Bibliotheca Polyglotta

BOOK V.

幾何原本第五卷之首

DEFINITIONS.

界說十九則
前四卷所論。皆獨幾何也。此下二卷所論。皆自兩以上、多幾何、同例相比者也。而本卷則總說完幾何之同例相比者也。諸卷中。獨此卷以虛例相比。絕不及線、面、體、諸類也。第六卷。則論線、論角、論圜界、諸類、及諸形之同例相比者也。今先解向後所用名目。為界說十九。

1. A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.

第一界
分者。幾何之幾何也。小能度大。以小為大之分。
以小幾何、度大幾何。謂之分。曰幾何之幾何者。謂非此小幾何。不能為此大幾何之分也。如一點無分。亦非幾何。卽不能為線之分也。一線無廣狹之分。非廣狹之幾何。卽不能為面之分也。一面無厚薄之分。非厚薄之幾何。卽不能為體之分也。曰能度大者。謂小幾何、度大幾何。能書大之分者也。如甲、為乙、為丙、之分。則甲為乙三分之一。為丙六分之一。無贏、不足也。若戊為丁之一、卽贏。為二、卽不足。己為丁之三、卽贏。為四、卽不足。是小不書大。則丁不能為戊、己、之分也。以數明之。若四於八、於十二、於十六、於(p. 二一二)二十、諸數。皆能盡分。無贏、不足也。若四於六、於七、於九、於十、於十八、於三十八、諸數。或贏、或不足。皆不能盡分者也。本書所論。皆指能盡分者。故稱為分。若不盡分者。當稱幾分幾何之幾。如四於六。為三分六之二。不得正名為分。不稱小度大也。不為大幾何內之小幾何也。

