a to_o a_o acute_a angle_n can_v not_o be_v find_v without_o the_o like_a trouble_n for_o in_o the_o triangle_n above_o as_o the_o sine_fw-la of_o the_o angle_n at_o b_o be_v to_o its_o opposite_a side_n a_o c_o or_o a_o d_o so_o be_v the_o sine_fw-la of_o the_o angle_n b_o c_o a_o or_o b_o d_o a_o to_o its_o opposite_a side_n b_o a_o where_o the_o reader_n may_v perceive_v that_o the_o same_o side_n have_v opposite_a to_o it_o both_o a_o acute_a and_o a_o obtuse_a angle_n the_o one_o the_o compliment_n of_o the_o other_o to_o 180_o the_o same_o sine_fw-la be_v common_a to_o both_o for_o the_o acute_a angle_n a_o c_o e_o be_v the_o compliment_n of_o the_o obtuse_a angle_n b_o c_o a_o but_o the_o angle_n at_o c_o be_v equal_a to_o the_o angle_n at_o d_o be_v subtend_v by_o equal_a side_n and_o the_o proportion_n of_o the_o sin_n of_o angle_n to_o their_o opposite_a side_n be_v already_o demonstrate_v in_o every_o book_n of_o trigonometry_n 3._o two_o side_n with_o the_o angle_n comprehend_v to_o find_v either_o of_o the_o other_o angles_n subtract_v the_o angle_n give_v from_o 180_o and_o there_o remain_v the_o sum_n of_o the_o two_o other_o angle_n then_o as_o the_o sum_n of_o the_o side_n give_v to_o tangent_fw-la of_o the_o half_a sum_n of_o the_o unknown_a angle_n so_o be_v the_o difference_n of_o the_o say_a side_n to_o tangent_fw-la of_o half_a the_o difference_n of_o the_o unknown_a angle_n if_o this_o half_a difference_n be_v add_v to_o half_a sum_n of_o the_o angle_n it_o make_v the_o great_a if_o substract_v from_o it_o it_o leave_v the_o lesser_a angle_n 4._o two_o side_n with_o the_o angle_n comprehend_v to_o find_v the_o three_o side_n by_o the_o former_a proposition_n one_o of_o the_o angle_n must_v be_v find_v and_o then_o as_o the_o sine_fw-la of_o the_o angle_n find_v be_v to_o its_o opposite_a side_n so_o be_v the_o sine_fw-la of_o the_o angle_n give_v to_o the_o side_n oppose_v thereto_o if_o two_o angle_n with_o a_o side_n opposite_a to_o one_o of_o they_o be_v give_v to_o find_v the_o side_n opposite_a to_o the_o three_o angle_n be_v no_o different_a case_n from_o the_o former_a because_o the_o three_o angle_n be_v by_o consequence_n give_v be_v the_o compliment_n of_o the_o two_o give_v angle_n to_o 180_o and_o the_o like_a if_o two_o angle_n with_o the_o side_n between_o they_o be_v give_v 5._o three_o side_n to_o find_v a_o angle_n the_o side_n subtend_v the_o angle_n seek_v be_v call_v the_o base_a as_o the_o rectangle_n or_o product_n of_o the_o half_a sum_n of_o the_o three_o side_n and_o of_o the_o difference_n of_o the_o base_a therefrom_o be_v to_o the_o square_n of_o the_o radius_fw-la so_o be_v the_o rectangle_n of_o the_o difference_n of_o the_o leg_n or_o contain_v side_n therefrom_o to_o the_o square_n of_o the_o tangent_fw-la to_o half_a the_o angle_n seek_v and_o by_o change_v the_o three_o term_n into_o the_o place_n of_o the_o first_o as_o the_o rectangle_n of_o the_o difference_n of_o the_o leg_n from_o the_o half_a sum_n of_o the_o 3_o side_n be_v to_o the_o square_n of_o the_o radius_fw-la so_o be_v the_o rectangle_n of_o the_o say_v half_a sum_n and_o of_o the_o difference_n of_o the_o base_a there_o from_o to_o the_o square_n of_o the_o tangent_fw-la of_o a_o angle_n which_o double_v be_v the_o compliment_n of_o the_o angle_n seek_v to_o 180_o or_o the_o compliment_n of_o this_o angle_n double_v be_v the_o angle_n seek_v to_o operate_v the_o former_a proportion_n by_o the_o table_n from_o the_o half_a sum_n of_o the_o three_o side_n subtract_v each_o side_n several_o then_o from_o the_o sum_n of_o the_o logarithme_n of_o the_o square_n of_o the_o radius_fw-la which_o in_o logarithm_n be_v the_o radius_fw-la double_v and_o of_o the_o difference_n of_o the_o side_n contain_v the_o angle_n seek_v subtract_v the_o sum_n of_o the_o logarithm_n of_o the_o half_a sum_n of_o the_o three_o side_n and_o of_o the_o difference_n of_o the_o base_a therefrom_o the_o half_a of_o the_o remainder_n be_v the_o logarithm_n of_o the_o tangent_fw-la of_o half_a the_o angle_n seek_v example_n in_o the_o triangle_n a_o b_o c_o let_v the_o three_o side_n be_v give_v to_o find_v the_o obtuse_a angle_n at_o c._n difference_n from_o the_o half_a sum_n b_o c_o 126_o leg_n 101_o 2,0043214_o a_o c_o 194_o leg_n 169_o 2,2278867_o a_o b_o 270_o base_n 25_o logarithm_n  _fw-fr 1,3979400_o sum_n with_o double_a radius_fw-la 24,2322081_o sum_v 590_o  _fw-fr  _fw-fr half_a sum_n 295_o 2,4698220_o  _fw-fr  _fw-fr  _fw-fr 3,8677620_o  _fw-fr  _fw-fr 3,8677620_o  _fw-fr  _fw-fr 20,3644461_o tangent_fw-la of_o 56_o 41′_n which_o double_v be_v 113_o 22′_n the_o angle_n seek_v 10,1822230_o half_n for_o the_o 2_o proportion_n for_o find_v a_o angle_n the_o

