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<span>六</span>欽定四庫全書蒼要卷萬貧章八壽湘製數理精蘊下編卷十五
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周脾曰圃出於方方出於矩矩者所謂直角即勾股也蔡因方易度而國難測方有盡而圓無盡故古人用割園之法內彈外切屢求勺股為無數多辜形以切隋園界使狐緣直練漸合為門而國周始得是則推因者以方推方者以矩英劉宋祖之以園容六邊起尊元趙伖銀以園容四邊起畀自明末西法入中國又<span>有</span>割園<span>八泉之系</span>六宗三要等說而圈度內外諸線相求之法始備要之圜內六遁起莫考園徑折半即園內陸邊之一乃用屢求勾股之法自六邊而十間邊自十二邊而一十四搢自訊十四邊而四十<span>八乂</span>邊如是累至億萬邊設徑為勿而周得三夙四夙五九二陸五<span>一</span>七咽餘圈四四遣起箕若則以圍徑為內容正方之斜弦目束折半開方而得四澹之卯亦用屢求勾股之法自四邊而<span>八矛</span>邊自<span>八又</span>邊而十六澹目十六造而三十一澹如是累至憶萬澹設徑為<span>川一</span>周亦得三因四因五九一川五一有餘圍外四蟲起算者圍徑即四唱之一圍徑目乘倍之開方即園外正方之斜弦減去圍徑即圜外兩角之餘又即園外<span>八矛</span>邊之以八濫之夙折半為勾半悍為股求得弦與半徑相減即股弦較又即小同式形之勾乃以<span>仈夕疊一</span>之一折半之勾為辜千半徑之肢為一率小同式形之勾為闕率推得四率鳥<span>卜仆</span>同式形之股倍之即十八澹之一如是累至憶萬邊設徑為尸洏周亦得三一四例<span>政一一九</span>一洲五戶有餘園外六<span>壺工</span>起羿老困徑為弦半徑為勾求得肢借之即囷外工澹之一取其一分之一即圜外入澹之夙以叺邊之嘉半為勾半徑為股求得弦與半徑相減即股弦較又即小同式形之勾乃以六邊之一折半之勾為崩<span>竿一</span>半徑之股為河率小回式形之勾為三率推得四率為小同式形之股倍之即十一邊之一如是累不憶萬邊設徑為一而周亦得一<span>一</span>四物五九一六五二有餘此<span>兩</span>法者或可園內容形少邊為勾股法便無數勿股小弦馮清圜周將輿圍周合而為一或自圜外切形之潛為勾股法使無數勺肢小股隔近園周亦將興圍周合而為內一法既立故凡園周圜徑諸法皆可以互相比例英割園<span>八乂</span>線則伊判周分為二百六十度先求狐度捕弦折半為正弦既得正弦而園內之正矢園外之正切正割由驟洏主至於餘弦餘矢餘切餘割則又由正而得名一百六十度平分四象限每如家限九度凡十度之中得其止角為正餘角為餘是以正餘相對而割圍<span>八乂</span>線之表以立門象限中成勾股形者五干四百故凡勾股三角測量諸法皆可以互相比例英目園內容形屢求勺股而得無數署過自園外切形屢求勾股而得無數多謚百外漆集則園周漸變為直線而設圍界為度分者內而正弦外而切線至於乘數則園周赤漸變為直線二者互相恭考俱為相荷可見悝之至看先後夙揆法之精者中外因理然則勾股即割圜步體而割園即勺股之用二者交相成而兩相得于圈內容耆遭起算
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設如園徑一兆用內容六澹起算間得園周幾博法以圜徑一兆折半得襄為園內所容六邊形之每乙邊乃以半徑夏為八折刈穴弦六邊之酒遁五折半得五千億為那末<span>一弓</span>易求得股<span>八又</span>干山百六十憶午千五百縱火<span>八</span>一四十萬二千七百<span>八父</span>十四<span>小六</span>酬<span>四八六七六一七七七五一旦貸卜聖日或冤一六八四此白十王七牙才力</span>棗鬥千一百二十九憶七干四百五十九萬六于二百四十五<span>小蛾真八一五虹六七六八凡山七口夏六刪甲六五九化</span>為勾六<span>詈亶</span>之辜進折半の五千億為肢求得弦五千八聊刈勿夙百七十六憶二干<span>八只</span>百零九萬零一測浹尸地百零五<span>小餘口四立四亡九火七九七六七五四八日九八六尺</span>晉箋<span>再七六六四</span>為囷內所容十二邊形之每勿邊如是屢求得園內二十四邊形之每澹為河干六百門十億五千門百二十<span>八又</span>萬四于四百四十零<span>几肺口二一八占九美八臥凹五立尤九口九七人口口八七</span>園內田十<span>八乂百蠆一芝</span>形之每列邊為引干品百零蒼<span>鑒丁</span>兵百二十五萬全十四百六卯零<span>尛一</span>幟什寥<span>夕裏乙五五</span>阡<span>夙亡口六五口八八八七九</span>盧內而<span>吏以斗八</span>元十六<span>畫三</span>形之每沮潛為人百五十四<span>八八</span>
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爪扎陛八一億二干<span>八父</span>百口十仈萬五干入百四十<span>一小餘五五八四一七嗜虹二八一區六口八六七八田二一</span>酬闢內一百几十一邊形之每壇迥為汨百開十七憶內千工百四十人萬四早二百五十田<span>小餘九七五六一沂八五九八五六五八</span>羸<span>几八几八一二</span>園內二百<span>八乂</span>十四邊形芝每遣為菁六十二憶六千一百亥亥油十七萬⑨千一百零七<span>小餘八七四一五心五七</span>冰<span>爻三言二九八四六五八九姜愚畫一六六七九九六四</span>園內七百六更<span>八奭以打</span>罵辜貧韋之每逼為半一憶<span>仈又</span>干帥百一十萬<span>八父</span>干零五十二<span>小師四九五七</span>狀<span>八九四八九九一目勺一青穹口夙五一一七匡尸二</span>五<span>片跌</span>百二十六邊形之每夙澹為四十憶充干零六十<span>二口</span>萬一千五百<span>八又</span>十二<span>唯餘一二一九日八八一六一瓜司夏九六八五八五九翅口一九匡</span>丙三千零七十二邊形之每直滯為二八安夕十億四干五百三十萬七千二百六十<span>箋八夕羹小餘洲七六六口九口八八五夕八勿邀瀆肩九兀一綴口七九心</span>勿介肌八一九園內入干一百四十㈣邊形之每一熏為手億一干二百入十五萬糾千八百一十四<span>小畔口一口</span>几<span>七二凡五口五九八九五</span>川豁左<span>四一七</span>園內玕馮二干二百<span>八夕</span>十<span>八壺父王</span>形之每涓遁為五憶乙十一百三市二萬六<span>干</span>九百門十三<span>小餘七四八田八八一兀夏</span>代<span>穴九口三一八四七六七九</span>囷內涓萬四式吠勿千五百七┿六邊形之每搜運為二憶<span>幣八炎</span>
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<span>恐夕</span>一<span>支孝戶憑</span>五千五百六可六萬三千四百六十三<span>夕八纍災小餘九五一三口九四八口五二四亦日史四如包㈣口六七六</span>圍內四萬九千田百五十二澹邢之每言遣為彊一干七百<span>八矛</span>即三萬夙千七百三十一<span>山敵六七六列六一八六九四七六四尚田</span>石<span>四几ロ几几先</span>囷內九萬<span>八又</span>千一百零四遭形之每邊為六千三百九十萬五干<span>八叉</span>百力十人<span>小餘口七</span><span>刀口一已一六口七口</span>訊勁<span>劣</span>一<span>八口末二六一八七口七五</span>囷內田十九萬八干<span>八巧</span>
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<span>八八多河少油</span>去百零禽遭形之每臺為二千一百及狐<span>一甘</span>災充十五萬七千九百三十三<span>小論口尤九五九虹</span>了禪<span>ロ九八江四一口六五八口口</span>圜府一十九萬二干二百門十六遙形之每邊寓享五百九十七萬<span>八乞</span>寸九百川十弋小餘五四口口五五八八六九<span>步一勁八七七九七二七五九亦七</span>園內七十<span>八父</span>萬六千四百內十二邊形之每豆埋<span>倘一</span>七百九<span>卜八一乂</span>萬九千四取逆夕百<span>八一父</span>二<span>小餘一尤口一一㈣八口六六六</span>心<span>以六五一口五</span>恥<span>一斗</span>木贈<span>甲已己一一四八</span>園內田百五十七萬二千<span>八叉</span>沉卜忤物八百六十四邊形之每乙邊為二百九十九當四芉七百四十田<span>川餘六立一一七四五二九</span>川<span>冗八六八口七哉六一一七九一九</span>園內三百內十四萬五干七百田十貧迥形之每重潛為百九<span>一二</span>九萬仈千一百七十零<span>小隸八一七五五九斛九六六六四口五司</span>五五四口陂八六七九六四往內人百二十九萬凶千四百五十六澹八支嘉之每辜迢為九┿丸萬全寸六百八<span>八尸五小餘四口八七七九六七八工一二</span>丑九七五五七五七四仙六三叔旺祀一脈焉閣內夙千二百五十<span>八父</span>萬二千九舌<span>小一</span>河邊形之每邊為四十九萬充千三百四<span>下一</span>二<span>小臨刀八亟老</span>如<span>山八川八</span>巨鷹<span>六九八一六五五</span>園內白干五百田十六萬五干<span>八父</span>百田十四邊形之烸項逆為駒十四萬九千六百七十一<span>小蘇五</span>砌<span>四凡尤札一七七六九八八口</span>刪<span>八八六五六</span>園內五襲姓尹千零因十調萬門千六百四十兮澤形<span>八一吔尸</span>
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冰州<span>一尸八</span>之每減過篇辜二萬四千<span>八父</span>百三十<span>勿二隻乞小餘六七六口九七四六四一照仁迎丑七一五四七日丸團八</span>圜內一憶零六十六萬彡千二百九十六邊形之每凸滑為人萬一千四百口十七<span>小佛八八一六五九</span>川<span>四六</span>蟄<span>兄似九五牛冽一</span>園內二億零百一十丙萬六干五百九十門邊形之每濕<span>且氏</span>為一萬<span>二八卜赤屯几口打</span>一奇二百雲尸先州則七以九工杞巡<span>門出一十凡八日口</span>園內四億零一百入十棄一五萬三干一百<span>八矛</span>十㈣邊形之毒盡<span>色史</span>尼寫一萬五千六百零四<span>川傑川丑九包一八口</span>三吠輿九禪九士蜀勺愧意譽五<span>石乘口力覆軒乂畫嘉</span>目三十萬六干<span>一曰</span>百六十<span>仈乂</span>邊形之每<span>一</span>邊為七干<span>乂乂</span>百零二<span>小餘五力</span>惶<span>伯七九五</span>曰
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<span>八八凡八一几五</span>列<span>其山團</span>圓內門十六憶尸千零六十峽禹一千七百一十六邊形之烸項<span>賈艾</span>為一千九百零一<span>川一</span>酬<span>一八</span>玉彥<span>七八心四打七九九川六九九六五入四八七什五</span>園內三<span>斗</span>
