Selected quad for the lemma: hand_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
hand_n	bear_v	left_a	right_a	5,448	5	6.6771	4	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A20103	Grammelogia, or, The mathematicall ring extracted from the logarythmes, and projected circular : now published in th[e] inlargement thereof unto any magnitude fit for use, shewing any reasonable capacity that hath not arithmeticke, how to resolve and worke, all ordinary operations of arithmeticke : and those that are most difficult with greatest facilitie, the extract on of rootes, the valuation of leases, &c. the measuring of plaines and solids, with the resolution of plaine and sphericall triangles applied to the practicall parts of geometrie, horo[l]ogographic, geographie, fortification, navigation, astronomie, &c, and that onely by an ocular inspection, and a circular motion / invented an[d] first published, by R. Delamain, teacher, and student of the mathematicks.	Delamain, Richard, fl. 1631.	1630 (1630)	STC 6543; ESTC S763	85,409	123

in_o the_o multiplicand_a and_o multiplier_n declaratio_fw-la declaratio_fw-la so_o if_o 38._o be_v multiply_v by_o 2._o the_o product_n will_v be_v but_o two_o place_n but_o if_o the_o say_v 38._o be_v multiply_v by_o 5_o the_o product_n will_v be_v three_o place_n for_o that_o 3._o by_o 2._o multiply_v do_v not_o cres●ere_o but_o the_o say_v 3._o by_o 5._o do_v bear_v excrescence_n viz._n more_o than_o 9_o this_o note_n be_v only_o to_o give_v domination_n to_o the_o first_o figure_n of_o the_o product_n towards_o the_o left_a hand_n for_o if_o the_o product_n have_v two_o figure_n than_o the_o first_o figure_n of_o that_o product_n towards_o the_o left_a hand_n be_v ten_o or_o ten_o if_o the_o product_n have_v three_o figure_n than_o the_o first_o figure_n of_o the_o product_n towards_o the_o left_a hand_n be_v hundred_o etc._n etc._n example_n 2_o to_o multiply_v 18._o by_o 5._o bring_v 1._o to_o 5._o then_o right_a against_o 18._o in_o the_o movable_a be_v 9_o in_o the_o fix_a which_o by_o the_o former_a note_n ☞_o of_o observation_n be_v 90._o which_o be_v the_o product_n of_o 18._o by_o 5._o but_o if_o 35._o be_v to_o be_v multiply_v by_o 4._o move_v 1._o to_o 4._o so_o right_a against_o 35._o in_o the_o move_v be_v 140._o by_o the_o last_o note_n ☞_o example_n 3_o to_o multiply_v fractionall_a as_o 40._o and_o 5._o 10._o by_o 7._o and_o 3._o 10._o bring_v 1._o to_o 7._o and_o 3._o 10._o so_o right_a against_o 40._o and_o 5._o 10._o in_o the_o movable_a be_v 295._o and_o 6._o 10._o in_o the_o fix_a the_o product_v require_v so_o to_o multiply_v 8._o 10._o by_o 5._o 10._o bring_v 1._o to_o 5._o 10._o so_o right_a against_o 8._o 10._o in_o the_o movable_a be_v 4._o 10._o in_o the_o fix_a use_v upon_o multiplication_n example_n 3_o 12._o monetht_fw-mi make_v a_o year_n bring_v 1._o unto_o it_o so_o right_a against_o any_o monthly_a expense_n in_o the_o movable_a you_o have_v the_o yearly_a expense_n in_o the_o fix_a according_a to_o the_o note_n ☞_o so_o if_o the_o monthly_a expense_n be_v 75._o li._n right_a against_o it_o in_o the_o fix_a be_v 9_o which_o by_o the_o former_a note_n ☞_o make_v 900._o if_z ●_o 