equidistant_a one_o from_o another_o but_o have_v determine_v the_o distance_n between_o the_o two_o extreme_a latitude_n to_o which_o they_o be_v fit_v for_o the_o the_o large_a sine_fw-la it_o will_v hold_v as_o the_o difference_n of_o the_o secant_v of_o the_o two_o extreme_a latitude_n it_o to_o the_o distance_n between_o the_o line_n fit_v thereto_o so_o be_v the_o difference_n of_o the_o secant_v of_o the_o lesser_a extreme_a latitude_n and_o any_o other_o intermediate_v latitude_n to_o the_o distance_n thereof_o from_o the_o lesser_a extreme_a and_o so_o for_o the_o lesser_a sine_fw-la continue_v the_o other_o way_n have_v place_v the_o two_o extreme_n under_o the_o two_o former_a extreme_n to_o place_v the_o imtermediate_v line_n the_o canon_n will_v be_v as_o the_o difference_n of_o the_o sin_n of_o the_o two_o extreme_a latitude_n be_v to_o the_o distance_n between_o the_o line_n fit_v thereto_o so_o be_v the_o difference_n of_o the_o sin_n of_o the_o lesser_a extreme_a latitude_n and_o of_o any_o other_o intermediate_v latitude_n to_o the_o distance_n thereof_o from_o the_o lesser_a extreme_a have_v fit_v the_o distance_n of_o the_o great_a sine_fw-la straight_a line_n draw_v through_o the_o two_o extreme_a sin_n shall_v divide_v the_o intermediate_v parallel_n also_o into_o line_n of_o sin_n proper_a to_o the_o latitude_n to_o which_o they_o be_v fit_v now_o for_o the_o lesser_a sin_n they_o be_v continue_v the_o other_o way_n at_o the_o end_n of_o the_o former_a parallel_n the_o line_n proper_a to_o each_o latitude_n shall_v be_v divide_v into_o a_o line_n of_o sin_n who_o radius_fw-la shall_v be_v equal_a to_o the_o sine_fw-la of_o the_o latitude_n of_o the_o other_o sine_fw-la whereto_o it_o be_v fit_v and_o so_o line_n trace_v through_o each_o degree_n to_o the_o extreme_n but_o by_o reason_n of_o the_o small_a distance_n of_o these_o line_n the_o difference_n be_v so_o exceed_v small_a that_o it_o may_v not_o be_v scruple_v to_o draw_v line_n diagonal_a wise_a from_o each_o degree_n of_o the_o two_o outward_a extreme_a sin_n for_o be_v draw_v true_a they_o will_v not_o be_v perceive_v to_o be_v any_o other_o then_o straight_a line_n whereas_o these_o line_n by_o reason_n of_o the_o latter_a proportion_n shall_v not_o fall_v absolute_o to_o be_v draw_v at_o the_o end_n of_o the_o former_a line_n whereto_o they_o be_v fit_v and_o then_o they_o will_v not_o be_v so_o fit_a for_o the_o purpose_n yet_o the_o difference_n be_v as_o we_o say_v so_o insensible_a that_o it_o can_v be_v scale_v they_o be_v notwithstanding_o there_o place_v and_o cross_v with_o diagonals_n draw_v through_o each_o degree_n of_o the_o extreme_n the_o use_v of_o the_o diagonal_a scale_n 1._o to_o find_v the_o time_n of_o sun_n rise_v or_o set_v in_o the_o parralel_n proper_a to_o the_o latitude_n take_v out_o the_o sun_n declination_n out_o of_o the_o lesser_a continue_a sin_n and_o enter_v one_o foot_n of_o this_o extent_n at_o the_o compliment_n of_o the_o declination_n in_o the_o line_n of_o sin_n and_o in_o the_o equal_a limb_n the_o thread_n be_v lay_v to_o the_o other_o foot_n will_v show_v the_o time_n seek_v in_o the_o latitude_n of_o york_n namely_o 54_o if_o the_o sun_n have_v 20_o of_o declination_n northward_o he_o rise_v at_o 4_o and_o set_v at_o 8_o southward_o he_o rise_v at_o 8_o and_o set_v at_o 4_o 2._o to_o find_v the_o hour_n of_o the_o day_n or_o night_n for_o south_n declination_n in_o the_o parralel_n proper_a to_o the_o latitude_n account_v the_o declination_n in_o the_o lesser_a continue_a sine_fw-la and_o the_o altitude_n in_o the_o great_a sine_fw-la and_o take_v their_o distance_n which_o extent_n apply_v as_o before_o to_o the_o cousin_n of_o the_o declination_n in_o the_o line_n of_o sin_n on_o the_o quadrant_n and_o lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n it_o show_v the_o time_n seek_v in_o the_o equal_a limb_n thus_o in_o the_o latitude_n of_o york_n when_o the_o sun_n have_v 20_o of_o south_n declination_n his_o altitude_n be_v 5_o the_o hour_n from_o noon_n will_v be_v find_v 45_o minute_n past_o 8_o in_o the_o morning_n or_o 15_o minute_n past_o 3_o in_o the_o afternoon_n feré_fw-la for_o north_n declination_n the_o declination_n must_v be_v take_v out_o of_o the_o lesser_a sine_z in_o the_o proper_a parralel_n and_o turn_v upward_o on_o the_o great_a sine_fw-la and_o there_o it_o show_v the_o altitude_n at_o six_o for_o the_o sun_n or_o any_o star_n in_o the_o northern_a hemispere_fw-fr the_o distance_n between_o which_o point_n and_o the_o give_v altitude_n must_v be_v enter_v as_o before_o at_o the_o cousin_n of_o the_o declination_n lay_v the_o thread_n to_o the_o other_o foot_n and_o it_o show_v the_o hour_n in_o the_o limb_n from_o six_o towards_o noon_n or_o midnight_n according_a as_o the_o sun_n or_o star_n altitude_n be_v great_a or_o lesser_a than_o its_o altitude_n at_o six_o so_o in_o the_o latitude_n of_o york_n when_o the_o sun_n have_v 20_o of_o north_n declination_n if_o his_o altitude_n be_v 40_o the_o hour_n will_v be_v 46_o minute_n past_o 8_o in_o the_o morning_n or_o 14_o minute_n