2. The greater is a multiple of the less when it is measured by the less.

第二界
若小幾何能度大者。則大為小之幾倍。
如第一界圖。甲與乙。能度丙。則丙為甲與乙之幾倍。若丁、戊、不能盡己之分。則己不為丁、戊、之幾倍。

3. A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

第三界
比例者。兩幾何以幾何相比之理。
兩幾何者。或兩數。或兩線。或兩面。或兩體。各以同類大小相比。謂之比例。若線與面、或數與線、相比。此異類。不為比例。又若白線與黑線、熱線與冷線、相比。雖同類。不以幾何相比。亦不為比例也。
比例之說在幾何為正用。亦有借用者。如時。如音。如聲。如所如動。如稱之屬。皆以比例論之。
凡兩幾何相比。以此幾何比他幾何。則此幾何為前率。所比之他幾何為後率。如以六尺之線、比三尺之線。則六尺為前率。三尺為後率也。反用之。以三尺之線。比六尺之線。則三尺為前率。六尺為後率也。比例為用甚廣。故詳論之。如左。
凡比例有二種。有大合。有小合。以數可明者、為大合。如二十尺之線、比十尺之線、是也其非數可明者、(p. 二一三)為小合。如直角方形之兩邊、與其對角線。可以相比、而非數可明者、是也。
如上二種。又有二名。其大合者、為有兩度之線。如二十尺、比八尺、兩線為大合。則二尺、四尺、皆可兩度之者、是也。如此之類。凡數之比例。皆大合也。何者。有數之屬。或無他數可兩度者。無有一數不可兩度者。若七比九。無他數可兩度之。以一、則可兩度之也。其小合線、為無兩度之線。如直角方形之兩邊、與其對角線、為小合。卽分至萬分、以及無數。終無小線、可以盡分、能度兩率者、是也。此論詳見　\\　十卷末題
小合之比例。至十卷詳之。本篇所論。皆大合也。
凡大合有兩種。有等者。如二十比二十。十尸之線、比十尺之線。是也。有不等者。如二十比十。八比四十。六尺之線比二尺之線。是也。
如上等者。為相同之比例。其不等者。又有兩種。有以大不等。如二十比十是也。有以小不等。如十比二十是也。大合比例之以大不等者。又有五種。一為幾倍大。二為等帶一分。三為等帶幾分。四為幾倍大帶一分。五為幾倍大帶幾分。
一為幾倍大者。謂大幾何內。有小幾何或二、或三、或十、或八也。如二十與四。是二十內。為四者五。如三十尺之線、與五尺之線。是三十尺內。為五尺者六。則二十與四。名為五倍大之比例也。三十尺與五尺。名為六倍大之比例也。倣此為名。可至無窮也。
二為等帶一分者。謂大幾何內。旣有小之一。別帶一分。此一分。或元一之半。或三分之一四分之一。以(p. 二一四)至無窮者。是也。如三與二。是三內旣有二。別帶一。一為二之半。如十二尺、之線。是十二內旣有九。別帶三。三為九三分之一。則三與二。名為等帶半也。十二尺與九尺。名為等帶三分之一也。
三為等帶幾分者。謂大幾何內。旣有小之一。別帶幾分。而此幾分、不能合為一盡分者。是也。如八與五。是八內旣有五。別帶三一。每一各為五之分。而三一不能合而為五之分也。他如十與八。其十內旣有八。別帶二一。雖每一各為八之分。與前例相似。而二一卻能為八四分之一。是為帶一分。屬在第二。不屬三也。則八與五。名為等帶三分也。又如二十二、與十六。卽名為等帶六分也。○四為幾倍大帶一分者。謂大幾何內。旣有小幾何之二、之三、之四、等。別帶一分。此一分。或元一之半。或三分、四分、之一、以至無窮者。是也。如九與四。是九內旣有二四。別帶一。一為四四分之一。則九與四。名為二倍大帶四分之一也。
五為幾倍大帶幾分者。謂大幾何內。旣有小幾何之二、之三、之四、等。別帶幾分。而此幾分。不能合為一盡分者。是也。如十一與三。是十一內旣有三三。別帶二一。每一各為三之分。而二一。不能合而為三之分也。則十一與三。名為三倍大帶二分也。
大合比例之以小不等者。亦有五種。俱與上以大不等五種。相反為名。一為反幾倍大。二為反等帶一分。三為反等帶幾分。四為反幾倍大帶一分。五為反幾倍大帶幾分。
凡比例諸種。如前所設諸數。俱有書法。書法中。有全數。有分數。全數者。如一、二、三、十、百、等。是也。分數者。(p. 二一五)如分一以二、以三、以四、等是也。書全數。依本數書之。不必立法。書分數。必有兩數。一為命分數。一為得分數。卽如分一以三而取其二。則為三分之二。卽三為命分數。二為得分數也。分一為十九而取其七。則為十九分之七。卽十九為命分數。七為得分數也。
書以大、小、不等各五種之比例。其一幾倍大以全數書之。如二十與四。為五倍大之比例。卽書五、是也。若四倍、卽書四。六倍、卽書六也。其反幾倍大。卽用分數書之。而以大比例之數、為命分之數。以一為得分之數。如大為五倍大之比例。則此書五之一、是也。若四倍、卽書四之一。六倍、卽書六之一也。
其二等帶一分之比例。有兩數。一全數。一分數。其全數恆為一。其分數。則以分率之數、為命分數。恆以一為得分數如三與二。名為等帶半。卽書一。別書二之一也。其反等帶一分。則全用分數。而以大比例之命分數、為此之得分數。以大比例之命分數、加一。為此之命分數。如大為等帶二之一。卽此書三之二也。又如等帶八分之一。反書之。卽書九之八也。又如等帶一千分之一。反書之。卽書一千○○一之一千也。
其三等帶幾分之比例。亦有兩數。一全數。一分數。其全數亦恆為一。其分數。亦以分率之數、為命分數。以所分之數、為得分數。如十與七。名為等帶三分。卽書一。別書七之三也。其反等帶幾分。亦全用分數。而以大比例之命分數、為此之得分數。以大比例之命分數、加大之得分數。為此之命分數。如大為等帶七之三。命數七。得數三。七加三為十。卽書十之七也。又如等帶二十之三。反書之。二十加三。卽書二(p. 二一六)十三之二十也。
其四幾倍大帶一分之比例。則以幾倍大之數、為全數。以分率之數、為命分數。恆以一為得分數。如二十二與七。二十二內。旣有三七。別帶一。一為七七分之一。名為三倍大帶七分之一。卽以三為全數。七為命分數。一為得分數。書三。別書七之一也。其反幾倍大帶一分。則以大比例之命分數、為此之得分數。以大之命分數、乘大之倍數。加一。為此之命分數。如大為三帶七之一。卽以七乘三、得二十一。又加一。為命分數。書二十二之七也。又如五帶九之一。反書之。九乘五、得四十五。加一、為四十六。卽書四十六之九也。
其五幾倍大帶幾分之比例。亦以幾倍大之數、為全數。以分率之數、為命分數。以所分之數、為得分數。如二十九與八。二十九內。旣有三八。別帶五一。名為三倍大帶五分。卽以三為全數。八為命分數。五為得分數。書三。別書八之五也。其反幾倍大帶幾分。則以大比例之命分數、為此之得分數。以大比例之命分數、乘大之倍數。加大之得分數。為此之命分數。如大為三帶八之五。卽以八乘三、得二十四。加五、為二十九。書二十九之八也。又如四帶五之二。卽書二十二之五也。
己上大小十種。足盡比例之凡。不得加一、減一。
第四界
兩比例之理相似。為同理之比例。(p. 二一七)
兩幾何相比。謂之比例。兩比例相比謂之同理之比例如甲與乙、兩幾何之比例。偕丙與丁、兩幾何之比例。其理相似。為同理之比例。又若戊與己、兩幾何之比例。偕己與庚、兩幾何之比例。其理相似。亦同理之比例。
凡同理之比例。有三種。有數之比例。有量法之比例。有樂律之比例。本篇所論。皆量法之比例也。量法比例。又有二種。一為連比例。連比例者。相續不斷。其中率、與前、後、兩率。遞相為比例。而中率旣為前率之後。又為後率之前。如後圖。戊與己比。己又與庚比。是也。二為斷比例。斷比例者。居中兩率一取不再用。如前圖。甲自與乙比。丙自與丁比。是也。