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人<span>以八</span>十田億目干百一十二萬五千四百穴鬼八七十一邊形之每掛<span>霞一</span>為一千九百五卜勇小餘五五史四一九口一八七<span>挺賣九五七四五九五一四四臥六八口</span>七園內八<span>卜一</span>四憶四干二百㈣十五萬零九百四十四竭形之每蒯喟為丸寶七十五<span>小餘山七八七琉五一刃一九七八七一九六曷九霆六</span>圜內夙百回十<span>八父</span>億合十四舌九十萬┅干<span>八一</span>百<span>八乂</span>十<span>八壺乂三</span>形之每熱劣一涓峨為四百<span>八艾</span>十七<span>小餘六三凡立九七五五七九八八炎一八几六七光四九一夕惱一八几口九凡口伍</span>圖內二百五十七江妃淺億六千九百<span>八下才</span>萬一千七百七十六臺形之每自遁為二百四十二州爛<span>別六犁九八七七八五九九四六八司弓五</span>冥州七四九四五四九五畫伐丑育一十五憶三千九百六十萬七千五百五十二澹形之每辜予為一百一十<span>二小餘九口九八九九八九二九</span>燭九比三四一㈣四七九人七几口九乃以五百一十五憶三千九百六<span>丁乙賈</span>叩妙<span>炙</span>萬七千五百五十二邊之數興其每一頰<span>乏墨刁迥三</span>菁開十內<span>小跡凡口九八九⑨一八九化九七四馭火怒裘合貶</span>恃<span>四凹七九匕近口尼</span>之數相采得六兆一千八百三十加憶<span>八刃</span>千五百二十萬七斗田百七十九<span>小慨五八六山七衤丘八口矧四憚宴五立口一八甘弋</span>馴八為園徑二兆之周數園內容四邊起算
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設如園徑二苑用內容四澹起算間得園周幾何<span>卜卜</span>
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讓以園徑二兆折半得剖兆目采得尸稷倍之開方得一兆四千夙百四十一八八尸父德乙丁一百五十六萬二干二百七十<span>禾馭五一小餘口九五口泣八八口六八州乂幻三二勾囚一口九洪九八口七汾五六</span>乙弋瓦為囷內所容四邊形之每濡迎乃以牢徑田兆為弦四邊之卿<span>簟己田丁一裏</span>訓百四十一憶乙丁門百五┿六萬二千三百七十三州酬<span>日列工六八八</span>牡<span>山一</span>勁勺<span>口九六九八斤夢尋牡于寡牡卜工比惆五六九尊往寄戶一虞鬥一二</span>億零六百七十<span>八少</span>萬乙寸百<span>八矛</span>十六<span>擊只肌餘五四七五四四口夕八凰配了八一多六口四八四九日九八山</span>師<span>一二</span>一<span>乙岑</span>為勾亦即為股<span>曰避折半所成之打股形其勺興股相等</span>與友匕心<span>八</span>半徑相減餘二千几百二十<span>八刃</span>憶九千三百臼十引萬<span>八之</span>千八百乙十二<span>小瀟四五</span>四七五五九九五五六一石夏高八九五五ロ兀六丸七六暴喜勾四澹芝一邊折半之七干零七十一億零列百七十<span>八又</span>萬一千州百<span>八一乂</span>穴<span>小餘五四七五四四八戰四</span>天門一口四八四九口⑨八山旁門股求得弦七干六百五十三億六千<span>患</span>八仆乂入百<span>八言矛劣</span>萬四千七百三十零<span>口簡一七八久兄丑四四五六九九九六八焉司久八八忘夫口七九七七五名匿</span>耨伉弋八百所容<span>八戶</span>澹形之每過<span>霆迂</span>復以半徑一窕為弦<span>八矛</span>邊之耳<span>巳一</span>折半得門干公曰午十六憶<span>八矛</span>千二百四十三萬②千三晉人十五<span>小綠口八几石犬一四五九九八四口</span>阡<span>乃一九八八八六七敞</span>為勾求得股九千二百三十穴<span>呂不</span>陋七干九百五十一萬一千五百田官山小餘一八六史五六八八<span>二矧八几九六七八八八六入八刁只火八只火</span>一<span>八</span>與半徑相減餘七百六十騰<span>思仁</span>一穴工<span>乂八</span>事零四十六萬七千四百<span>八卜八小伽乂鄱又七役二悉四八七八六八叔夏焉乙買央口土既七八知木蓍</span>容勺公潛之托遣折半之刃千<span>八艾</span>百門十六憶<span>八父</span>千一百四十涓萬一千二百八尸五小餘舟八九七七七八四五<span>二丑九九八田愆口九八八大亡</span>具<span>六</span>為股求得弦夙干九百零鳩侍<span>八又</span>一干零六十四萬四千零一十②刺<span>館一六五五六九六五六九七六九舄日了軍四日四四四八八五五以篇樞稷</span>勿乂一所容十丸<span>壺三</span>形之每洎巡如是屢求得八上<span>一奠</span>囷內三十┅邊形之每百哨為一丁九<span>二已</span>
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<span>鸞</span>旅史菁八十億內干四百二十<span>八乂</span>萬零六百<span>亡丙乙小餘口一九八八互井力九州七七七七八六</span>羸<span>七一</span>園內八十㈣遙形之每峭階為冠百<span>八又</span>十州憶一千五百三十四萬八千六百五十四<span>小餘八八口口九九二五口</span>匕<span>八五一五門九囚八口田五八犬</span>園內夙百二┿舍造形之每耳睹為四百九十憶<span>八才</span>千二百四八八刀乂十五萬七千零四十五<span>小礪八甲四丘七六口六四</span>僻<span>仁迄八</span>此<span>慎六口六以丘丑五田</span>圜內②百五十<span>以災一八八</span>
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乃忤恪八六漏形之每稟霆暫晉田十五憶四千二百零以萬八千五百七十因<span>小四</span>訛<span>氾八五五八八七八口五日了</span>八七口七六ロ遭憚杯五百刀丁一邊形之每夏熏百二干門憶七千夙百七十六管九千二百<span>訂乙馬小餘一逾氏凡五口七九</span>力井尸八口九八凡七五一几尸慚<span>口┅七</span>園內因千零開十四澹形之每八<span>一</span>遣為六印一憶刃千五百九十拙局小<span>尼一忠</span>三干五百一十五<span>小仰亦四八一八四口目九五六一</span>淚<span>一二元五陸一畏八五口</span><span>八八八</span>圍內二千零四十<span>八父壺三</span>形之每重過為同十憶六千七百九十穴萬零土百<span>究一</span>十二<span>小證立六几五一四六口七五互口四八伍五五七八口五田</span>圜內四干零九十六邊形之每濕妃為門十五憶二千一百九十<span>八只</span>萬零六百肉十七<span>小餘四八五四忒九逆五八七七六八司斬八</span>ロ六九八口五勿六五二九一僅同久八八力以<span>乂</span>干申日九十二鳴形之每一邊為七憶如<span>八乙迫</span>知千六百九十凡萬零三百七十五<span>小餘覆尸八八四┅七九七八四四九六日銘八一四口七九已八八乙市匡</span>內一萬六千三百<span>八久</span>十四邊形之每一邊為三億<span>八又</span>千一百四十九萬五干夙百九十四<span>小餘六四口六入一山八七九八三九四六七五北一</span>圜丙<span>一</span>萬開干內人巾<span>八乂鼻口憚一</span>形之每重<span>日夫</span>為蔭九千吾七<span>口一</span>百萬七干五百九十<span>八小又伍</span>酬<span>九六九</span>酬<span>凡七八區一畫</span><span>日四四四七田七</span>園肉六萬五千<span>一只斟八</span>豆百胡十六<span>壺三</span>形之每遭迥為九千五凶紅育<span>八少</span>十七萬山千七百九十几<span>卜眼則六三一七六九口九八血九司弓一文六六八四九五八口七匿付</span>手囚萬享零七十二遣形之每邊兄寫四千七百九十工萬六干<span>八矛</span>百九十乙尛餘六一六八六四七四五八夕一七五六五七一七七四八七園射肉一十六萬二千一百四十四澹形之每逼為一干三百九十六萬<span>八又</span>史以<span>火</span>干四百四十九<span>川餘八口二比四一昭田口曰迫二火一</span>
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武山細<span>七山六一入</span>卿<span>五</span>圈內五十二萬四干二<span>一八八</span>
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<span>八</span>巴取八百<span>八柄八艾乂</span>邊形之每判邊為凶┿一百九十<span>八乂</span>萬四千二百肉十四<span>小師月口五八四八五立大八丘七六口口四九一凡五五四六八八</span>園內一百零四萬<span>八乂</span>千五百七十六邊形之烸因邊為五百九十九萬門干閒百糾十二<span>肌餘田五一六孤九五迢口九</span>口加九八吏六亡勳六口六五圍甲四一百零兀萬七干州百五十一遣八午形の每<span>舌一宣</span>為開百九十丸萬六千零<span>八王卜弋小餘六一八丑四畝旦一頎五北七口八七四口火</span>二憤比八九荊內四百月十九萬四千三百零四邊形の每濕了為因百四十九萬<span>廷一二亡八卜謙菽方鬥四</span>力苛雲一十尸三<span>亢一</span>壓阻一六一門<span>七丘一</span>陋<span>凹丈四一</span>園內<span>八矛</span>百夙十公羈<span>八乂</span>干洲百零盆<span>蠆旦</span>形之每項贈為七十四苗究干零辜四<span>小謙逆其七六八</span>岫<span>五八山七八口七尤四六五一五五口七七</span>園內一寸六百七十八文火七萬七千間百┅十六<span>壺一</span>形之每至八<span>川</span>如<span>匕忒</span>為惆十七萬四千五百零八<span>小餘八</span>似叔悠八心<span>九口六八力七七六七口六六八口口一</span>六圈內一干門百五十五萬㈣千四百門十一擅嘉之每辜<span>迴一</span>為辜<span>八又</span>萬七千二百五十<span>一小餘五四四六九六一六五</span>二五九八四四五豆迫八
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區園內六干七百甲十萬公寸<span>八乂</span>百亦丰四邊形之每腹昭為九萬夙吁六百<span>守小餘七五七翅七口九八</span>一寸畎八丑九口五九阻六炊怒吐削一劉內夙億三干四百二十洲尚<span>一鬥</span>
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邦仈<span>了八</span>石于七百二十八濫形之每重<span>只上</span>馮四八亡八<span>八徙王</span>禹入<span>口八一叉</span>百一十三<span>小餘上七八五二六五四凡口五</span>二<span>九口二四一韓四六氏</span>園內┅憶川千<span>八又</span>晉四十工萬五干四百五十六<span>董玉</span>形之毒閔邊為夙萬三千四百零六<span>小餘六八九</span>猷<span>八七四弄五四一六四丸口九九七</span>岫<span>四到四几</span>內伍露戶千人百<span>八只</span>十七萬零九百內十二運形之每湛予為仆尚門干七百零三<span>三京軸一四四六四七一七七三</span>卜玉火文一央一口九夙四人<span>八一尹仈</span>園內辜憶七干三百七十四萬凶干及蒸氏尺百夙十四邊形之每壇<span>宦冀</span>為五千<span>八乂</span>言王卜一小餘六七七弓六人<span>毫丑一六八七五八五六七六</span>酷<span>屾六閒六四九</span>申內一十內憶四干乳百團平<span>八矛</span>萬開千六<span>一</span>四十盆暗形之每胡壇為一千凡百一十五<span>小敝八戊一五八五四一凡六了瓦几七百八伍夙九四</span>園內四十二憶尤千四百九十六萬七千一百九十六憲刀乂一邊形之每踏蒿一千四百六十夙嗣乍咳一九八口北九六七五九六企目<span>少岑仈亶九日九六七七四五一九慢尸火少</span>可<span>娶</span>百<span>八一</span>十五憶<span>八只</span>干九百九十因萬四千五百九十二澹形之每沮逍為七百三<span>尸邇小餘四五九口一九陸一三五七二九八四咫五口四口六所</span>宙一圜內因百七十一憶七千九百<span>八乂</span>十人萬九千阿百<span>八乂</span>十四邊形之每一<span>量三</span>為二百六十五<span>小臘七几伍刺九八六七八九几迴二五九判八四大九內入六</span>圈內一<span>面一</span>四十三憶<span>逢</span>十<span>之</span>五千九百七十三萬<span>八父</span>千一百六十<span>八乂八二</span>