past_o 3_o in_o the_o afternoon_n 4._o the_o converse_n of_o the_o former_a proposition_n will_v be_v to_o find_v the_o altitude_n of_o the_o sun_n at_o any_o hour_n of_o the_o day_n or_o of_o any_o star_n at_o any_o hour_n of_o the_o night_n i_o need_v not_o insist_v on_o this_o have_v show_v the_o manner_n of_o it_o on_o the_o small_a quadrant_a only_o for_o these_o scale_n use_v the_o limb_n instead_o of_o the_o lesser_a sin_n for_o star_n the_o time_n of_o the_o night_n must_v first_o be_v turn_v into_o the_o star_n hour_n and_o then_o the_o work_v the_o same_o as_o for_o the_o sun_n 5._o to_o find_v the_o amplitude_n of_o ehe_fw-ge sun_n or_o star_n take_v out_o the_o declination_n out_o of_o the_o great_a sine_z in_o the_o parralel_n proper_a to_o the_o latitude_n and_o measure_v it_o on_o the_o line_n of_o sin_n on_o the_o lesser_a quadrant_n and_o it_o show_v the_o amplitude_n seek_v so_o in_o the_o latitude_n of_o york_n 54_o when_o the_o sun_n have_v 20_o of_o declination_n his_o amplitude_n will_v be_v 35_o 35′_n 6._o to_o find_v the_o azimuth_n for_o the_o sun_n or_o any_o star_n in_o the_o hemisphere_n for_o south_n declination_n account_v the_o altitude_n in_o the_o lesser_a sine_fw-la continue_v in_o the_o proper_a parralel_n and_o the_o declination_n in_o the_o great_a sine_fw-la and_o take_v their_o distance_n enter_v one_o foot_n of_o this_o extent_n at_o the_o cousin_n of_o the_o altitude_n on_o the_o quadrant_n and_o lay_v the_o thread_n to_o the_o other_o according_a to_o near_a distance_n and_o in_o the_o limb_n it_o show_v the_o azimuth_n from_o east_n or_o west_n southwards_o so_o in_o the_o latitude_n of_o york_n when_o the_o sun_n have_v 20_o of_o south_n declination_n his_o altitude_n be_v 5_o the_o azimuth_n will_v be_v find_v to_o be_v 44_o 47′_n to_o the_o southwards_o of_o the_o east_n or_o west_n for_o north_n declination_n account_v the_o altitude_n in_o the_o lesser_a sine_z continued_z and_o apply_v it_o upward_o on_o the_o great_a sine_fw-la and_o it_o find_v a_o point_n thereon_o from_o whence_o take_v the_o distance_n to_o the_o declination_n in_o the_o say_v great_a sine_z in_o the_o parralel_n proper_a to_o the_o latitude_n of_o the_o place_n and_o enter_v one_o foot_n of_o this_o extent_n at_o the_o cousin_n of_o the_o altitude_n on_o the_o line_n of_o sin_n and_o the_o thread_n be_v lay_v to_o the_o other_o foot_n according_a to_o near_a distance_n show_v the_o azimuth_n in_o the_o limb_n from_o east_n or_o west_n so_o in_o the_o latitude_n of_o york_n when_o the_o sun_n have_v 20_o of_o north_n declination_n and_o 40_o of_o altitude_n his_o azimuth_n will_v be_v 23_o 16′_n to_o the_o southwards_o of_o the_o east_n or_o west_n when_o the_o hour_n or_o azimuth_n fall_v near_a noon_n for_o more_o certainty_n you_o may_v lay_v the_o thread_n to_o the_o compliment_n of_o the_o declination_n for_o the_o hour_n or_o the_o compliment_n of_o the_o altitude_n for_o the_o azimuth_n in_o the_o limb_n and_o enter_v the_o respective_a extent_n parralel_o between_o the_o thread_n and_o the_o sin_n and_o find_v the_o answer_n in_o the_o sin_n we_o may_v have_v fit_v one_o scale_n on_o the_o quadrant_a to_o give_v both_o the_o hour_n and_o azimuth_n in_o the_o equal_a limb_n by_o a_o lateral_a entrance_n and_o have_v enlarge_v upon_o many_o more_o proposition_n which_o shall_v be_v handle_v in_o the_o great_a quadrant_n mr_n sutton_n be_v willing_a to_o add_v a_o backside_n to_o this_o scale_n and_o therefore_o have_v put_v on_o particular_a scale_n of_o his_o own_o for_o give_v the_o requisite_n of_o a_o upright_a decliner_n

this_o whole_a extent_n be_v enter_v at_o 12_o deg_n 13′_n in_o the_o scale_n of_o entrance_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n and_o it_o will_v intersect_v the_o equal_a limb_n at_o 40_o deg_n and_o so_o much_o be_v the_o sun_n azimuth_n from_o the_o east_n or_o west_n because_o the_o scale_n of_o entrance_n can_v not_o be_v continue_v by_o reason_n of_o the_o projection_n the_o residue_n of_o it_o be_v put_v on_o a_o little_a line_n near_o the_o amanack_n the_o use_n whereof_o be_v to_o lay_v the_o thread_n to_o the_o altitude_n in_o it_o when_o the_o azimuth_n be_v seek_v and_o in_o the_o limb_n it_o show_v at_o what_o ark_n of_o the_o sin_n the_o point_n of_o entrance_n will_v happen_v which_o may_v likewise_o be_v find_v by_o prick_v down_o the_o coaltitude_n on_o the_o line_n of_o sines_n out_o of_o the_o fit_v hour_n scale_n on_o the_o right_a edge_n how_o to_o find_v the_o hour_n and_o azimuth_n general_o in_o the_o equal_a limb_n either_o with_o or_o without_o tangent_n or_o secant_v have_v be_v also_o show_v and_o how_o that_o those_o two_o point_n for_o any_o latitude_n may_v be_v there_o prick_v and_o may_v be_v take_v off_o either_o from_o the_o limb_n or_o from_o a_o line_n of_o sines_n or_o best_a of_o all_o by_o table_n for_o half_a the_o natural_a tangent_fw-la