50._o li._n for_o a_o month_n right_a against_o it_o in_o the_o fix_a be_v 1800._o li._n the_o yearly_a charge_n or_o expense_n other_o use_n upon_o multiplication_n example_n 4_o 60._o minute_n make_v a_o hour_n bring_v 1._o to_o 60._o so_o right_a against_o any_o number_n of_o hour_n in_o the_o movable_a be_v the_o minute_n of_o those_o hour_n in_o the_o fix_a further_o upon_o multiplication_n example_n 5_o admit_v land_n be_v sell_v at_o 14._o year_n purchase_n bring_v 1._o to_o 14._o so_o against_o any_o rent_n in_o the_o movable_a you_o may_v at_o one_o instant_a see_v the_o purchase_n thereof_o in_o the_o fix_a have_v regard_n to_o the_o former_a note_n ☞_o how_o to_o square_v a_o number_n example_n 6_o to_o square_a 18._o bring_v 1._o to_o 18._o so_o right_a against_o 18._o in_o the_o movable_a be_v 324._o in_o the_o fix_a the_o square_a of_o the_o say_v 18._o in_o like_a manner_n may_v you_o square_v whole_a number_n and_o fraction_n as_o to_o square_v 13._o and_o 5._o 10._o facit_fw-la 182._o and_o 25._o 100_o how_o to_o cube_n a_o number_n example_n 7_o as_o to_o cube_n 6._o and_o 2._o 10._o bring_v 1._o to_o 6._o and_o 2._o 10._o so_o right_a against_o 6._o and_o 2._o 10._o in_o the_o movable_a be_v 38._o and_o 4._o 10._o in_o the_o fix_a and_o right_a against_o 38._o and_o 4._o 10._o in_o the_o movable_a be_v 228._o in_o the_o fix_a the_o cube_n of_o 6._o and_o 2._o 10._o again_o to_o cube_n 6._o bring_v 1._o to_o 6._o for_o right_a against_o 6._o in_o the_o movable_a be_v 36._o in_o the_o fix_a and_o right_o against_o this_o 36._o in_o the_o movable_a be_v 216._o in_o the_o fix_a the_o cube_n of_o 6._o etc._n etc._n to_o find_v number_n in_o continual_a proportion_n unto_o any_o two_o number_n assign_v constructio_fw-la constructio_fw-la bring_v the_o first_o number_n to_o the_o second_o then_o right_o against_o the_o second_o upon_o the_o movable_a be_v the_o three_o number_n in_o the_o fix_a and_o against_o this_o three_o number_n in_o the_o movable_a be_v the_o four_o number_n in_o the_o fix_a etc._n etc._n declaratio_fw-la declaratio_fw-la so_o if_o the_o number_n to_o be_v continue_v in_o proportion_n be_v 2._o to_o 4._o move_v 2._o to_o 4._o so_o right_a against_o 4._o in_o the_o movable_a be_v 8._o in_o the_o fix_a and_o 8._o in_o the_o movable_a give_v 16._o in_o the_o fix_a and_o those_o number_n 2._o 4._o 8._o 16_o etc._n etc._n be_v say_v to_o be_v in_o continual_a proportion_n example_n 2_o again_o it_o i_o will_v continue_v a_o proportion_n as_o 2._o to_o 3._o move_v 2._o to_o 3._o then_o 3._o in_o the_o movable_a shall_v point_v out_o 4._o &_o 5._o 10._o in_o the_o fix_a and_o 4._o &_o 5._o 10._o in_o the_o movable_a shall_v give_v 6_o &_o 7._o 10._o i●_n in_o the_o fix_a and_o so_o on_o if_o need_n be_v to_o find_v other_o and_o those_o number_n be_v say_v to_o be_v in_o continual_a proportion_n one_o unto_o another_o ☞_o ☞_o the_o increase_n or_o interest_n of_o mon●y_n from_o this_o ground_n be_v easy_o find_v see_v the_o increase_n of_o the_o money_n must_v be_v in_o continual_a proportion_n to_o the_o principal_a as_o 100_o li._n be_v to_o his_o interest_n example_n 3_o as_o if_o the_o proportion_n be_v