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八如助<span>豎一</span>形之每┅邊為門百<span>八卜乂</span>一削酬<span>久七少干八五九几口八凡四九六會至乂一衫長一尤六心六八六口上七口只似一</span>開四十一億五千九百几<span>卜一</span>三萬全<span>丅八</span>一百六十盆運之數與其每葛二百<span>心己小餘八六山七五九九位八尸丹一九酬丸六口几六口六八</span>柄刊七之數相乘得六兆門十八百三十沂億仈千五百二十萬光十門百七一乙小餘五八曰毛六八六八斗父一口六七五五丸什三乃已六為囷經二兆之周數
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園外切六<span>宣上</span>起真設如圍徑一兆鼡外切六邊起耳問得園周幾何法以因徑肉兆為弦半徑襄為勾求得股冤七千三百二十憶五千零<span>八乂八岫乂</span>十萬七千五百八十<span>八小餘八石七一叒九遯一七四</span>氏門八<span>一四六四五口丘八八克一六六九田</span>取其三分之田老人得菲一千五百四十七憶零五十三萬<span>八二</span>丁三百比十九<span>小辟五立一⑨疏內九上</span>醴<span>六九</span><span>川日凡一工九五</span>即園外六邊形之每莘<span>只蓋因往為弦牛偉為勾所側股上即罔外一邊形之每邊之半倍之厲圈外一邊形之每一邊具每一遙之一分之一印嗣閭外六邊形之義一邊今以八邊起算故眉求一邊止以所保庠股取其一分之偽七邊形之每八</span>瓜<span>一邊二也</span>乃以六邊形の每毫凶兆乙十<span>工近</span>
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人<span>一八一左畢</span>五百四十七億零五十三萬<span>八父</span>千三百<span>尸父一甘卜乙小餘一五五九限八乙六一文九七五六句口一九四九一⑨五</span>折半得五千七百七十三憶五干零一十六萬九千門百<span>八一又</span>九<span>小蹄六五七六四五口几四八七八口高勺五心九五七四五五人四七乏二蓬</span>一半徑山兆為股即用六邊之辜為弦<span>南內六邊與半律等因外六邊亦與本酌半徑寺故即田六邊易邊馬弦也</span>真半徑相減餘共丁五百四十不億零八<span>火</span>伍十三萬八干三百七十九<span>川餘五一五一九尸富八九七五六八口口吏至交九囚兀九五戶屈監草</span>一八吏即小同式形之勺復以六邊形之一邊折半の勺五干七百七十三憶<span>江一</span>千零韋六萬九千凶百<span>八矛</span>十九<span>小跡六一五七卉川五口九四八七八乃五真九五七四五五六四七</span>為乙率半徑之股罔兆為同率小同式形之勾一十五百四十七億零五十三萬<span>八乂</span>千<span>一</span>百七十九<span>小保五扛引八九七五六二口</span>幻<span>一三式</span>爻<span>舟四九九五</span>為二率推得四率②千六<span>八數一八</span>百七十九億四千九百因十九萬二干姑四百三十尸<span>小餘二七洲四一五五六五八四九</span>酬<span>凶七六口五七</span>為小同式形之股倍之得五芉工百五十<span>八父</span>憶九千<span>八乂</span>百山十<span>八少</span>萬四干<span>八父</span>百六千門<span>小餘四嫌可上一九四五自七</span>霆<span>六九八八五一大六乙乙囚</span>為圜外一遣形乏每逼<span>口奻</span>是屢求得園外軾十四擔嘉亡母邊為回干六百二十一憶零八只<span>火</span>百百九十九萬五千一百七十四<span>小隸七几贄史七戶六九四運五九四司卜八兵哀几四四口七八四匱夕乂</span>愈八百十坌遁形之每逾唱為回干門百夙事億兮十入百九十目萬五于六百三牢零<span>小餘四七八四五七凡口八七四四九七五九八八五五八九文一</span>四園外化十六邊形之每辜厲六百五十四億七干三百由十胡萬零<span>八乂</span>百一十五<span>小洲九四立二七工八七瓜五七八九七七川六九几一乞</span>岫七園外一百九<span>一山</span>二邊形之每八<span>一</span>百階為一百門十七億一千七百<span>八卜矛一八八遏芬鼻</span>四萬四干曰百七十零<span>小餘六更一六伍口六</span>折八訓<span>一丘七一九八八九七</span>莊<span>五六</span>比園外門百<span>八一</span>十四遣形之每重血為引百六十三憶人干一百<span>八乂</span>十二萬六干八百零七<span>小師五八七七五七四口七五口四司卜</span>璽六几口五五口匡夕
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七百六十<span>八父</span>邊形之每辜丁為<span>八楚</span>十憶<span>八父</span>千囚百一斗七萬六千五百零四<span>山餘五七田七四唯伍八孤</span>五倒七日口六七<span>八炰</span>田四六圃刀外門干五百一十六邊形之每毫八<span>運八一</span>畔為四十憶九千零六十<span>乙一</span>萬享一百乂三十<span>八小餘四又七八</span>娛<span>凡四八七一五七六五</span>例<span>七口几了八六七口</span>囷外三干零七十四邊形之每裏囂一十憶四千五百三十萬令寸四百三十零<span>小餘八升六八口凡代七尤八九口四九口七口四八</span>園外六千門百四十四邊形之每自造馬州十億二千<span>一</span>百人十五萬一千九百四十七<span>小蹴七一互九四</span>畎<span>五六</span>熒<span>門七力六氏田口</span>吐<span>六凡口七六八</span>園外苗曹享火<span>八</span>已育<span>八卜八若以一乂三</span>形之每府<span>暨王</span>為五憶夙匕<span>八</span>干菁三十二萬六干九百四十零嗣<span>四一五几七口乙陸田八司几八六九六七七八六一匿卜夕</span>田萬四千五百七十六邊形功每一遣為一億五千五百六印六萬三午四旨六六<span>小縱心山口口口五四五一七</span>洲<span>六六四祀九五</span>峭<span>立日一</span>園外四萬九千一百五十以邊形之每向連為一憶一千七百<span>八乂</span>十坑汀穹畜河乙寸七百夙十一<span>小餘四列七八七七七仈山口口江六口七尤山勺口田六田八</span>園外九萬合十<span>刻</span>
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不<span>八</span>言百零四邊形夕每沼邊為六干戶百石十抉禹五千<span>八乂</span>百六<span>山一</span>六<span>小獻八一六大口㈣口五六四日瓢卜一乙七七六七八田八四陛夕一井</span>充萬八千六百零<span>八墨乂三</span>形之每濕<span>里了一梁</span>軸享判百几十五萬七千九百三十一<span>小餘口八陸七翅七七運六八九</span>午面四北七五日五六九澗
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<span>山闢卜一夕</span>三十九萬明千一百十六<span>蚩丘</span>形八尸畏菅口澹為田干五百九<span>一己</span>七萬<span>八乂</span>千外亢百陸<span>可一</span>六<span>小繡五田心匕止四一八四一七門口一七凡</span>火<span>淹一一口几四一</span>園外七十<span>八又</span>萬八干四右隔十二澹嘉之每羸<span>色具</span>為七百九十<span>八乂</span>蒿九芉四百<span>八又</span>十三<span>小翩七口八逼一五八</span>薪<span>乙口八太右八口四妙口</span>園外百五十九萬二千<span>八父</span>百六<span>卜一</span>四邊形之每邊<span>駟一</span>三百丸十九萬四千七百㈣十一<span>小師六豆四六九六凡六五六九司卜純</span>一八四八七口川五五八耽夕羲一中舟十四萬五千七百田十兮追形<span>八</span>乏每卿連為一百丸十九轅七芉二百<span>二卜崇八匕互丙汁姒屯七</span>運升零一輔即卉把悅七四九把七光<span>西四一五四</span>園外六百一<span>巾一</span>九萬乙寸四百五十六<span>實三</span>形之每山過為九十⑨萬<span>八以</span>享六百<span>八尸</span>十五<span>小雕山妃塹</span>酣<span>八止人上七氏六</span>蟲<span>七兀七山一五厄入</span>園外一千二百五十<span>八乂</span>姑尚一千凡百一十二澹霧之每邊為百十⑨萬九干工百四十二<span>小師八四三八</span>似<span>文六七五四六七七七八司卜一青穿工戊住几四六四匱夕一一</span>內<span>薛川</span>九百用十入蒿五千<span>八又</span>百一十四擅形<span>尸七</span>乏每豆啼為閑十四萬九<span>早一</span>六百惆<span>小餘五几四九九八八五以秋一六八八一八八田八八奭一</span>嗣因外五千零洲十三萬一千幻百四事蓄嘉の每潛為享一萬寧
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<span>一八一艾</span>百二十<span>戶小檢判七六了凡七田幻一四四五四九婁一九八</span>畢吁<span>二七一氏</span>圜外液隅零六十六萬三千二百九十六邊形の每濫蒸人萬一享四<span>一</span>享七削<span>欄凋尸俎尸一六六六五八</span><span>兵四二五七心九六九七六</span>園外一憶零<span>一</span>百三一十一萬六干五百几沖回邊形政每瞻為彡萬因千二百零<span>八又</span>川<span>館二</span>酬艾刑剽止七五七八人七一司卜刀意享<span>八八七吠五四八泄夕日比碧</span>了百六五萬三千昭百<span>八乂</span>十四<span>蓄軍</span>形川脾慘
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の每<span>一一</span>邊為苗阿五千六百零四剛酬<span>厄五一八心六九四司卜</span>五八五五胡六舟八臥而量欲八憶零五百三十萬六<span>十一</span>山百六十<span>八以乎</span>邊形亡毋嘉囂七千<span>八乂</span>百零二<span>小跡二弋</span>酬弘岫<span>六研几陳五八曰一尸九九七口</span>圖外<span>八</span>辜六憶亭零六十畜何田千七<span>出一</span>二十六遯形之每鴻<span>只一</span>為二干九百零乙小餘四八七八口四五七瓦涓<span>乙丑四四七四田八凹五六一</span>園外一十一憶一千一百一十一萬五千四百七十一邊形之叩羽階<span>為</span>一手几百五┿零<span>小餘五立七四几一一入七九五七五</span>印蠡<span>六四口一六氏</span>園外六十四憶四千胡<span>眇</span>首四十五萬零九百四十四邊形尼征叫堪為九百七十五<span>卜踰仈七一姬五四九</span>筆一此<span>八引四山阿七二丙口</span>園外一百二十<span>八乂卜且一</span>