of_o the_o latitude_n of_o london_n be_v equal_a to_o the_o sine_fw-la of_o 〈◊〉_d 39_o deg_n and_o half_o the_o secant_fw-la thereof_o equal_a to_o the_o sine_fw-la of_o 〈◊〉_d 53d_o 30_o against_o which_o arke_n of_o the_o limb_n the_o tangent_fw-la and_o secant_fw-la of_o the_o latitude_n be_v graduate_v but_o of_o this_o enough_o have_v be_v say_v in_o the_o description_n of_o the_o small_a quadrant_a of_o the_o quadrat_fw-la and_o shadow_n the_o use_n thereof_o be_v the_o same_o as_o in_o the_o small_a quadrant_a only_o if_o the_o thread_n hang_v over_o any_o degree_n of_o the_o limb_n less_o than_o 45d_o to_o take_v out_o the_o tangent_fw-la thereof_o out_o of_o the_o quadrat_fw-la count_v the_o arch_n from_o the_o right_a edge_n of_o the_o quadrant_a towards_o the_o left_a and_o lay_v the_o thread_n over_o it_o the_o prick_n be_v repeat_v in_o the_o limb_n to_o save_v this_o trouble_n for_o those_o eminent_a part_n of_o the_o equal_a limb_n we_o have_v before_o show_v that_o a_o sine_fw-la tangent_fw-la and_o secant_fw-la may_v be_v take_v off_o from_o it_o and_o that_o have_v a_o sine_fw-la or_o secant_fw-la with_o the_o radius_fw-la thereof_o the_o correspondent_a ark_n thereto_o may_v be_v find_v &_o that_o a_o chord_n may_v be_v take_v off_o from_o concentric_a circle_n or_o by_o help_n of_o a_o bead_n but_o if_o both_o be_v want_v enter_v the_o semidiameter_n or_o radius_fw-la whereto_o you_o will_v take_v out_o a_o chord_n twice_o down_o the_o right_a edge_n from_o the_o centre_n and_o lay_v the_o thread_n over_o half_a the_o and_o lay_v the_o thread_n over_o half_a the_o arch_n propose_v take_v the_o near_a distance_n to_o it_o and_o thus_o may_v a_o chord_n be_v take_v out_o to_o any_o number_n of_o degree_n less_o than_o a_o semicircle_n it_o have_v be_v assert_v also_o that_o the_o hour_n and_o azimuth_n may_v be_v find_v general_o without_o protraction_n by_o the_o sole_a help_n of_o the_o limb_n with_o compass_n and_o a_o thread_n example_n for_o find_v the_o hour_n the_o first_o work_n will_v be_v to_o find_v the_o point_n of_o entrance_n take_v out_o the_o cousin_n of_o the_o latitude_n by_o take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o the_o say_a ark_n from_o the_o concurrence_n of_o the_o limb_n with_o the_o right_a edge_n and_o enter_v it_o down_o the_o right_a edge_n line_n and_o take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o the_o compliment_n of_o the_o declination_n count_v from_o the_o right_a edge_n this_o extent_n enter_v down_o the_o right_a edge_n find_v the_o point_n of_o entrance_n let_v it_o be_v note_v with_o a_o mark_n next_o to_o find_v the_o sine_fw-la point_n take_v out_o the_o sine_fw-la of_o the_o declin_n &_o enter_v it_o dowh_o the_o right_a edge_n &_o from_o the_o point_n of_o termination_n take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o the_o ark_n of_o the_o latit_fw-fr count_v from_o the_o right_a edge_n this_o extent_n enter_v from_o the_o centre_n and_o it_o find_v the_o sine_fw-la point_n let_v it_o be_v note_v with_o a_o mark_n three_o take_v out_o the_o sine_fw-la of_o the_o altitude_n &_o in_o winter_n add_v it_o in_o length_n to_o the_o sine_fw-la point_n in_o summer_n enter_v it_o from_o the_o centre_n &_o take_v the_o distance_n between_o it_o &_o the_o sine_fw-la point_n which_o extent_n enter_v upon_o the_o point_n of_o entrance_n if_o the_o thread_n be_v lay_v to_o the_o other_o foot_n show_v the_o the_o hour_n from_o 6_o in_o the_o equal_a limb_n before_o or_o after_o it_o as_o the_o sine_fw-la of_o the_o altitude_n fall_v short_a or_o beyond_o the_o sine_fw-la point_n example_n in_o the_o latitude_n of_o 39_o d._n the_o sun_n have_v 23d_o 31′_n of_o north_n declination_n and_o altitude_n 51_o deg_n 32′_n the_o hour_n will_v be_v find_v to_o be_v 33_o deg_n 45′_n from_o six_o towards_o noon_n note_v the_o point_n of_o entrance_n and_o sine_fw-la point_n vary_v not_o till_o the_o declination_n vary_n after_o the_o same_o manner_n may_v the_o azimuth_n be_v find_v in_o the_o limb_n by_o proportion_n deliver_v in_o the_o other_o great_a quadrant_a also_o both_o or_o any_o angle_n when_o three_o side_n be_v give_v may_v be_v find_v by_o the_o last_o general_a proportion_n in_o the_o small_a quadrant_a which_o find_v the_o half_a verse_v sine_fw-la of_o the_o ark_n fight_v which_o will_v be_v too_o tedious_a to_o insist_v upon_o &_o be_v more_o proper_a to_o be_v protract_v with_o a_o line_n of_o chord_n to_o find_v the_o azimuth_n universal_o the_o proportion_n use_v on_o the_o small_a quadrant_a for_o find_v it_o in_o the_o equal_a limb_n wherein_o the_o first_o operation_n for_o the_o vertical_a altitude_n be_v fix_v for_o one_o day_n by_o reason_n of_o its_o excursion_n will_v not_o serve_v on_o a_o quadrant_a for_o the_o sun_n or_o star_n when_o they_o come_v to_o the_o meridian_n between_o the_o zenith_n and_o the_o elevate_a pole_n but_o the_o proportion_n