to_o be_v continue_v to_o 40._o li._n as_o 100_o to_o 108._o constructio_fw-la constructio_fw-la move_v 100_o to_o 108._o then_o against_o 40._o in_o the_o movable_a be_v 43._o li._n 2._o 10._o in_o the_o fix_a the_o first_o year_n interest_n and_o its_o principal_a &_o against_o this_o 43._o li._n 2._o 10._o in_o the_o movable_a be_v 46._o li_n 8._o 10._o in_o the_o fix_a which_o be_v the_o second_o year_n principal_a and_o interest_n in_o like_a manner_n may_v you_o proceed_v to_o other_o year_n ☞_o ☞_o the_o instrument_n be_v at_o this_o stay_n the_o eye_n may_v denote_v out_o at_o one_o instant_a the_o interest_n of_o any_o sum_n of_o money_n for_o right_a against_o your_o number_n in_o the_o movable_a be_v both_o principal_a and_o interest_n in_o the_o fix_a example_n 4_o as_o if_o it_o be_v 27._o li._n 14._o s._n that_o be_v 27._o li._n 7._o 10._o right_a against_o it_o be_v 30._o li._n ferè_fw-la and_o so_o much_o do_v 27._o li._n 14._o s._n come_v to_o at_o the_o year_n end_n and_o so_o all_o other_o sum_n of_o money_n do_v offer_v themselves_o at_o one_o instant_a to_o the_o eye_n in_o their_o resolution_n to_o find_v a_o mean_a proportion_n or_o many_o between_o any_o two_o number_n give_v mark_v what_o number_n of_o equal_a part_n in_o the_o fix_a be_v against_o each_o of_o the_o give_v number_n which_o equal_a part_n represent_v the_o logarythme_n of_o these_o number_n if_o the_o logaryt●mall_a index_n be_v put_v unto_o they_o which_o be_v a_o unity_n less_o than_o the_o place_n of_o any_o give_a number_n and_o add_v these_o logarythme_n or_o equal_a part_n together_o then_o take_v the_o half_a of_o that_o sum_n which_o seek_v out_o in_o the_o former_a circle_n of_o equal_a part_n right_a against_o it_o in_o the_o circle_n of_o number_n be_v the_o mean_a proportional_a require_v or_o the_o half_a difference_n of_o these_o two_o logarythmall_a number_n or_o equal_a part_n be_v add_v to_o the_o lesser_a logarythme_n will_v give_v the_o same_o or_o sub_fw-la etc._n etc._n but_o if_o many_o mean_a proportional_o be_v require_v divide_v the_o differentiall_a logarythme_n of_o the_o two_o number_n or_o number_n of_o equal_a part_n by_o a_o unity_n more_o than_o the_o number_n of_o mean_a proportional_o which_o quotient_a be_v by_o succession_n add_v to_o the_o logarythme_n or_o equal_a part_n belong_v to_o the_o lesser_a number_n do_v show_v the_o several_a mean_a proportional_o require_v so_o if_o between_o the_o cubic_n number_n 27._o and_o 64._o two_o mean_a proportional_o be_v require_v the_o three_o part_n of_o the_o difference_n of_o equal_a part_n between_o these_o number_n be_v 125._o which_v be_v add_v unto_o 431._o the_o equal_a part_n against_o 27._o make_v 556._o against_o which_o in_o the_o circle_n of_o number_n be_v 36._o the_o first_o mean_v proportional_a unto_o which_o 556._o again_o add_v successive_o the_o say_v 125._o which_o make_v 681._o against_o these_o equal_a part_n in_o the_o circle_n of_o number_n be_v 48._o the_o other_o mean_v proportional_a according_a to_o the_o same_o manner_n between_o 243._o and_o 1024._o four_o mean_a proportional_o may_v be_v find_v to_o wit_n 324._o 432._o 576._o and_o 768._o or_o proportion_n may_v be_v find_v to_o a_o term_n assign_v between_o two_o number_n either_o by_o augmentation_n from_o