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毫<span>八一</span>干四百九十萬一十八百<span>八尸</span>十<span>八乂取</span>
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運嘉之每邊為四百<span>八卜才佽</span>七則刪<span>尹一享几七五五七九八兀刀六九八五五八</span>比<span>一六囿卜一几夕</span>二百五十七憶六千九百<span>八矛</span>十萬二千七百七十六邊形之每痕階為二百㈣十一小餘八九六七九八七七八五九<span>一九團六八四六七七六口</span>吝六園外五百十五憶二千九百六十八萬七千五百五十二邊形之每莘寫亡竇二卜小餘九川九八九九<span>口了八凡九九七田口七九一五</span>麟<span>八</span>乃以五百因十五憶二千九百入十萬七千五百五十一邊之數與其每裏<span>只土</span>書胡十一惻酬<span>九以九九九八九九几七山一口七七六八五六步</span>寡相乘得八兆片千<span>八一</span>百二千夙億八千五百二十萬七千調百七十九<span>小鋒五八壬門匕弋尼三江四六口七七八一八三九六口八沂</span>為圜徑二兆之周數
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設如園徑一兆用外切四邊起莫閫得園周幾何藻以國徑二苑為外切四邊形之每四逼乃以園徑二兆瓦股亦即為勿求得藝彊二兆八寸一百<span>八卜尸</span>四億一千七百<span>一八</span>
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矢一十四萬四于七百四十六川唯<span>九九七八六口一七七四凶八四一乃司卜刃九九六五七列八蕃偪夕己</span>道埋形之斜弦輿園徑相減餘全十胡百又十四億一千七百口十夙萬四千七爭百四十六<span>小怖九口一七七四四</span>刪<span>幾孤六口三四凡九幽七一八</span>即園外舍遣形之每洎阻又以八遣形之每項<span>晝八丑叉</span>千二百<span>八一</span>判一叉八億開于七百一十二萬四于七百曰十<span>咸博尛餘一九口斛九七六口七七</span>以八三妙四巴八四九凡六五七艾八一州折半得四干百四十二億門千一百五十六萬一千三百七十三側酬帆<span>四八八ロ六八八七四口九六凡八口七八五大九</span>一為勿半徑襄為股求得弦因兆零<span>八才</span>百田十一憶丸干一百一十萬零二百九十一<span>五餘一九一九六八人⑨九四四入四斫七一七七八八四口</span>勇半徑相減餘<span>八又</span>百一十三憶九千二八巽育夙十萬零一百九十一<span>小餘九一九六八八九</span>桑八乂<span>冤四山列四臥七一夏比八八四日勿</span>即股弦較又赴八<span>八</span>肅小同式形之勾復以<span>八量乂王</span>形之咽進孫半之勾四干夙百四十二億一寸一旨五十六萬目干二百七┿一刪<span>開五</span>似區八八魚六八樹王土質一嵩一率<span>霆尤六九八日七八五六九卓</span>罕徑之股一兆為一率小同式形之勿不百一十三億九于二百二十萬零一竇九十田<span>小餘二九一刀六八八九升四四六四勺七勾七克一丁一</span>一繫<span>川二</span>篇門率推得四率辜九<span>一</span>八十凡億一十由百二十<span>賣尹</span>萬七千涓乞於首七十九<span>小諭六五八口口六升一五尤七六一六酉四六七其五</span>帳<span>八七</span>為小同式形之股倍之得開享九百七十<span>八意乂于</span>門千四百七十二萬圖千七百五十兀<span>小讎王六口一八几五一四</span>龜<span>八川五七</span>幄<span>九一曰</span>為圈<span>卜夕</span>十人邊形之毒胡邊如是屢求得園外穎十二搢形乏每疆浮為乙寸九百八十⑨憶<span>八又</span>千乏二百<span>八卜父</span>萬六千七百享四馴爛<span>一五八口六五四九五迫四五日甘馬丁</span>八天六五四八六四五八四匱夕尹以八干四邊形之每一邊為九百<span>八又</span>十夙憶五千二百八十九萬九千五百二十<span>八叉小餘尤一四五五八色六八六六賈割五四六七四五八</span>園外塵百一十<span>八又</span>邊形之每毫烹㈣百九十憶九千七百一十四萬四千閩首一十以<span>川餘八立口八八八口一五尤五口七九八</span>巽<span>山四一八刀</span>園外一百五十六邊形之每一斗固一邊為洇百四十五億四千四百九十一干<span>一泣一</span>萬四干七百五十九<span>小蹄立五且四六玉</span>三悼<span>七火口六坦六八五四五九八几口</span>圜外五百口十印<span>畜宮一三</span>形之每唱為一亨手籌七享引百萬零二百州十<span>小餘四亦一八喻凡</span>床<span>立性八七乙以六七口五八</span>園外千零一十四邊形之每晦<span>貫工</span>為六叩<span>意一尹</span>三早五百九十四萬二千四百零二<span>小躡八四五九九七四四七八一目了一六四四四七六五八四面匠夕一</span>弩了灌<span>一</span>享零四十公遯形之每澹罵一十<span>八奚立</span>
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洲琴<span>什一</span><span>支惠丁丁</span>六干七百九十六萬訊千九百<span>八乂</span>十<span>介卜八一小餘七七五六九八五正四勁六一六七了口八七田杞凶</span>盂園外四千零九十囚邊形之每鳩夷為辜五憶<span>一</span>干山百九十公禹一千牽<span>八又</span>十<span>八小傑六八卜八五口乂田六口五四一五五八山七五八</span>園外<span>八乂</span>干招百<span>送九</span>十二擔嘉之每畜<span>匿上</span>為七憶六千六百九十几萬零四百三十一<span>小餘互山八八凡七六六九四六八</span>可逝<span>夙八工四一九口</span>讎外焉享三百<span>八乂</span>師<span>支</span>址十四澹形之每臺為一憶<span>八少</span>千二百天<span>一</span>八<span>一</span>四十九萬五干二百零因<span>小剛八七一四七七七口九七六</span>恤<span>口具一</span>二<span>一</span>園外二萬二干七百六十鷺遭形之每裏寫億九千菁七十四萬七干五百九十九小餘口七口六口入口口九九六口九四五團州口囷外入萬五千五百一十入邊形之每夙邊為九千五百八┿七繁三千七百乙蟲乙小餘一八九口九四
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一八寸什<span>甲八</span>五口六五五入王六<span>一刀父互八</span>園外一十一萬一干零七十一八擒遁形之每河邊為四于七百九十一萬入千八百九十九<span>小餘亡占口石九九口犬丘几暑五兀八八九四州几四四</span>園外二十六萬二十夙百四十四邊形之每一邊為一千三<span>富刻</span>
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百九十六扞八干四百四十几<span>山?八一八穴口六口六几五七口一昌卜王入九五入九口口匡夕丑</span>十二軒四干二百八十八邊形之每一門邊為一幹四百几十八萬四千判百口<span>日小餘九口五五口口口日四九五</span>引可趍口口口四八五口口什<span>一八</span>六圈外一百零四萬八于五百七十月邊形之每舟邃為五百九十北忙一干凶百田牛二<span>山餘川五六凡六五五八九几六八口一口</span><span>八口五四</span>圜外二百零兀萬七千山百五十一澹形之每膽寫明百几平⑨萬六千零五十打<span>小八六四代山失山几六娛遍八四八</span>麟<span>一</span>園外四百厄十迂肖元萬四千三百零四彈場形之每鄉虹為卜刀八八一胡百四十九萬<span>仈之</span>千零一十<span>八川肺戶</span>語八入<span>凡八五旱六五口六八一五五心六四七五</span>帆<span>七</span>園外<span>點叉</span>寶三十全高<span>八乂</span>千六百零運形夕毋夙澹寫七十四萬九千零享四<span>口借一五</span>辜征<span>八以八一八五</span>馴<span>七八山四八兀四五一</span>屬外一干入百七十七萬七干二百工十六澹形之每至寞倚二十七萬四干五百<span>雲亡小餘方八九九七五</span>真斗五五七几一七四了四川百斗凶<span>少一口</span>圈外鬥千三百五十五萬四干四<span>內邇庚</span>陣百三十二邊形之每御<span>聲王</span>為<span>一一</span>十貧嗣悴怨七千二百五十三<span>川蹄五凹四引一九六九八六</span>州<span>四四四五七口一五七四</span>園外六千七百因十萬掌<span>八父</span>百六十四邊形之每毫為元萬三千六百一┿六<span>小餘七</span>毗麟<span>八口氏</span>帆<span>七比八几四七且八七八大</span>園外鹿憾三于四百二十萬七千七百二十盆坦形之每內澹為四萬六干<span>八乂</span>百一十三嗣<span>一匕仈五六五四九凶一心五司</span>寸矛一慮山大醜五五五一七六汴匿<span>一繫</span>外一憶六千<span>八父</span>百四十一萬五千四百尸芒五十六邊形以每卿<span>責旦</span>為二萬三芉四晉零八<span>小餘六八几六八七五五厄六五四七九六</span>咽<span>立五荊土凡一六</span>園外五憶三干六百<span>八矛</span>十七萬豪九百開十二邊刑之每重嵩內萬一干七百零二<span>小餘四山六川九一七一七七五八力匹尹大乃山恂九大</span>園外一十憶七千三百霍十四萬一千八百二十四邊形淺母叫謀乙岫邊罵五千<span>八乂</span>百五十因<span>小麻六人乙七姒六瀟八艷八六八七四口九丑一司卜二卜內</span>一八八囚七七五四四几區夕一斗卜八憶四千七百四十<span>八尸</span>萬昭干六百四┿八澹嘉晏每浪字為曰干九百二十五小餘八一入五八五田九<span>裏四八九八凡八七八凡四</span>真聞外四十二憶九千四百九十入萬七辜二百九十六邊形之每咽遁為仍千西百六十閱<span>小繡凡八久几判七五九六八九八巳引几八五貞五</span>園外<span>八坊</span>十五憶<span>八坎</span>干九百夷頂刷充十三萬四千五百几十由邊形之每<span>一八一三</span>一科邊為七百一十<span>小餘四立几川九六五七九</span>姿也八<span>文四口六六一六七</span>叫<span>四六六</span>圈外明百七十<span>辜隅花</span>七干九百<span>八父</span>十六萬⑨千<span>一乙</span>百<span>八欣</span>門西邊形之每壇進為門百入十五<span>小慚七</span>五憶岫<span>兀八亦七八山几了四四八八八</span>園外二百四十內憶五干九百七十二萬<span>八又</span>干三百六十<span>八乂</span>澹形少每澹為菁八卜一小餘八六四七五九九洞八<span>父一九巴九六口四一六九九</span>八昭又<span>五四四五口</span>乃以一百四十田憶五千九百俎尊尛七十三萬<span>八牙</span>干三百六十<span>八又</span>邊之數與畏賞<span>八</span>具每百延一百<span>八矛</span>十門<span>小餘八五九九</span>傾<span>口七八</span>妣<span>四几一九</span>財<span>口已四戶六四四五口</span>之歎相東嘚<span>丁封</span>兆一干<span>八之</span>百門十夙憶<span>八又</span>干五百三十萬七干叫百七十九<span>小佩打八六川七七艾七八昭川五一七荒</span>仙間園徑二兆之周數