there_o use_v for_o find_v the_o hour_n apply_v to_o other_o side_n will_v serve_v for_o the_o azimuth_n universal_o and_o that_o be_v as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o altitude_n to_o a_o four_o sine_fw-la again_o as_o the_o cousin_n of_o the_o altitude_n be_v the_o secant_fw-la of_o the_o latitude_n or_o as_o the_o cousin_n of_o the_o latitude_n be_v the_o secant_fw-la of_o the_o altitude_n so_o in_o declination_n towards_o the_o elevate_v pole_n be_v the_o difference_n but_o towards_o the_o depress_v pole_n the_o sum_n of_o the_o four_o sine_fw-la and_o of_o the_o sine_fw-la sine_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n from_o the_o vertical_a in_o declination_n towards_o the_o depress_v pole_n the_o azimuth_n be_v always_o obtuse_a towards_o the_o elevate_a pole_n if_o the_o declination_n be_v more_o than_o the_o four_o arch_n it_o be_v acute_a if_o less_o obtuse_a example_n for_o the_o latitude_n of_o the_o barbados_n 13_o deg_n altitude_n 27_o deg_n 27′_n declination_n 20_o deg_n north._n lay_v the_o thread_n to_o 27_o deg_n 27′_n in_o the_o limb_n and_o from_o the_o sine_fw-la of_o 13_o deg_n tahe_fw-la near_a distance_n to_o it_o which_o enter_v on_o the_o line_n of_o sin_n from_o the_o centre_n and_o take_v the_o distance_n between_o the_o limit_a point_n and_o the_o sine_fw-la of_o 20_o deg_n the_o declination_n this_o latter_a extent_n enter_v twice_o down_o the_o line_n of_o the_o sin_n from_o the_o centre_n and_o take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o the_o secant_fw-la of_o 27_o deg_n 27′_n this_o extent_n enter_v at_o the_o sine_fw-la of_o 77_o deg_n the_o compliment_n of_o the_o latitude_n and_o lay_v the_o thread_n to_o the_o other_o foot_n it_o will_v lie_v over_o 16_o deg_n in_o the_o equal_a limb_n the_o sun_n azimuth_n to_o the_o northwards_o of_o the_o east_n or_o west_n otherways_o another_o example_n for_o the_o same_o latitude_n and_o declination_n the_o altitude_n be_v 52_o deg_n 27′_n lay_v the_o thread_n to_o it_o in_o the_o limb_n and_o take_v the_o near_a distance_n to_o it_o from_o the_o sine_fw-la of_o 13_o deg_n as_o before_o and_o enter_v it_o down_o the_o line_n of_o sin_n from_o the_o centre_n and_o from_o the_o point_n of_o the_o limitation_n take_v the_o distance_n to_o the_o sine_fw-la of_o 20_o deg_n the_o sun_n declination_n this_o

particular_a sine_fw-la on_o the_o foreside_n by_o the_o intersection_n of_o the_o thread_n and_o for_o this_o example_n will_v be_v 42_o deg_n 53′_n 4_o the_o angle_n of_o 12_o and_o 6._o account_v the_o compliment_n of_o the_o reinclination_n in_o the_o peculiar_a hour_n scale_n as_o a_o sine_fw-la and_o just_a against_o it_o in_o the_o annex_v tangent_fw-la stand_v the_o compliment_n of_o the_o angle_n seek_v in_o this_o example_n the_o angle_n of_o 12_o and_o 6_o be_v 68_o deg_n 20′_n in_o other_o latitude_n the_o operation_n must_v be_v perform_v by_o proportional_a work_n with_o the_o compass_n of_o the_o line_n derive_v from_o mr._n gunter_n sector_n such_o be_v the_o line_n of_o superficies_n solid_n etc._n etc._n of_o the_o line_n of_o superficies_n or_o square_n the_o chief_a use_n of_o this_o line_n joint_o with_o the_o line_n of_o line_n in_o the_o limb_n be_v when_o a_o square_a number_n be_v give_v to_o find_v the_o root_n thereof_o or_o a_o root_n give_v to_o find_v the_o square_a number_n thereto_o these_o line_n place_v on_o a_o quadrant_a will_v perform_v this_o some_o what_o better_a than_o a_o sector_n because_o it_o be_v give_v by_o the_o intersection_n of_o the_o thread_n without_o compass_n the_o property_n of_o the_o quadrant_a cast_v these_o line_n large_a where_o on_o a_o sector_n they_o will_v be_v narrow_a to_o find_v the_o square_a root_n of_o a_o number_n the_o root_n be_v give_v to_o find_v the_o square_a number_n of_o that_o root_n in_o extract_v the_o square_a root_n prick_n must_v be_v set_v under_o the_o first_o three_o five_o and_o seven_o figure_n and_o so_o forward_o and_o as_o many_o prick_v as_o fall_v to_o be_v under_o the_o square_a number_n give_v so_o many_o figure_n shall_v be_v in_o the_o root_n and_o according_o the_o line_n of_o line_n and_o superficies_n must_v vary_v in_o the_o number_n they_o represent_v i_o be_o very_o unwilling_a to_o spend_v any_o time_n about_o these_o kind_n of_o line_n as_o be_v of_o small_a performance_n and_o by_o myself_o and_o almost_o by_o all_o man_n account_v mere_a toy_n if_o a_o number_n be_v give_v in_o the_o superficies_n the_o thread_n in_o the_o line_n show_v the_o root_n of_o it_o and_o the_o contrary_a if_o a_o number_n be_v give_v in_o the_o line_n the_o thread_n lay_v over_o it_o intersect_v the_o square_a thereof_o the_o performance_n thereof_o by_o these_o line_n be_v so_o deficient_a that_o i_o shall_v give_v no_o example_n of_o it_o when_o a_o number_n be_v give_v to_o find_v the_o square_a thereof_o if_o not_o to_o large_a