as_o admit_v the_o side_n of_o the_o least_o cube_n be_v 4._o inch_n and_o can_v not_o be_v make_v under_o 12._o li._n what_o may_v a_o cube_n of_o the_o same_o mettle_n be_v worth_a that_o be_v but_o one_o inch_n more_o in_o the_o side_n viz._n 5._o inch_n bring_v 4._o to_o 5._o so_o right_a against_o 12._o in_o the_o movable_a constructio_fw-la constructio_fw-la be_v 15._o in_o the_o fix_a and_o right_a against_o this_o 15._o in_o the_o movable_a be_v 18._o and_o 75._o 100_o in_o the_o fix_a and_o right_a against_o this_o 18._o and_o 75._o 100_o in_o the_o movable_a be_v 23._o li._n and_o 4._o 10._o in_o the_o fix_a and_o so_o much_o will_v the_o second_o cube_n cost_v this_o may_v be_v apply_v to_o the_o weight_n worth_n or_o quantity_n of_o other_o solid_n pro._n 2_o a_o pe●ce_n of_o 5._o inch_n boar_n or_o diameter_n require_v for_o her_o charge_n 16._o pound_n of_o powder_n what_o quantity_n of_o powder_n will_v serve_v another_o piece_n of_o 4._o inch_n in_o the_o boar_n bring_v 5._o to_o 4._o so_o right_a against_o 16._o in_o the_o movable_a be_v 12._o and_o 8._o 10._o in_o the_o fix_a constructio_fw-la constructio_fw-la and_o right_a against_o 12._o and_o 8._o 10._o in_o the_o movable_a be_v 10._o and_o 24._o 100_o in_o the_o fix_a and_o right_o against_o this_o 10._o and_o 24._o 100_o in_o the_o movable_a be_v 8._o and_o 2._o 10._o in_o the_o fix_a the_o answer_n of_o powder_n according_a to_o cubick_a proportion_n but_o canonier_n do_v somewhat_o qualify_v this_o proprotion_n to_o find_v what_o proportion_n in_o quantity_n there_o be_v between_o two_o or_o more_o solid_n pro._n 3_o there_o be_v two_o bullet_n globe_n or_o cylender_n the_o diameter_n of_o the_o one_o be_v 10._o inch_n and_o the_o other_o the_o diameter_n be_v 4._o inch_n what_o proportion_n be_v there_o between_o the_o solid_n or_o how_o often_o do_v the_o great_a contain_v the_o lesser_a bring_v 10._o to_o 4._o so_o right_a against_o 100_o in_o the_o movable_a be_v 40._o in_o the_o fix_a constructio_fw-la constructio_fw-la against_o this_o 40._o in_o the_o movable_a be_v 16._o in_o the_o fix_a and_o right_o against_o this_o 16._o in_o the_o movable_a be_v 6._o and_o 4._o 10._o in_o the_o fix_a so_o the_o proportion_n between_o the_o solid_n be_v as_o 100_o to_o 6._o and_o 4._o 10._o but_o how_o often_o the_o great_a do_v contain_v the_o lesser_a the_o rule_n ensue_v do_v teach_v pro._n 1_o how_o to_o divide_v one_o number_n by_o another_o constructio_fw-la constructio_fw-la move_v the_o divisor_n to_o 1._o so_o right_a against_o the_o dividend_n in_o the_o movable_a be_v the_o quotient_a in_o the_o fix_a declaratio_fw-la declaratio_fw-la so_o if_o it_o be_v demand_v how_o many_o day_n there_o be_v in_o 216._o hour_n because_o a_o day_n natural_a contain_v 24._o hour_n that_o therefore_o be_v the_o divisor_n move_v then_o 24._o to_o 1._o and_o right_n against_o the_o say_v 216._o in_o the_o movable_a be_v 9_o in_o the_o fix_a and_o so_o many_o day_n be_v 216._o hour_n ☞_o ☞_o here_o note_v that_o in_o all_o division_n by_o how_o many_o figure_n