4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

第五界
兩幾何。倍其身而能相勝者。為有比例之幾何。
上文言為比例之幾何。必同類。然同類中。亦有無比例者。故此界顯有比例之幾何也。曰倍其身而能相勝者。如三尺之線、與八尺之線。三尺之線。三倍其身。卽大於八尺之線。是為有比例之線也。又如直角方形之一邊、與其對角線。雖非大合之比例。可以數明。而直角方形之一邊。一倍之。卽大於對角線。兩邊等三角形。其兩邊幷。　\\　必大於一邊。見一卷二十。是亦有小合比例之線也。又圜之徑。四倍之、卽大於圜之界。則圜之徑與界。(p. 二一八)亦有小合比例之線也。圜之界、當三徑七分徑　\\　之一弱。別見圜形書。又曲線與直線。亦有比例。如以大小兩曲線相合。為初月形。別作一直角方形。與之等六卷三十三　\\　一增題今附卽曲直兩線相視。有大、有小。亦有比例也。又方形與圜。雖自古至今。學士無數。不能為相等之形。然兩形相視。有大、有小。亦不可謂無比例也。又直線角與曲線角。亦有比例。如上圖。直角、鈍角、銳角。皆有與曲線角等者。若第一圖。甲乙丙直角。在甲乙、乙丙、兩直線內。而其間設有甲乙丁、與丙乙戊、兩圜分角等。卽於甲乙丁角、加甲乙戊角。則丁乙戊曲線角。與甲乙丙直角等矣。依顯壬庚癸曲線角。與己庚辛鈍角等也。又依顯卯丑辰曲線角。與子丑寅銳角。各減同用之子丑、丑辰、內圜小分。卽兩角亦等也。此五者。皆疑無比例。而實有比例者也。他若有窮之線、與無窮之線。雖則同類。實無比例。何者。有窮之線。畢世倍之。不能勝無窮之線故也。又線與面。面與體。各自為類。亦無比例。何者。畢世倍線。不能及面。畢世倍面。不能及體。故也。又切圜角、與直線銳角。亦無比例。何者。依三卷十六題所說。畢世倍切邊角。不能勝至小之銳角。故也此後諸篇中。每有倍此幾何。令至勝彼幾何者。故備著其理。以需後論也。