the_o reader_n may_v correct_v the_o last_o figure_n of_o it_o by_o multiply_v it_o in_o his_o memory_n to_o three_o number_n give_v to_o find_v a_o four_o in_o a_o duplicate_v proportion_n that_o be_v to_o work_v a_o proportion_n between_o number_n and_o square_n example_n if_o the_o diameter_n of_o a_o circle_n who_o area_n be_v 154_o be_v 14_o what_o shall_v the_o diameter_n of_o that_o circle_n be_v who_o area_n be_v 616._o example_n lay_v the_o thread_n over_o 616_o in_o the_o superficies_n and_o from_o 14_o in_o the_o equal_a part_n take_v the_o near_a distance_n to_o it_o then_o lay_v the_o thread_n to_o 154_o in_o the_o superficies_n and_o enter_v the_o former_a extent_n between_o the_o thread_n and_o the_o scale_n and_o the_o foot_n of_o the_o compass_n will_v rest_v upon_o 28_o the_o diameter_n seek_v to_o find_v a_o proportion_n between_o two_o or_o more_o like_a superficies_n admit_v there_o be_v two_o circle_n and_o i_o will_v know_v what_o proportion_n their_o areas_n bear_v to_o each_o other_o in_o this_o case_n the_o proper_a use_n of_o a_o line_n of_o superficies_n will_v be_v to_o have_v it_o on_o a_o ruler_n and_o to_o measure_v the_o length_n of_o their_o like_a side_n for_o circle_n the_o length_n of_o their_o diameter_n upon_o it_o and_o then_o i_o say_v the_o number_n find_v on_o the_o superficies_n bear_v such_o proportion_n each_o to_o other_o as_o the_o areas_n or_o superficial_a content_n and_o for_o small_a quantity_n may_v be_v do_v on_o the_o quadrant_a by_o enter_v down_o the_o large_a extent_n of_o the_o compass_n on_o the_o line_n of_o line_n from_o the_o centre_n and_o mind_n the_o point_n of_o limitation_n enter_v then_o the_o other_o extent_n on_o the_o point_n of_o limitation_n and_o lay_v the_o thread_n to_o the_o other_o foot_n find_v what_o number_n it_o cut_v in_o the_o superficies_n and_o the_o great_a shall_v bear_v such_o proportion_n to_o the_o lesser_a as_o 100_o etc._n etc._n the_o length_n of_o the_o whole_a line_n do_v to_o the_o part_n cut_v the_o proportion_n that_o two_o superficies_n bear_v each_o to_o other_o be_v the_o same_o that_o the_o square_n of_o their_o like_a side_n and_o therefore_o their_o side_n may_v be_v measure_v either_o in_o foot_n or_o inch_n measure_n and_o then_o the_o square_n take_v out_o as_o before_o show_v the_o line_n of_o superficies_n serve_v for_o the_o reduce_n of_o plot_n to_o any_o proportion_n admit_v a_o plot_n of_o a_o piece_n of_o ground_n be_v cast_v up_o contain_v 364_o acre_n and_o it_o be_v require_v to_o draw_v another_o plot_n which_o be_v cast_v up_o by_o the_o same_o scale_n shall_v contain_v but_o a_o quarter_n so_o much_o and_o let_v one_o side_n of_o the_o say_a plot_n be_v 60_o inch_n against_o 60_o in_o the_o line_n the_o square_a of_o it_o will_v be_v find_v to_o be_v 3600_o and_o the_o four_o part_n hereof_o will_v be_v 900_o which_o account_n in_o the_o superficies_n and_o you_o will_v find_v the_o square_a root_n of_o it_o to_o be_v 30_o and_o so_o many_o inch_n must_v be_v the_o like_a side_n of_o the_o lesser_a plot_n if_o be_v cast_v up_o by_o the_o same_o scale_v it_o shall_v contain_v but_o ¼_n of_o what_o it_o do_v before_o if_o the_o line_n of_o superficies_n be_v on_o a_o straight_a ruler_n then_o to_o perform_v such_o a_o proposition_n as_o this_o will_v be_v to_o measure_v therewith_o the_o side_n of_o the_o plot_n give_v mind_v what_o number_n it_o reach_v to_o in_o the_o superficies_n the_o four_o part_n of_o the_o say_a number_n be_v reckon_v on_o the_o superficies_n and_o thence_o take_v shall_v be_v the_o length_n of_o the_o side_n in_o the_o proportion_n require_v of_o the_o line_n of_o solid_n if_o a_o number_n be_v due_o estimate_v in_o the_o say_a line_n and_o the_o thread_n lay_v over_o it_o it_o will_v in_o the_o line_n of_o line_n show_v the_o cube_fw-la root_n of_o that_o number_n and_o the_o converse_v the_o root_n be_v assign_v the_o cube_n may_v be_v find_v but_o by_o reason_n of_o the_o sorry_a performance_n of_o these_o line_n i_o shall_v spend_v no_o time_n about_o it_o if_o this_o line_n be_v place_v on_o a_o loose_a ruler_n and_o the_o like_a side_n of_o two_o like_a solid_n be_v measure_v therewith_o those_o solid_n shall_v bear_v such_o proportion_n in_o their_o content_n each_o to_o other_o as_o the_o measure_a length_n on_o the_o solid_n three_o number_n be_v give_v to_o find_v the_o four_o in_o a_o duplicate_v proportion_n example_n if_o a_o bullet_n of_o 4_o inch_n diameter_n weigh_v 9_o pound_n what_o shall_v a_o bullet_n of_o 8_o inch_n diameter_n weigh_v answer_v 72_o pound_n in_o this_o case_n let_v the_o whole_a line_n of_o solid_n represent_v 100_o always_o the_o solid_a content_n whether_o give_v or_o seek_v must_v be_v account_v in_o the_o line_n of_o solid_n and_o the_o side_n or_o diameter_n in_o the_o equal_a part_n lay_v the_o thread_n to_o 9_o in_o the_o line_n of_o solid_n and_o from_o 8_o in_o