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御製數理精蘊丅編卷十五
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關關磬醴繁已總校宮正苗小停張能照讀攝釐孰亡
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蟲<span>亡還</span>酣恥北狄對官中隕斥臣郭長發<span>一運夏一</span>
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驪<span>鮮一口川川一似一霆三具</span>川阻臉煙竦臥血生臣江潮避烏謂貫門蟲九剛
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<span>謂孟</span>左<span>一</span>一畎<span>以昨一怒</span>禪州川爛燈錄監生臣劉國永醉一召<span>豐一</span>二一賢<span>一奉</span>口蛤慚圖監生臣周韓<span>呵</span>
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欽萣四庫全書蒼要鯉蓋讀毫章反欽定四庫全書蒼要卷毫書一九喜漏假數理精蘊下編卷十八
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<h2 id="匱荀六"></h2>
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八只台邑割囷八線川宗一要一簡法<span>莊勤匕堅昌日八線相求求象限內各線總法</span>附邯喜
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<h2 id="割囷八根"></h2>
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圖周定為三百六十度大而周天小而寸許皆如之蓋園有大小而度分隨之其為數則同句囷心平汾因周為四分名曰四象限每戶象限九十度州象限之中提為正弦餘弦止矢餘矢正切餘切正割餘割名夕曰割園八線
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<span>八几</span>設如甲乙丙丁之園自園惢戊平分全了我霄因為甲乙乙丙丙丁丁甲四象限其每<span>大牙一</span>鶴尸黼等月象限皆九十度乃旬園心戊任作引戊已半徑則將甲丁九十度之孤分為甲己己丁二段己丁為巳戊丁角所對之孤甲己為甲戊已角所對之弧如命<span>八屢</span>一一己戊丁為正角則甲戌已為餘角甲戊<span>戶</span>
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言鬲亶為正角則已戊丁為餘角正角所對<span>逞慎幸</span>
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略下一為正弧餘角所對為餘弧今以已丁為王孤故甲己為餘弧又旬已與甲丙全住平行作已辛綠謂之通弦其對巳丁止孤洏立於戍丁半徑者目正弦又與茂丁半徑平行作壬已線謂之餘弦以五萬甲已餘孤之所對也於戊丁半徑丙減塵庚餘屢丁謂之正矢於甲戊半乙<span>一②</span>住內減士戌餘甲壬謂之餘矢自圍界跂禹一盥苗止岱下徑平行立於戊丁半徑之末邢一作垂線仍與已戊丁角相對者曰正切縛已戊半徑引長與囸切相遇於癸成茂葵綠謂之正割又自圍界與戊丁半產平行作甲子綠謂之餘切戊癸正割疲甲子餘切截莽子所分戊子謂之餘罰每刀用凶弧即有囸弦餘弦正矢餘奚已成四疏於園界之內復引出半徑乙一於圜界之外而成正切餘切正割餘割烹丘怛川事西線內外共為八線故曰割圍八繡<span>八</span>
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<span>只</span>惢遑遠虔逐分正弦之餘即為餘弧之正餘菰之正即為正弧之餘是以前四十五蔓之八線正餘互相對雷為用不必復襄後四十五度之八線也凡北八線皆充十度以內鏡角之所歲若直角九十夏考則不能成八線蓋因半徑即九十度之正弦甲戊半徑即甲丁弧之弦而刺而線割線仙于<span>丁布</span>終無相遇の處也若洲憂角過九十度以外者則於半周一百怵<span>曰</span>異八十度內減二角度用其餘度之八線節如已庚為己丁弧之止弦亦即乙已孤之正弦也要之仈綠以正弦為琴有至弦則諸練皆由此生故月宗二要皆係正弦之法
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<h2 id="八宗三要一藩澄辟"></h2>
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西洋座嘗轉來作割圜八緣表始自圍內容八邊四邊十邊三邊五邊十五邊名曰八宗蓋圉圍徑求各等邊形之鬥邊為相當弧之通弦以為立表之原故謂之宗然八者實本於二如八邊形之言<span>竇輩</span>即園之半徑不藉他求數無零餘而理最易見此其夙也四邊形之耳沓則為半徑所作正方形之對角斜弦此又其夙也十邊形之耳邊則為牛徑所作連比例三率之中率西法謂之科肩蹇稟理分中末線此又其開也至於三邊形則出於幻邊五邊形則出於十邊十五邊形則又出於三邊及五邊非別自立引法也既得此〣種形之亶透各半之即得乃種孤之各正孤爰命此川種弧為本弧按法可求本弧之餘弦可求倍本弧之正弦餘弦亦可求半本弧之正弦餘弦是為三偠又以不等兩弧之正弦餘弦求相加相減弧之正弦又兩弧距八十度前後之度等得其兩正弦之較即得距弧之正弦是又名為三閒法由此錯綿之可嘚正弦夙百二十其中最小者為四十五分之弦其次亶冬二十分又次為二度十五分又次為三度如此每越四十五分而得四弦其自州分至四十四分の弦則以比例求之因弧分甚微輿直線所差無幾故以弦求弦而得之此西法立割圍八線表之大綱也爾表西法對數表內有讓連比例四率以求圍內嫆七邊九邊白法因推廣其理於川宗之外增求圍內容十八邊形十四邊形之法俱以半徑為首率求連比例四率于縞
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之第引率即十會瞑形十四邊形の每夙邊而九邊七邊又因之以生亦猶三邊之出於六邊五邊之出於十邊也有北二形輿月宗相恭伍可得正弦三百六十其中最小者為干玉分之正弦又增判法求十五分之三分之引五分之正弦所少考止判分至四分之正弦較之四十五分為尤密可知奚今以八宗二要河簡湛理分中末線并新增數法皆按類具例於左旨賣圖囚谷片邊形四鷹近形一邊安胥形十邊形<span>牡一</span>邊形十五邊形設如園仲二十萬求內容八邊形之司邊幾何法以囷徑二┿萬折半得半徑十萬即圈內容六邊形之每亶建也如甲園內八斗及容朴邊形每邊之弧得圍甲八分之同<span>汴川一</span>門卞<span>二一</span>皆八十度試自圜心甲至圍界乙丙二<span>詳一</span>
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<span>戶蔡巧</span>忙處作甲乙甲丙門半徑綠成甲乙丙三角形則甲角所對之弧為八十度而甲乙甲丙兩腰俱為半徑既相等則乙角高育亦必楿等而各為川十度英三角昆寸則一邊亦必相等故乙丙邊即與沃牙夕甫乙甲丙半徑相等也乙丙弧既為八<span>八</span>
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<span>汁一叩乂紅乂</span>手度則乙丙邊十萬為〣十度之通弦<span>一之進哭</span>薪半得乙丁五萬即乙戊狐一十度之<span>岐鯉一</span>
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一正弦也此即八邊起莫之理前設囷徑玄壇窕者所以求其密合今設圍杵為呵┿萬所以取其便於用也
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設如圍徑二印萬求內容三邊形之亶遭幾何法以園徑二十萬為弦自乘得四百億又以半徑十萬為勾自乘得菁億相減餘三百憶開方得股凶十七萬三千八道乂二百零五<span>吐輪口川口七五六八</span>即圜內容三邊<span>八</span>
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<span>八久</span>毗<span>一</span>形之每亶遙也如甲圜內容三邊形每靜眇忙渥違弧嘚園周三分之畔官乙百二十度為州邊形每邊弧之料倍試有乙角馮園心至對界作乙丁全徑線又自丁依半徑度至丙作丁丙綠則咸丹邊形之每增透其丙丁弧即為二邊形之每邊弧之重<span>一二</span>而丙角立於圍界之夙半必為置角故半往為勾全徑為弦求得八仆杲肢即三邊形之每一邊也乙丙弧既為工邸以汾百二十度則乙丙邊夙十七萬三千判語<span>八</span>內百零五<span>小餘口八口七丑六八</span>為判百二十度之通弦折半得乙戊八萬八千幻百零二<span>小餘五㈣口一七入四</span>即乙已弧月十度之正弦也
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設如園柯二丁萬求內容四澤形之鋪遭幾何法以園徑二十萬折半得半徑十萬白乘得門百億倍之得鬥百億開方得一<span>部一</span>四萬享四百二十一<span>小餘一五六一蹤七一</span>八刁乂即圍內容四邊形之每判邊也如甲圖<span>八乂火</span>
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<span>八鼻之桑旦</span>內容四邊形每邊之弧得圜周四分之降似<span>八八夜八</span>囚皆九十度試句因心甲至圍界乙丙箕叫
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工處作甲乙甲丙門半徑綠成甲乙丙勾股形若命甲乙半往為股則甲丙半佩為笥碧命甲丙半徑為股則甲徑為蜀茵勾肢皆為翠柯故以半徑句乘倍之開方而得弦即如勾股各旬乘<span>八八火乂</span>
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<span>八八矛乂爻乙</span>以即之開矛而得弦也乙丙弧既為九十<span>大遺軸</span>
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拘頌<span>午迄</span>八咫度則乙丙邊手四萬享四百納十羈八<span>卜餘一五六一乙一弋一</span>為九十度之通弦折半得乙丁七萬零七百一十<span>尛一</span>聊憔<span>七八弋</span>即乙戊弧四十五度之正弦也理分申末線<span>此西法名也因命線為百率將此首卒分為大小兩分犬分為中辛小</span>分為末牟典原線妾為楿連比例一率故謂之理分中末線也設如以十萬為首率作相連比例三率使中率末率相加與首率等求中率末率各幾何法以十萬自乘得凶百億晦長方積以十萬為長闊之較用帶縱較數開方法<span>書啼言畫昌</span>
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翠六荅一箕之得圈萬辜八<span>百零</span>嘻即相<span>聖式季衰葵比例之中</span>率以中率與昔率十萬相減餘三萬八千占百九十七即相連比例之末率也此法蓋因連比例三率之首湘卒末率相乘之長方積典中率句來之正方積等而首率之中有竇中率山爪率之數故首率句乘之帥正方積中有<span>鵠八</span>首率中率相來之一長方又有首率末率相秉之鬥長方即如甲乙為音率丙一己為中率甲丙為求率丙乙Φ率旬采<span>軒</span>