5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

第六界
四幾何。若第一與二。偕第三與四。為同理之比例。則第一、第三、之幾倍。偕第二、第四、之幾倍。其相視。或等。或俱為大。俱為小。恆如是。
兩幾何。曷顯其能為比例乎。上第五界所說是也。兩比例。曷顯其能為同理之比例乎。此所說是也。其術通大合、小合。皆以加倍法求之。如一甲、二乙、三丙、四丁、四幾何。於一甲三丙。任加幾倍。為戊、為己。戊倍甲己倍丙。其數自相等。次於二乙四丁。任加幾倍。為庚、為辛。庚倍乙。辛倍丁。其數自相等。而戊與己。偕庚與辛。相視。或等。或俱大。或俱小。如是等、大、小、累試之恆如是。卽知一甲與二乙。偕三丙與四丁。為同理之比例也。
如初試之。甲幾倍之戊。小於乙幾倍之庚。而丙幾倍之己。亦小於丁幾倍之辛。又試之。倍甲之戊。與倍乙之庚等。而倍丙之己。亦與倍丁之辛等。三試之。倍甲之戊。大於倍乙之庚。而倍丙之己。亦大於(p. 二二○)倍丁之辛。此之謂或相等。或雖不等、而俱為大。俱為小。若累合一差。卽元設四幾何。不得為同理之比例。如下第八界所指是也。
下文所論。若言四幾何為同理之比例。卽當推顯第一、第三、之幾倍。與第二、第四、之幾倍。或等。或俱大、俱小。若許其四幾何、為同理之比例。亦如之。
>以數明之。如有四幾何。第一為三。第二為二。第三為六。第四為四。今以第一之三。第三之六。同加四倍。為十二。為二十四。次以第二之二。第四之四。同加七倍。為十四。為二十八。其倍第一之十二。旣小於倍第二之十四。而倍第三之二十四。亦小於倍第四之二十八也。又以第一之三。第三之六。同加六倍。為十八。為三十六。次以第二之二。第四之四。同加九倍。為十八。為三十六。其倍第一之十八。旣等於倍第二之十八。而倍第三之三十六。亦等於倍第四之三十六也。又以第一之三。第三之六。同加三倍。為九。為十八。次以第二之二。第四之四。同加二倍。為四。為八。其倍第一之九。旣大於倍第二之四。而倍第三之十八。亦大於倍第(p. 二二一)四之八也。若爾。或俱大、俱小。或等。累試之、皆合。則三與二。偕六與四。得為同理之比例也。
以上論四幾何者。斷比例之法也。其連比例法倣此。但連比例之中率。兩用之。旣為第二。又為第三。視此異耳。

6. Let magnitudes which have the same ratio be called proportional.

第七界
同理比例之幾何。為相稱之幾何。
甲與乙。若丙與丁。是四幾何、為同理之比例。卽四幾何、為相稱之幾何。又戊與己。若己與庚。卽三幾何、亦相稱之幾何。

7. When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.

第八界
四幾何。若第一之幾倍。大於第二之幾倍。而第三之幾倍。不大於第四之幾倍。則第一與二之比例。大於第三與四之比例。
此反上第六界。而釋不同理之兩比例。其相視。曷顯為大。曷顯為小也。謂第一、第三、之幾倍。與第二、第四、之幾倍。依上累試之。其間有第一之幾倍。大(p. 二二二)於第二之幾倍。而第三之幾倍。乃或等、或小、於第四之幾倍。卽第一與二之比例。大於第三與四之比例也。如上圖。甲一、乙二、丙三、丁四。甲與丙。各三倍、為戊、己。乙與丁。各四倍、為庚辛。其甲三倍之戊。大於乙四倍之庚。而丙三倍之己。乃小於丁四倍之辛。卽甲與乙之比例。大於丙與丁也。若第一之幾倍。小於第二之幾倍。而第三之幾倍。乃或等、或大、於第四之幾倍。卽第一與二之比例。小於第三與四之比例。如是等、大、小、相戾者。但有其一。不必再試。
以數明之。中設三、二、四三、四幾何。先有第一之倍。大於第二之倍。而第三之倍。亦大於第四之倍。後復有第一之倍。大於第二之倍。而第三之倍。乃或等、或小於第四之倍。卽第一與二之比例。大於第三與四也。若以上圖之數反用之。以第一為二。第二為一。第三為四。第四為三。則第一與二之比例。小於第三與四。

8. A proportion in three terms is the least possible.

第九界
同理之比例。至少必三率。
同理之比例。必兩比例相比。如甲與乙。若丙與丁。是四率。斷比例也。若連比例之戊與己。若己與庚。則(p. 二二三)中率己、旣為戊之後。又為庚之前。是以三率當四率也。

9. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.