or_o place_n the_o dividend_n exceed_v the_o divisor_n so_o many_o place_n or_o figure_n shall_v the_o quotient_n have_v but_o if_o the_o figure_n of_o the_o divisor_n may_v be_v take_v from_o as_o many_o of_o the_o first_o figure_n or_o place_n towards_o the_o left_a hand_n of_o the_o dividend_n than_o the_o quotient_n shall_v have_v one_o place_n more_o example_n 2_o so_o if_o it_o be_v further_o require_v how_o many_o day_n there_o be_v in_o 360._o hour_n or_o any_o other_o number_n the_o instrument_n not_o move_v from_o his_o first_o set_v they_o be_v all_o give_v at_o one_o instant_a for_o right_a against_o the_o number_n in_o the_o movable_a be_v the_o answer_n in_o the_o fix_a so_o right_a against_o 360._o in_o the_o movable_a be_v 15._o in_o the_o fix_a and_o so_o many_o day_n be_v there_o in_o 360._o hour_n this_o note_n serve_v only_o to_o know_v the_o number_n of_o figure_n or_o place_n in_o the_o quotient_a by_o which_o the_o denomination_n of_o the_o first_o figure_n of_o the_o quotient_n may_v be_v have_v example_n 3_o so_o if_o it_o be_v demand_v how_o many_o year_n there_o be_v in_o 14600._o day_n there_o be_v 365._o day_n in_o the_o year_n this_o therefore_o be_v the_o divisor_n bring_v then_o 365._o to_o 1._o so_o right_a against_o 14600._o in_o the_o movable_a be_v 4._o in_o the_o fix_a but_o by_o the_o former_a note_n ☞_o it_o must_v be_v 40._o and_o so_o many_o year_n be_v there_o in_o 14600._o day_n the_o instrument_n not_o move_v right_o against_o any_o number_n of_o day_n as_o 5000._o 10000_o 20000._o etc._n etc._n in_o the_o movable_a be_v the_o year_n in_o the_o fix_a with_o the_o same_o expedition_n and_o facility_n may_v you_o divide_v by_o fractionall_a number_n further_a use_n upon_o division_n example_n 4_o in_o a_o year_n be_v 52._o week_n or_o 365._o day_n if_o i_o will_v know_v the_o weekly_a expense_n of_o any_o yearly_a sum_n of_o money_n bring_v 52._o to_o 1._o then_o right_a against_o any_o sum_n of_o money_n in_o the_o movable_a you_o have_v the_o weekly_a expense_n in_o the_o fix_a constructio_fw-la constructio_fw-la but_o if_o you_o move_v 365._o to_o 1._o then_o right_a against_o any_o sum_n of_o money_n in_o the_o movable_a you_o have_v the_o daily_a expense_n in_o the_o fix_a so_o if_o the_o expense_n yearly_a be_v 1000_o li._n or_o the_o charge_n of_o a_o certain_a company_n of_o soldier_n declaratio_fw-la declaratio_fw-la right_o against_o it_o according_a to_o the_o note_n ☞_o of_o division_n be_v 2._o li._n 7._o 10._o the_o daily_a charge_n the_o instrument_n not_o remove_v you_o may_v see_v at_o one_o instant_a the_o daily_a charge_n of_o 20000._o li._n a_o year_n 50000._o li._n or_o 100000._o li._n a_o year_n for_o right_a against_o the_o charge_n or_o expense_n in_o the_o movable_a be_v the_o answer_n in_o the_o fix_a more_o use_n upon_o division_n example_n 5_o it_o be_v say_v that_o land_n be_v buy_v after_o the_o rate_n of_o 14._o year_n purchase_n if_o 14._o be_v therefore_o bring_v to_o 1._o right_a against_o any_o sum_n of_o money_n in_o the_o movable_a you_o have_v the_o annual_a rent_n in_o the_o fix_a answerable_a to_o that_o money_n and_o thus_o you_o have_v lie_v before_o you_o