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之正方為丁戊乙丙甲丙末率與甲乙首率相乘之長方為甲丙庚辛<span>甲景與甲乙等</span>此正方問長方之積等而甲乙首率自乘之正方為甲乙巳辛丙乙中率輿甲乙首率相采之長方為丙乙已廣<span>丙襄丁庾</span>一<span>輿甲乙等</span>夫甲丙庚辛之長方既輿丁戊乙一丙之正方等則甲乙己辛之正方亦必<span>巾肌</span>
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孟子戊已庚之長方暫是以丁戊已庚<span>一虞</span>便方形之閨即中率具長比闊之較即轉庖<span>己</span>菁率故以首率白乘為長方積仍以首華為長比閭之較用帶縱平方法開之簿閣為中率也
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吏法以首率十萬為肢首率十萬折半得五萬為勾求得弦十畜間革人庚<span>戊</span>一百零三內減勾五萬餘幻萬刑千八百一零彡為相連比例之中率以中率興首<span>尸一</span>
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八率相減餘三萬八千州百九十七即為岫<span>山日一上季</span>連比例之末率也如圖甲乙與乙丙小辛謹昏為首率今鉯甲乙為股乙丙折半得乞丁為勾求得甲丁弦試依甲丁弦度潛丁乙勾引長至踐傑丁乙戍織仍旬甲至戊作一圈界則甲丁戊丁同為半徑且皆為弦於戊丁弦內減乙丁勾所餘乙戍與已乙等即中率於甲乙首率八一內減去典乙戊相等之己乙中率所餘尸丸甲已即末率也此法與前法理實相同<span>尸</span>
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<span>竝</span>帶縱較數開方法有以半較句乘與原斗積相加開方得半和於半和內減半較得閭者今此法以首率柄肢目乘得甲乙丙壬正方形即與庚戊丙辛長方形積等乙丙即長闊乏較乙丁即半較戊轅一哭丁即半和今以乙丁為勾自乘甲乙為荒已乙股自乘相加開方得甲丁弦即如乙丁孩尸了半較自乘與甲乙旬乘原積相加開方下門而得甲丁興戊丁等戊丁內減乙丁餘戊乙即半和內減半較得閭為中率也設如囷徑二十禹求內容十邊形之專雖幾哬法用連比例三率有首率求中率末率使中率末率相加與首率等之法以圍徑二十萬折牛得十萬為首率自乘得二百億為長方積以十萬為長聞之較用帶縱較數開方法莫之得六萬一千又人百零三<span>小餘九八入毛固尤</span>為連比例之中甲<span>迓</span>亡六率即圍內容十邊形之每判邊也如甲改一園內容十邊形每邊之弧得園周十分乏一皆三十八度其通弦即園內十邊彭之南遷試目圍心甲至園界己丙二處作甲乙甲丙召半徑線遂成甲乙丙同角形復旬園界乙至圓界戊作拍乙戍練則載甲丙綠於丁又成乙丙丁三角形而乙茂遽為引百零八度之通弦之此乙丙丁三角形與甲乙丙一角形為寅川<span>竟霄</span>同圓式形<span>乙丙丁一角形之乙角富成內孤為乙丙孤之陪則乙丙丁次八角形之乙角與甲乙丙一角形之甲角邸忿等又同用丙角其餘角赤必著故為</span>用式其相當各邊俱成相連比例故甲乙與乙丙之比同於乙丙興丙丁之比為相連比例一率而甲乙為首率乙丙差苗委平丙丁為萊率也又甲乙丙彡角形其甲角既居全園十分之酬為三十工斧前乙角必比甲角大巳倍為七十淹夕引度<span>一角形之角丹百八十角既馬一十天度則乙丙兩</span>酬<span>甲必迂┅為百四十四度平分之谷得形二蕭伊</span>兵紅葛十度比甲角為大陪也汙屯日火大凶王一角形之乙角與甲乙丙三角形之事角等則甲丁乙三角形之乙角亦必輿甲角寸是則甲丁乙二角形為兩邊箱寸之三角形而乙丙丁三角形亦為禹邊相等之一角形也夫甲丁既與丁乙等而丁乙又與乙丙中率等則甲丁亦必皇革率等妾是以甲丁中率與丁丙末率相加與甲丙首率等故用連比甲例三率有首率求中率法莫之得中率為十邊形之重隨也
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又法鉯圍徑二十萬折半得半徑十萬為股句乘得州百億又以半徑十萬折半得五萬為勾句萊得二十五億相加得一百二十五憶開方得弦十己萬夙千八百零三<span>辟館九八八七四九</span>於弦數內減姜勾數餘川萬因千八百零三<span>吐篤一九丙</span>夕弧帆<span>七</span>即圍內容十邊形之每百遭也<span>一嘻牡</span>縱如甲園內容十邊形每邊之孤得圍周<span>己午八</span>
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尹恥鬥災仙十分之胡皆三十六度試自圍心甲至<span>且一</span>
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圍界乙作甲乙半往線為股又旬圖心甲取直角作甲丙半往線折半嘚甲丁為勾求得乙丁弦內減踵革丁相等之縕高判惡戊丁餘乙戊即與乙已等為圍內容十貳只邊形之每亶違也乙已弧既為三十川<span>蒸及</span>
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夙<span>之及</span>度則乙已邊列萬內千八百零三<span>小餘一九千什一蓍</span>
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判卑<span>一八八八七一八日屯</span>為一十川度之通弦折半得乙<span>及也寒哭二內</span>麥庚一萬零九百零州<span>小餘陸九九目一弋囚</span>即乙辛弧十八度之正弦也
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設如囷徑門十禹求內容五邊形之一邊幾何法以半徑十萬為底仍以半往十萬與園內容十邊形之重沓〣萬一十八百胎令引州<span>笛一力八七日把</span>為兩腰用判角形求中委綠法莫之得中垂線五萬八千七膏七十八<span>小佛五五昔一化予</span>之得鬥十夙萬七千伍百玉十七則<span>繇口土了五入固</span>即圍以內容五邊形之每鬥邊也如甲園內虎一五邊形每邊之弧得圍周五分之州皆宅十二度試白園心甲至圜界乙丙一富體苗乞甲丙夙半徑線遽成甲乙丙一畜口形其乙丙邊為七十二度之逼弦如以乙丙孤七十門度折半於丁作乙丁綠即圜內容十邊形之釘邊仍自圖心甲至圍界丁作甲丁半徑線黍成甲八乞丁三角形而甲丁綠平分乙丙線於山事化此乙箴線為甲乙丁三角形之中垂玉丁顯即五邊形每邊の拙千故以甲丁半蘊為底甲乙半徑為大腰乙丁十邊形芝門邊為小腰求得乙茂甲垂線倍之焉五邊形導毋凶邊也
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爻法以半粹十萬為股旬乘得一百憶圍內容十邊形之州邊月萬州千八百零三<span>小戊</span>踰<span>乙九八田瓦</span>山勾旬萊得夙十八億享九百川十八萬零白百一十一酥<span>餘四八凡刀几川八丑戊伍口口乙</span>相加得一百三十天億工九百朴十月萬零內百所十土<span>小騰四八九力刀川五八五八五口口</span>開方得弦一十川萬七千五百五十七<span>小餘凸土ロ囚丑戊曰</span>即圍內容五邊形之每百遭也此法蓋因羊徑旬乘十邊形之亶沓自栗兩旬乘矛積相併即與五邊形之副邊旬乘之署積等故用勾股求弦の法箕之如甲以一園內容五邊形將乙丙弧折丰於丁作叩一乙丁綠即劉內容十邊形之別邊仍自臥玳亮心甲至丁作甲丁半徑綠遂成甲乙予三角形又依乙丁線度裁申丁半徑霧已作乙已綠成乙已丁一角形與甲乞丁三角形為同式形故甲乙差觀百率乞丁為中率巳丁翩求率甲已亦與乙于等為中率而乙丙邊平分已丁末率於戊又成乙戊丁勾股形乙戊五邊形每邊之半為股丁戊末率之半為勻乙<span>迫鬥乙</span>丁中率為弦試傑甲丁半徑度作甲庚<span>兩</span>
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丁驪市<span>忤</span>辛丁正方形又依乙丙五邊形之言達<span>刮</span>
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徊度作乙丙癸壬正方形其甲庚辛丁正方形內甲子丑巳為乙丁弦自乘之一正方<span>甲巳既與乙丁弦等故甲子丑巳為弦自乘之正方</span>已富莘丁長方形亦輿乙丁弦自乘之一正方臺丁辛原與甲丁首率等已丁末導與<span>羔丁辛首率祖秉自與乙丁中┅自乘</span>態黃夕<span>之正方等</span>而子廣寅丑長方形為乙一弦旬<span>鶩夕</span>
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呵<span>坤斗艾儀二</span>乘之一正方內少勾自萊之四正方<span>蓋子乞糞三座翠丁卯長方形為首卒與末辛相秉之祚體訣幻長方與乙丁中率自來之正方等內卻可善少丑寅辛卯正方形而丑寅辛卯正方奭形實為戊了勾自采之四正方故于庚仁船寅丑長方形為乙丁弦目秉之亢呈</span>正方少勾有東之四正方也署貝書丁半徑自乘之甲庚辛丁正方形內有弦旬乘之二正方而少勾句萊之四正芳再加乙丁弦自乘之一正方共得弦苞果之四正方而少勾自果之四正方夏凡弦自果之正方內原有勾句乘之<span>乙討汀</span>
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<span>八</span>刀沈一一正方股目乘之正方今弦旬乘之<span>二伊</span>
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耳打西正方內少勾自果之四正方即與腹<span>造乙拜</span>
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<span>一</span>坤婦事旨乘之四正方等而乙丙一邊匍乘之<span>即一斤</span>卜已丙癸壬正方形實為乙戊股自乘之西正方然則甲丁半徑會乘方與乙丁牢邊形之百遽目乘方相併既與乙戊股旬乘之四正方等而乙丙哀運旬乘之正方豈不與甲丁半徑自塖乙丁十近夕咽運形之耳避自果之兩正方等乎故以<span>卜只夕</span>
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打斟<span>庚</span>甲丁半柯為股乙丁十邊形之咽運為<span>尸八一福臥地畢</span>軸<span>二童</span>蠅勾求得弦而為伍邊形之享遜也<span>址一</span>亡又法以半徑十萬旬乘得菁億為長打佑方積仍以半粹十萬為長闊之較用帶縱較數開方法莫之得長享八萬一千八百零二<span>幾魚九八八北四九</span>折半得八萬零九百零門<span>小晦六凡几田毛酉</span>為自圍心至五邊形每邊之垂線乃以半徑十萬為弦圍心至五邊形每邊之垂線為股求乂尸得勾五萬八千七百七十八則<span>句九乂</span>
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<span>八一</span>倍之得一十州萬七千五百五十七尸壯川鄉卜以汁即圍內併五邊形之每一<span>才巴互</span>邊也如甲圉內麤五邊形將乙丙弧折半於丁作乙丁線即圍內六十邊形之山邊仍自圖心甲至丁作甲丁半徑綠成甲乙丁一角形又依乙丁線度載甲丁半徑於已作乙已線成乙已丁一角形輿甲乙丁二角形為固式形故甲乙八襄以乂為首率乙丁山中率巳丁為求率甲已<span>火勿</span>
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州呻亦尺乙丁等為中率而乙丙邊平汾己<span>山</span>
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脈恬亂丁末率於零定以巳戊與戊丁俱為半末率而甲戊旬困心至邊之垂線則為裏臺車末率之共數今以半徑首率自乘為長方積鹽布縱平方得長乃首率典中率之和苴丙有兩中率十本率攝半得辜率半末率即甲戊自園心室邊之垂線既得甲茂晏線乃以甲乙八尸童丁徑為弦甲茂垂綠為股求得乙戊勾叩一倍之得乙丙即圍內容五邊形之一邊降區或以已丁中率為弦戊丁半末率為宥翼水得乙戊股倍之亦即圍內六五晏彥之耳避吔乙丙弧既為七十一度則乞丙邊手開萬七十五百五十七嗣几口<span>仕八</span>恥四為七十一度之通弦折半得尸己戊五萬八干七百七十八酬<span>藏朴几門</span>牡<span>┅清</span>即乙丁弧一十八度之正弦也設如圍徑二十禹求內容十五邊形之夙邊樂倚法以半徑十萬為弦圍內容五邊形之忘<span>取</span>鏘半五萬八千七百七十仈<span>几貔王立一九一</span>土軌鉤勾求得股公禹零九百零烈<span>心行六九几囚只八少</span>戴八大迂一內減半徑之半五萬餘一萬零凡百零夙<span>小飾夫九九囚一七醜</span>為股次以圍內容二邊形之重造一十七萬一千二百零五<span>小餘口八口七五光八</span>內減圍內容五邊形之白邊烈十喜刊七千五百五十七<span>川簫心五</span>八仈獸助<span>口四五人目</span>餘五萬五千川百四十八<span>小餘口八況一</span>灼只<span>一</span>印<span>九日</span>折半得州萬七千八百一十四<span>火矛一一江以桐</span>