the_o inch_n take_v the_o near_a distance_n to_o it_o enter_z one_o foot_n of_o that_o extent_n at_o 4_o in_o the_o inch_n and_o lay_v the_o thread_n to_o the_o other_o foot_n and_o it_o will_v lie_v over_o 72_o in_o the_o solid_n for_o the_o weight_n of_o the_o bullet_n seek_v a_o example_n of_o the_o converse_n if_o a_o bullet_n who_o diameter_n be_v 4_o inch_n weigh_v 9_o pound_n another_o bullet_n who_o weight_n be_v 40_o pound_n what_o shall_v be_v the_o diameter_n of_o it_o lay_v the_o thread_n to_o 40_o in_o the_o solid_n and_o from_o 4_o inch_n in_o the_o line_n take_v the_o near_a distance_n to_o it_o then_o lay_v the_o thread_n to_o 9_o in_o the_o solid_n and_o enter_v the_o say_a extent_n at_o the_o equal_a scale_n so_o that_o the_o other_o foot_n turn_v about_o may_v but_o just_o touch_v the_o thread_n and_o it_o it_o will_v rest_v at_o 6½_n inches_n near_o which_o be_v the_o diameter_n seek_v of_o the_o line_n of_o inscribe_v body_n this_o line_n have_v these_o letter_n set_v to_o it_o d_o signify_v the_o side_n of_o a_o dodecahedron_n s_o signify_v the_o side_n of_o a_o ______o i_o signify_v the_o side_n of_o a_o icosahedron_n

the_o boul_a at_o the_o same_o time_n the_o shade_n in_o the_o boul_a be_v the_o hour_n line_n and_o if_o the_o boul_a be_v full_a of_o water_n or_o any_o other_o liquor_n you_o may_v draw_v the_o hour-line_n which_o will_v never_o show_v the_o true_a hour_n unless_o fill_v with_o the_o say_a liquor_n again_o reflect_v dial_v to_o draw_v a_o reflect_v dial_n on_o any_o plain_a or_o plains_n be_v they_o never_o so_o gibous_a and_o concave_a or_o convex_v or_o any_o irregularity_n whatsoever_o the_o glass_n be_v fix_v at_o any_o reclination_n at_o pleasure_n provide_v it_o may_v cast_v its_o reflex_n upon_o the_o place_n propose_v together_o with_o all_o other_o necessary_a line_n or_o furniture_n thereon_o viz._n the_o parallel_n of_o declination_n the_o azimuth_n line_n the_o parallel_n of_o altitude_n or_o proportion_n of_o shadow_n the_o planetary_a hour-line_n and_o the_o cuspis_fw-la of_o those_o house_n which_o be_v above_o the_o horizon_n etc._n etc._n 1._o if_o the_o glass_n be_v place_v horizontal_n upon_o the_o transome_n of_o a_o window_n or_o other_o convenient_a place_n how_o upon_o the_o wall_n or_o ceiling_n whereon_o that_o glass_n do_v reflect_v to_o draw_v the_o hour-line_n thereon_o although_o it_o be_v never_o so_o irregular_a or_o in_o any_o form_n whatsoever_o constructio_fw-la first_o draw_v on_o pasteboard_n or_o other_o material_a a_o horizontal_n dial_n for_o the_o latitude_n propose_v then_o by_o help_n of_o the_o azimuth_n or_o at_o the_o time_n when_o the_o sun_n be_v in_o the_o meridian_n or_o by_o know_v the_o true_a hour_n of_o day_n whereby_o may_v be_v draw_v several_a line_n on_o the_o ceiling_n floor_n and_o wall_n of_o the_o room_n so_o as_o in_o respect_n of_o the_o centre_n of_o the_o glass_n they_o may_v be_v in_o the_o true_a meridian-circle_n of_o the_o world_n for_o if_o right_a line_n be_v extend_v from_o the_o centre_n of_o the_o say_a glass_n by_o any_o point_n though_o elevate_v in_o any_o of_o those_o line_n so_o draw_v it_o will_v be_v direct_o in_o the_o meridian_n circle_n of_o the_o world_n now_o all_o reflective_a dial_n be_v perform_v from_o that_o principle_n in_o optic_n which_o be_v that_o the_o angle_n of_o incidence_n be_v equal_a to_o the_o angle_n of_o reflection_n and_o as_o any_o direct_a dial_n may_v be_v make_v by_o help_n of_o a_o point_n find_v in_o the_o direct_a axis_n so_o may_v any_o reflect_v dial_n be_v also_o make_v by_o help_n of_o any_o point_n find_v in_o the_o reflect_v axis_n and_o in_o regard_n the_o reflect_a axis_n for_o the_o most_o part_n will_v fall_v above_o the_o horizon_n of_o the_o glass_n without_o the_o window_n so_o that_o no_o point_n there_o can_v be_v fix_v therefore_o a_o point_n must_v be_v find_v in_o the_o say_v reflect_v axis_n continue_v below_o the_o horizontal_n of_o the_o say_a glass_n until_o it_o touch_v the_o ground_n or_o floor_n of_o the_o room_n in_o some_o part_n of_o the_o meridian_n former_o draw_v which_o point_n will_v be_v the_o point_n in_o the_o reverse_a axis_n desire_v and_o may_v be_v find_v as_o follow_v one_o end_n of_o the_o thread_n be_v fix_v at_o or_o in_o the_o centre_n of_o the_o say_a glass_n move_v the_o other_o end_n thereof_o in_o the_o meridian_n former_o draw_v below_o the_o say_a glass_n until_o the_o say_v reverse_v axis_n be_v depress_v below_o the_o horizon_n as_o the_o direct_a axis_n be_v elevate_v above_o the_o horizon_n which_o may_v be_v do_v by_o apply_v the_o side_n or_o edge_n of_o a_o quadrant_n to_o the_o say_a thread_n and_o move_v the_o end_n thereof_o to_o and_o fro_o in_o the_o say_a meridian_n until_o the_o thread_n with_o a_o plummet_n cut_v the_o same_o degree_n as_o the_o pole_n be_v above_o the_o horizontal_n glass_n and_o then_o that_o point_n where_o