a_o whole_a circularity_n of_o number_n by_o which_o at_o one_o instant_a do_v but_o speak_v the_o sum_n of_o money_n right_a against_o it_o be_v his_o rent_n but_o if_o the_o rent_n be_v give_v and_o the_o purchase_v require_v it_o be_v the_o inverse_n of_o this_o and_o be_v proper_a to_o multiplication_n and_o the_o rule_n follow_v in_o the_o next_o page_n other_o use_n upon_o division_n to_o find_v the_o scale_n to_o divide_v the_o meridian_n line_n in_o a_o sea_n chartley_n according_a to_o any_o breadth_n &_o to_o a_o latitude_n assign_v example_n 6_o let_v the_o breadth_n of_o the_o chart_n extend_v from_o the_o latitude_n of_o 30._o unto_o 40._o the_o degree_n of_o the_o equator_fw-la answerable_a to_o the_o difference_n of_o those_o latitude_n according_a to_o m._n wright_n projection_n be_v 12._o &_o 24_o 100_o bring_v this_o 12._o &_o 24._o 100_o to_o 1._o so_o right_a against_o the_o breadth_n of_o the_o card_n in_o the_o movable_a you_o have_v the_o inch_n or_o part_n of_o inch_n in_o the_o fix_a to_o make_v your_o scale_n by_o to_o divide_v the_o meridional_a line_n so_o if_o the_o breadth_n or_o the_o card_n be_v 33._o inch_n declaratio_fw-la declaratio_fw-la right_a against_o it_o in_o the_o fix_a be_v 2._o inch_n 7._o 10._o the_o largeness_n of_o a_o degree_n of_o the_o equator_fw-la if_o the_o breadth_n be_v 24._o &_o 5._o 10._o right_a against_o it_o be_v 2._o inches_n if_o 20._o &_o 8._o 10._o then_o the_o breadth_n of_o a_o degree_n be_v 1._o &_o 7._o 10._o if_o 14._o &_o 7._o 10._o then_o 1._o &_o 2._o 10._o if_o 8._o &_o 5._o 10._o then_o 7._o 10._o etc._n etc._n to_o multiply_v one_o number_n by_o another_o or_o to_o find_v the_o product_n of_o two_o number_n constructio_fw-la constructio_fw-la move_v 1._o to_o the_o multiplier_n then_o right_o against_o the_o multicand_a in_o the_o movable_a circle_n you_o have_v the_o product_n in_o in_o the_o fix_a circle_n ☞_o ☞_o here_o note_v that_o the_o product_n of_o any_o multiplication_n be_v ever_o as_o many_o figure_n or_o place_n as_o there_o be_v place_n or_o figure_n contain_v in_o the_o multiplicand_a and_o m_n ltiply_o if_o the_o two_o first_o figure_n towards_o the_o left_a hand_n be_v multiply_v together_o have_v excrescence_n that_o be_v if_o the_o product_n exceed_v 9_o otherwise_o the_o product_n shall_v be_v one_o figure_n or_o place_n less_o than_o there_o be_v figure_n or_o place_n contain_v both_o

present_a if_o it_o be_v to_o be_v receive_v 7._o year_n hence_o facit_fw-la 156._o li._n 6_o 10._o and_o so_o much_o be_v the_o say_a lease_n of_o 40._o li._n per_fw-la annum_fw-la worth_n which_o be_v to_o begin_v 7._o year_n hence_o and_o then_o to_o continue_v unto_o 10._o year_n thus_o i_o may_v have_v go_v further_o in_o those_o matter_n but_o i_o intend_v not_o to_o be_v large_a in_o this_o tract_n only_o show_v what_o weighty_a and_o difficult_a matter_n in_o this_o kind_n by_o the_o grammelogia_n or_o mathematical_a ring_n may_v be_v easy_o and_o speedy_o resolve_v conclusion_n if_o there_o be_v compose_v three_o circle_n of_o equal_a thickness_n a._n b._n c._n so_o that_o the_o inner_a edge_n of_o d_o and_o the_o outward_a edge_n of_o ●_o be●_n answerable_o