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<span>乙</span>縣川堪<span>曹輪口五一曰凡</span>為勾求得弦四萬烈千五百八十二<span>且倚具洲一弋五</span>即圍內容十五邊形之每亶建也如甲囷內容十五邊形每邊之弧得圈周十五分之一皆一十四度試從圍界乙作圍內容一邊形又作園內容五邊形將三邊形妻母一邊弧分五段五邊形之每言邁弧分一佐段即得十五邊形之每百覺弧如箴戊芒岐與已丁二段皆為十五邊形之弧故以湮甲丁半徑為弦丁丙五邊之半為勾求得甲丙肢內減甲辛目圍心至一角底邊之垂綠為半徑之半餘辛丙與癸丁或壬庚等復於一邊形之戊已淳甫減五邊形之庚丁邊即如戊已綠內減壬癸餘戊壬典癸已一段折半得癸已或戌壬今任以癸丁或壬庚為股癸已戓思亢神勿戊壬為勾求得巳丁弦或茂芙弦即圍<span>烹八八酢</span>
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<td>
禁汴卑<span>匕一一前</span>內容十五邊形之每亶違也己丁弧既<span>二巧一八</span>
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不祚州逾為二十四度則巳丁邊四萬夙千五百八十二<span>虯笛人一六五</span>為一十四度之遺弦折半得已子一萬零七百九十一嗣<span>九口八毛</span>即已丑弧十二度之正弦也新增按分作楿連比例四率法設如以十萬為半牟作相連比例四率使熏牛輿四率相加與二準一倍等問鬥率三率四率各幾何法以引率十萬自乘再乘得二止此馴縱泗牡方為實又以門率十萬旬采一因之<span>口野一寄</span>懈<span>一亞</span>川<span>言長作亡</span>得二百億<span>戰一乎勁績</span>為法以除原實一千<span>矣一乏</span>
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<span>喜窒二尸三震</span>兆得三萬乃以三萬自乘再束得一十畫毒鑄七兆益於原實州千兆內得一千零一<span>一言</span>
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十七兆為共實按除法以所得三萬與法三百億相因得九百兆與共實相減餘一百囚十七兆為第二位實以法之三百億除之得四千乃以首位所得一一<span>午一</span>萬谷次位所得四千共一萬四千旬乘<span>切一</span>
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<span>一二一丑亡</span>再乘得一┿九兆三千零四十億仍益<span>吟</span>
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<span>一亡</span>吁船於原實玉非簫禪逞千零一十九兆<span>戶</span>
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三干零四十億為共實按除法減首位所得三萬與法一百億相困之九百兆又減交位所得四千與法三百億相因之罔百二十兆餘一十九兆一千零四十億為第三位實以法之一百億除之得月百所餘太多困益積故取塞大の<span>一叫一乙一</span>倒斟取為七百合前兩位所得一萬四千共<span>一乃一弄</span>賣三萬四千七百自乘再栗得四十一兆<span>奢聶擅一</span>
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荒蹇<span>七喜</span>七千八百一十九億一幹一百萬璽血<span>一陋一</span>
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<span>勿</span>翠於原實引千兆內得帕千零四十兆七千八百一十九億一千一百苗申高共實按除法減首位所得一萬與法一百億相因之⑨百兆又減次位所得四千與法三百億相因之州百二十兆又減三位所得七百與法州百億相因少二<span>尸</span>
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柳沖<span>心了一</span>唯十夙兆餘七千八百州十九億②干一<span>一川</span>
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川<span>以塚</span>精隋虞百萬為第四位實以法之一百億除之<span>勿尸必一</span>
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工工面得二十合前三位所得一萬四千七百<span>直以一一切</span>
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<td>
青一止公云禹四芉七百一十自秉再乘得四十一兆八千丑百四十二億同千零四萬八千仍孟於原實門千兆內得一千豪四十鬥兆八千五百四十二億一干豪四萬八芉為共實按除法減首位所淳二萬與法三百億相困之丸百兆又<span>尸二二</span>
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<span>怡冗笛</span>觴礪輕匝菽堤位所得四千與法一百億相因之<span>玄一二一冰</span>
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<td>
<span>二二</span>仁<span>了鼉</span>歸<span>一造</span>酬百二十兆又減一位所得七百血藩<span>諱蟲品馘亞</span>
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<td>
口蠶一<span>亢玄必</span>一三百億相因之一十一兆又減四位所<span>以彗一</span>
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一得二十與法一百億相因の月千億餘一千五百四十一億門干零四萬八千為策獻貝以法之一百億除之得八所餘亦太多因蓋積仍取客大之數為九合前四位所得一萬四干七百一十共二三萬四千七百州十九自乘再束得四<span>尸</span>
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<td>
<span>口一葵一</span>式十鬥兆八千八百月十七億朴千川百<span>八二旦二一</span>
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<td>
<span>一足</span>墜蠻驚<span>三一</span>四十萬零左十㈣百八十九仍益於原<span>一已工</span>
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<td>
覽<span>二嵩行一</span>畫一山實在丁兆內得夙千零四十夙兆八十<span>出善一</span>
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<td>
八百川十七憶八千八百四十萬一千四百八十九為其實按除法以五次所得之數與法相困之數遁減之仍餘四百八十七億八千八百四十萬一千四百八十九不盡是共奈得一萬四千七喜百二十九為相連比例之一率也以一<span>尸一</span>
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就毒丁<span>一一了一工一</span>率之一萬四千七百一十九自乘得一<span>一上害二</span>
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<td>
八扞驥鑿出十一億零六百戶十萬三干四百四十<span>一②一二</span>
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<td>
拂八崑州以山率之十萬除之得畜擅享零<span>同一</span>
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東霆八十臥為一率以一率之一萬四千七百二十九三倍之得十萬四千一百八十七內減去裏咑之十萬餘四千夙百八十七為四率如以一率之同萬二千零川十占日秉以一率之三萬四千七百二十九除之亦得四千內百八十七王三乏乙為四率也此為盍實歸除之法蓋因此享一豐秀法止有半率之數作相連比例四率便<span>襲</span>
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<td>
單聖宅年率與四率之共數與二率二倍等而連比例四率之理一率洎乘用四率再乘輿二率句乘再乘之數掌今立法以開率自乘再來為原實較之一倍一率與來率旬乘之面積相乘之數卻少月二率自東再乘之數故鉯累除所得之數屢次目乘再乘益入原實然復按法馴即弁門姚除之始足二率三倍之數也如圖甲乙<span>一求高</span>
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<td>
雕仙輟尚山率庚子子辰辰乙皆為一率庚甲<span>百</span>
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<td>
<span>一</span>沒立未共向四率庚乙為判率四率之共數又為<span>干</span>
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<td>
一率之二倍甲乙丙丁戊已為辜<span>一一</span>自萊再來之正方體廣乙丙丁壬癸為一倍二率與裏<span>┅</span>目乘面積相采之長方豐一率自采困之傳一平面如川<span>骨辛采之戎一扁方體合之即威倍一率來員責之</span><span>卒自乘匕長方體上</span>鬥率自乘再乘之正方體多山庚甲酉戊壬癸扁方體此<span>襄青</span>
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<td>
<span>匐一</span>民扁方體即一率旬乘用四率再栗之數<span>言一本洋一</span>
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<td>
盥公中牟自乘再乘之積等若於享沐自卞末乘再乘之囸方體內加入拍率目采再<span>一封可制</span>
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<td>
乘之正方體即如於甲乙丙丁戊巳正方體上加閩庚甲酉戊壬癸之扁方體成庚乙丙丁壬癸之長方體而以一率百乘之乙丙丁申方面除之必得庚乙蒿一率之一倍苟合乙丙丁申與辰巳字木及子丑寅卯一方面余之必得廣<span>句尺</span>
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<td>
<span>一</span>亡一孚戰子辰或辰乙為一率若不加積止<span>三了</span>
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<td>
鞅三方面除之則所得仍為裏十之一<span>一竹一甘一</span>
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<td>