the_o end_n of_o the_o thread_n touch_v the_o meridian_n either_o on_o the_o floor_n or_o wall_n of_o the_o room_n be_v the_o point_n in_o the_o reflect_v reverse_v axis_n seek_v for_o now_o if_o the_o reverse_v axis_n can_v be_v draw_v from_o the_o glass_n by_o reason_n of_o the_o jet_n of_o the_o window_n or_o other_o impediment_n that_o point_n in_o the_o reverse_n axis_n may_v be_v find_v by_o a_o line_n parallel_v thereto_o by_o fix_v one_o end_n of_o it_o on_o the_o glass_n and_o the_o other_o end_n in_o the_o meridian_n so_o as_o that_o it_o may_v be_v parallel_n to_o the_o floor_n or_o wall_n in_o which_o the_o reverse_a axis-point_n will_v fall_v and_o find_v the_o axis_n point_n from_o that_o other_o end_n of_o the_o lath_n so_o if_o the_o same_o distance_n be_v set_v from_o that_o point_n backward_o in_o the_o meridian_n on_o the_o floor_n as_o be_v the_o lath_n the_o point_n will_v be_v find_v in_o the_o reverse_v axis_n desire_v thus_o have_v find_v a_o point_n in_o the_o reflect_v reverse_v axis_n it_o be_v not_o hard_o by_o help_n whereof_o and_o the_o horizontal_n dial_n to_o draw_v the_o reflect_a hour-line_n on_o any_o ceiling_n or_o wall_n be_v it_o never_o so_o concave_a or_o convex_a to_o do_v which_o first_o note_n that_o all_o straight_a line_n in_o any_o projection_n on_o any_o plain_a do_v always_o represent_v great_a circle_n in_o the_o sphere_n such_o be_v all_o the_o hour-line_n place_v the_o centre_n of_o this_o horizontal_n dial_n in_o the_o centre_n of_o the_o glass_n the_o hour-line_n of_o the_o say_a dial_n be_v horizontal_a and_o the_o meridian_n of_o the_o say_a dial_n in_o the_o meridian_n of_o the_o world_n which_o may_v be_v do_v by_o plumb_n line_n let_v fall_v from_o the_o meridian_n on_o the_o ceiling_n then_o fix_v the_o end_n of_o a_o thread_n or_o silk_n in_o the_o say_a centre_n of_o the_o dial_n or_o glass_n and_o draw_v it_o direct_o over_o any_o hour-line_n on_o the_o dial_n which_o you_o intend_v to_o draw_v and_o at_o the_o further_a side_n of_o the_o room_n and_o there_o let_v one_o hold_v or_o fasten_v that_o thread_n with_o a_o small_a nail_n then_o in_o the_o point_n former_o find_v on_o the_o reverse_a axis_n on_o the_o ●oor_a fix_v another_o thread_n there_o as_o former_o be_v do_v in_o the_o centre_n of_o the_o dial_n then_o take_v that_o thread_n and_o make_v it_o just_a touch_n the_o thread_n on_o the_o hour-line_n of_o the_o horizontal_n dial_n extend_v in_o any_o point_n thereof_o it_o matter_n not_o whereabouts_o and_o mark_v where_o the_o end_n of_o that_o thread_n touch_v the_o wall_n or_o ceiling_n and_o there_o make_v some_o mark_n or_o point_n then_o again_o move_v the_o same_o thread_n high_o or_o low_a at_o pleasure_n till_o it_o as_o former_o touch_v the_o say_v same_o hour_n thread_n and_o mark_v again_o whereabouts_o on_o the_o wall_n or_o ceiling_n the_o end_n of_o the_o say_a thread_n also_o touch_v in_o like_a manner_n may_v be_v find_v more_o point_n at_o pleasure_n but_o any_o two_o will_v be_v sufficient_a for_o the_o project_v or_o draw_v any_o hour-line_n on_o any_o plain_a how_o irregular_a soever_o for_o if_o you_o move_v a_o thread_n and_o also_o your_o eye_n to_o and_o fro_o until_o you_o bring_v the_o say_a thread_n direct_o between_o your_o eye_n and_o the_o point_n former_o find_v you_o may_v project_v thereby_o as_o many_o point_n as_o you_o please_v at_o every_o angle_n of_o the_o wall_n or_o ceiling_n whereby_o the_o reflect_a hour-line_n may_v be_v exact_o draw_v again_o in_o like_a manner_n remove_v the_o say_a thread_n fasten_v in_o the_o centre_n of_o the_o horizontal_n dial_n which_o also_o be_v the_o centre_n of_o the_o glass_n on_o any_o other_o hour-line_n desire_v to_o be_v draw_v and_o as_o before_o fasten_v the_o other_o end_n of_o the_o thread_n by_o a_o small_a nail_n or_o otherwise_o at_o the_o further_a side_n of_o the_o room_n but_o so_o that_o the_o say_a thread_n may_v lie_v just_a on_o the_o hour-line_n propose_v to_o be_v draw_v on_o the_o horizontal_n dial._n then_o as_o before_o take_v the_o thread_n fasten_v in_o the_o point_n on_o the_o reflect_a axis_n and_o bring_v it_o to_o touch_v the_o thread_n of_o the_o hour-line_n in_o any_o part_n thereof_o and_o mark_v where_o the_o end_n of_o that_o thread_n touch_v the_o say_a wall_n or_o ceiling_n then_o again_o as_o before_o move_v the_o say_a thread_n so_o as_o that_o it_o only_o touch_v the_o say_a thread_n of_o the_o hour-line_n in_o any_o other_o part_n thereof_o and_o also_o mark_v where_o the_o end_n of_o that_o thread_n touch_v the_o say_a wall_n or_o ceiling_n so_o be_v there_o find_v two_o point_n on_o the_o wall_n or_o ceiling_n be_v in_o the_o reflect_a hour-line_n desire_v by_o help_n of_o which_o two_o point_n the_o whole_a hour-line_n may_v be_v draw_v for_o if_o as_o before_o a_o thread_n be_v so_o situate_v that_o it_o may_v interpose_v between_o the_o eye_n and_o the_o say_v