graduate_v with_o logarithmall_a sign_n and_o the_o outward_a edge_n of_o b_o and_o the_o inner_a edge_n of_o a_o with_o logarithme_n and_o then_o on_o the_o backside_n be_v graduate_v the_o logarithmall_a tangent_n and_o 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britain_n france_n and_o ireland_n defender_n of_o the_o faith_n etc._n etc._n to_o all_o our_o love_a subject_n who_o it_o may_v concern_v greeting_n whereas_o richard_n delamain_n teacher_n of_o the_o mathematics_n have_v present_v unto_o we_o a_o instrument_n call_v grammelogia_n or_o the_o mathematical_a ring_n together_o with_o a_o book_n so_o entitle_v express_v the_o use_n thereof_o be_v his_o own_o invention_n we_o of_o our_o gracious_a and_o princely_a favour_n have_v grant_v unto_o the_o say_v richard_n delamain_n and_o his_o assign_n privilege_n licence_n and_o authority_n for_o the_o sole_a make_v printing_n and_o sell_v of_o the_o say_a instrument_n and_o book_n straight_o forbid_v any_o other_o to_o make_v imprint_v or_o sell_v or_o cause_n to_o be_v make_v or_o imprint_v or_o sell_v the_o say_a instrument_n or_o book_n within_o any_o our_o dominion_n during_o the_o space_n of_o ten_o year_n next_o ensue_v the_o date_n hereof_o upon_o pain_n of_o our_o high_a 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account_v most_o facile_a and_o for_o expedition_n most_o excellent_a and_o upon_o that_o i_o write_v show_v some_o use_n of_o those_o circle_n among_o themselves_o and_o conjoin_v with_o other_o in_o the_o resolution_n of_o such_o question_n which_o be_v ordinary_o practise_v in_o astronomy_n horolographie_n in_o plain_a triangle_n apply_v to_o dimension_n navigation_n fortification_n etc._n etc._n as_o a_o preparative_n ground_n for_o a_o more_o ample_a work_n and_o as_o a_o declaration_n of_o the_o admirable_a and_o excellent_a use_n of_o this_o ring_n in_o expedition_n and_o facility_n but_o before_o i_o come_v to_o construction_n i_o have_v think_v it_o convenient_a by_o way_n introduction_n to_o examine_v the_o truth_n of_o the_o graduation_n of_o those_o circle_n which_o may_v be_v from_o the_o ensue_a table_n and_o direction_n of_o the_o examination_n of_o the_o graduation_n of_o the_o circle_n of_o the_o ring_n which_o may_v serve_v as_o a_o inducement_n and_o furtherance_n to_o the_o learner_n to_o fit_a and_o acquaint_v he_o how_o with_o promptness_n to_o conceive_v of_o opposite_a number_n in_o the_o answer_n of_o question_n follow_v first_o to_o examine_v the_o circle_n of_o number_n bring_v any_o number_n in_o the_o movable_a to_o half_n of_o that_o number_n in_o the_o fix_a so_o any_o number_n or_o part_n in_o the_o fix_a shall_v give_v his_o double_a in_o the_o movable_a and_o so_o may_v you_o try_v of_o the_o three_o fourth_n &c._n &c._n of_o number_n vel_fw-la contra_fw-la 2._o bring_v 2._o in_o the_o movable_a unto_o 3._o in_o the_o fix_a so_o against_o 3._o in_o the_o movable_a be_v 4._o and_o 5._o ten_o in_o the_o fix_a against_o 4._o and_o 5._o ten_o in_o the_o movable_a be_v 6._o and_o 75._o 100_o in_o the_o fix_a and_o so_o may_v you_o go_v on_o in_o try_v the_o division_n of_o the_o circle_n of_o number_n in_o continual_a