<span>烹八</span>汁扎<span>一扎</span>芬之一比一率數必小故以屢除所得乏數屢次自乘再宋盍入原積則積漸壇而得數亦漸大瞻及末位則所少之覆已足而除得之數即為引率之全數馬
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<td>
設如園徑工丁揭求內容十八邊形之曰邊幾何法用連比例四率囿一率求一率使一率與四率相加與二率三倍等之法以八圜柯二十萬折半得十萬為冽率自乘八獻再乘得享疵為實又以半徑十萬自攻之乘三因の得三百億為法接盍盧貝歸除之法除實得三萬四千七百一十九<span>小黔大五五一四</span>為二率即圍內十八邊形之<span>一寶節一</span>
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喜州邊也如甲囷內虎十八邊形每邊之弧得圖周十八分之一皆二十度其運弦即圍內十八邊形之言遭試苴圜了乂忌至圍界乙丙作甲乙甲丙二半徑線甲川敵石榮成甲乙丙彡角形復自圍界乙至圖橫界廣作斷乙廣線則截甲丙綠於戊又箴己丙戊三角形而乙廣為六十度之逼弦復自因界丙按丙戍綠度至乙庚蘊之丁作┅丙丁線則又成丙丁戊三育形此一三角形皆為同式形<span>巳丙成一角形芝乙角當庚丙弧為乙丙孤之倍則乙曷戊一角形之乙角與甲乙丙一角形乏甲角等又輿甲乙丙一角形同用丙蓋羹莘禹二亳門形之丁丙綿那與甲辛半崖平行則丙丁戊角形之丙角與甲乂一丙辛一角形之甲角為相肘錯箱〣亦必七稟又與乙丙戊一角形同用戊角是此邊內一冉形之各角互相等而為同式形之</span>鶴豆其相當各邊俱成相連比例故甲乙勇乙丙之比同於乙丙與丙戊之比乙高典丙戊之比又同於丙戊與戊丁之比為相連比例四率而甲乙為一率乙丙為二率丙戊為一率戊丁為四率也又乙庚為幻十度之通弦與甲乙叫率等而乙戊丁已已庚工段皆與乙丙二乂若率等是乙庚工竿中有乙丙一率之三<span>吹認</span>甲一築倍而少白丁戊四率也必以乙庚冽率<span>二</span>
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<td>
藩輿丁戊四率相加方與乙丙十率之三倍等故用連比例四率有拍率求二率法莫之得可率為十八邊形之例邊就乙丙孤既為二十度乙丙邊一萬四幹瓦一七百二十九<span>小餘六土五四</span>為一十度之摶什點邇弦折半得非禹七千三百八十四<span>心餘</span>火<span>一蠹八一吐氏弋弋</span>怯七即十度之正弦也設如囷徑┅十禹求內容九邊形之片邊幾何法以半往十萬為底仍以半徑十萬與乂乂蜀內容十八邊形之夙邊一萬四千七百二十九<span>小餘六五五四</span>為兩腰用┅角<span>一八勿憚</span>形求中垂線法莫之得中垂線一萬四干二百零二訓輪<span>卦四普一六于</span>之得六萬八干四百零四<span>小飴日印八弋受最</span>即因內容九邊形之烸夙邊也如甲園容九邊形每邊之弧得圜周九分之片皆四十度八蘭友勿試創圜心甲至圍界乙丙二處作甲乙<span>勢縱</span>疇測章堯甲丙目半往綠遂成甲乙丙三角形其<span>之入二</span>易絡咫淺乙丙邊為四十度之通弦如以乙丙弧四十度折半於丁作乙丁綠即圍內容千八邊形之耳沓仍旬圜心甲至圈界丁作甲丁半徑線又成甲乙丁一角形而甲丁線平分乙丙綠於戊此乙戊綠為甲乙丁二角形之中垂綠即九邊形每邊之了十故以甲丁半徑為底甲乙臥必半徑為大腰乙丁十八邊形之引邊為<span>夕</span>
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<td>
<span>事中</span>叫小腰求得中垂線倍之為九邊形之毋<span>乏</span>
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<td>
蝦門迂引邊也乙丙弧既為四十度乙丙邊為四十度之通弦其乙咸甲垂綠三萬四千二百零一<span>小餘一</span>別<span>四六</span>即乙丁弧夙十度之正弦也
</td>
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<td>
按分作相建比例四率又法
</td>
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<td>
設如以十萬為州率作相連比例四率使叫率輿㈣率相加與二王兩倍再加因三率之數等問一率三率四率各幾何
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<td>
川法以饗率十萬苞乘再乘得以十兆刊碧性方為蟲貝又以率十萬旬乘州因之<span>二ㄖ堪</span>認得二百億<span>威一乎面積</span>為法以除原實一十娣兆得五萬為盡數因減寶大於蓋實故取略小之數為四萬乃以四萬自采再采得八十四兆蓋於原實夙千兆內得一于千零八十四兆為益實復以所得四切一
</td>
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一太萬旬乘得內十六億以內率十萬再乘詣焉恨侍一百朴十兆於蓋實內減之餘九百<span>一姒</span>
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寸零四兆為正實接除法以所得四萬與主法內百億相因得八百兆與正實相咸餘因百零四兆為第州位實以法之一行什百偪除之得五千仍取略尛之數為四<span>瀆八巴</span>
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千乃以首位所得四萬合次位所得四千共四萬四干倉乘再東得八十五兆千八百四十億益於碌實肉干兆內惡得夙千零八十五兆夙千八百四十憶訂<span>八殺</span>為蓋實復以所得四萬四千自乘得夙<span>搜川</span>
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二鳶十九億二千八百萬以烈率十萬再乘<span>一八一</span>
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茫片得門百九十二兆川干億於蓋實內減<span>二石</span>
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步鶩餘八百九十訊兆五千八百四十億為正實按除法減首位所得四萬與法二百憶相因之八百兆又減笑位所得四千與法二百憾楿因之八十兆餘夙<span>刪一譬一</span>一二十一兆五干八百四十億為第三位實<span>尸三一屢</span>
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巍乙山以法之二百億除之得五百合前兩位<span>召血弓門</span>
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示<span>蠶忌一</span>所嘚四萬四千共四萬四千五百句乘<span>芸豐</span>
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一三再采得八十八兆千二百夙十一樞動虹沮二干五百萬盍於原實判千兆內得夙<span>二</span>
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里巍<span>一</span>千零八十八兆州千二百十憶二<span>一言一</span>
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蹈宙謂亡下五百萬為蓋實復以所得四萬四干<span>亡一害</span>
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<span>一</span>主馴五百旬來得享九憶八千零判十五萬以蓬竿十萬再乘得因百⑨十八兆零鬥百五十億於益實內減之餘八百一二九十兆零九百八十一億<span>二一</span>乙自萬<span>二二二</span>
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<span>羨幼三尸</span>仙為正實按除法減首位所得四萬與法<span>蓋盟熊</span>
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<span>一苴一一託一</span>夙百億相因之八百兆又減次位所得<span>幻一責</span>
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<span>鬥什見忘</span>仙四千與法二百億相困之八十兆又減<span>一一</span>
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<span>一</span>晉三位所得五百與法二百億相因之十兆餘九百八十沮億夙千五百萬為第四位實以法之一百億除之實不是三法乃以第四位為空位而第五位得四<span>繫幻一之</span>
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乞映<span>一仁一豈</span>籌故以四為末位合前四位所得四萬四<span>一上多一</span>
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<span>量臺乙墮</span>著鸞千五百空十共四萬四千五百零四自<span>一二一門一</span>
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蠶憲菁灑乘再乘得八十八兆干四百四十八<span>云乃丈一</span>
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<span>八</span>懽曲隱九千零革三萬八干零八十四盍<span>喜逼彊</span>
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<span>二巖汀元</span>於原實夙干兆內得一千零八十八兆<span>二瓦夏</span>
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山灌川一千四百四十八憶九干零工王書何<span>言一</span>
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禪<span>賣詠翟了</span>月千零月十四為益實復以所得四萬<span>定完芸</span>
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啼二試四干五百零四甸乘得夙十九區八千鵲芻而而岑八十萬洲芉零州十六以十萬口乘<span>二甚冗納一富</span>得州百九十八兆零列百零八憶零州<span>以丘頡</span>
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獻卿佩百列十萬於益實內減之餘八百九十<span>蠹蓋乙</span>
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<span>了逼悲忘</span>面兆零八百四十工億八千八百五十一<span>丘文</span>
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一<span>了</span>苗甸八千零八十四為正實按除法以五<span>一乃一一舌</span>
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岫<span>引一上讎一一了工了</span>次所得之數與法相因之數遁咸之仍<span>一睡毫謹墮丘</span>
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甕蠹誥罰<span>又藪</span>臥隸四十判憶<span>八叉</span>千八百五十三萬八干蒯揣鵲鵲<span>驢</span>零八十四不盡是共除得四萬四千五<span>云亡工一</span>
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貢言百零四為相連比例之二率也以二率了西萬四千五百