第十界
三幾何。為同理之連比例。則第一與三。為再加之比例。

10. When four magnitudes are proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.

四幾何。為同理之連比例。則第一與四。為三加之比例。倣此以至無窮。
甲、乙、丙、丁、戊、五幾何。為同理之連比例。其甲與乙。若乙與丙。乙與丙。若丙與丁。丙與丁。若丁與戊。卽一甲與三丙。視一甲與二乙。為再加之比例。又一甲與四丁。視一甲與二乙。為三加之比例。何者。甲、丁、之中。有乙、丙、兩幾何。為同理之比例、如甲與乙。故也。又一甲與五戊。視一甲與二乙。為四加之比例也。若反用之。以戊為首。則一戊與三丙為再加。與四乙為三加。與五甲為四加也。
下第六卷二十題。言此直角方形、與彼直角方形。為此形之一邊。與彼形之一邊再加之比例。何者。(p. 二二四)若作三幾何、為同理之連比例。則此直角方形、與彼直角方形。若第一幾何、與第三幾何。故也。以數明之。如此直角方形之邊、三尺。而彼直角方形之邊、一尺。卽此形邊、與彼形邊。若九、與一也夫九與一之間。有三。為同理之比例。則九、三、一、三幾何之連比例。旣有三與一、為比例。又以九比三。三比一。為再加之比例也。則彼直角方形。當為此形九分之一。不止為此形三分之一也。大略第一與二之比例。若線相比。第一與三。若平面相比。第一與四。若體相比也。第一與五。若算家三乘方。與六。若四乘　\\　方。與七。若五乘方。倣此以至無窮。

11. The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.

第十一界
>同理之幾何。前與前相當。後與後相當。
上文巳解同理之比例。此又解同理之幾何者。蓋一比例之兩幾何。有前、後。而同理之兩比例四幾何。有兩前、兩後。故特解言比例之論。常以前與前相當。後與後相當也。如上甲與乙。丙與丁。兩比例同理。則甲與丙相當。乙與丁相當也。戊己、己庚、兩比例同理。則己旣為前。又為後。兩相當也。如下文有兩三角形之邊相比。亦常以同理之兩邊相當。不可混也。
上文第六、第八、界說幾何之幾倍。常以一與三同倍。二與四同倍。則以第一、第三、為兩前。第二、第四、為兩後。各同理故。(p. 二二五)

12. Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.

第十二界
有屬理。更前與前。更後與後。
此下說比例六理。皆後論所需也。
四幾何。甲與乙之比例。若丙與丁。今更推甲與丙。若乙與丁、為屬理。　下言屬理。皆省曰更。
此論未證證。見本卷十六。
此界之理。可施於四率同類之比例。若兩線、兩面。或兩面、兩數等。不為同類。卽不得相更也。

13. Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.

第十三界
有反理。取後為前。取前為後。(p. 二二六)
>甲與乙之比例。若丙與丁。今反推乙與甲。若丁與丙。為反理。
>證見本篇四之系。
此界之理。亦可施於異類之比例。

14. Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.

第十四界
有合理。合前與後為一、而比其後。
甲乙與乙丙之比例。若丁戊與戊己。今合甲丙為一、而比乙丙。合丁己為一、而比戊己。卽推甲丙與乙丙。若丁己與戊己。是合兩前、後、率、為兩一率。而比兩後率也。
證見本卷十八。(p. 二二七)

15. Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.

第十五界
有分理。取前之較、而比其後。
甲乙與丙乙之比例。若丁戊與己戊。今分推甲乙之較甲丙、與丙乙。若丁戊之較丁己、與己戊。
證見本卷十七。

16. Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.

第十六界
有轉理。以前為前。以前之較為後。
甲乙與丙乙之比例。若丁戊與己戊。今轉推甲乙與甲丙。若丁戊與丁己。(p. 二二八)
>證見本卷十九。

17. A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes;
Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms. An ordered proportion arises when, as antecedent is to consequent, so is consequent to something else.¹

1. Interpolation after Theon‘s time, cf. Heath vol. 2, p. 137.

第十七界
有平理。彼此幾何。各自三以上。相為同理之連比例。則此之第一與三。若彼之第一與三。又曰。去其中。取其首尾。
甲、乙、丙、三幾何。丁、戊、己、三幾何。等數。相為同理之連比例者。甲與乙、若丁與戊。乙與丙、若戊與己也。今平推首甲、與尾丙。若首丁、與尾己。(p. 二二九)
平理之分。又有二種。如後二界。第十八界
有平理之序者。此之前與後。若彼之前與後。而此之後與他率。若彼之後與他率。
甲與乙。若丁與戊。而後乙、與他率丙。若後戊、與他率己。是序也今平推甲與丙。若丁與己也。此與十七界　\\　同‧重宣序義‧以別　\\　後界也(p. 二三○)
證見本卷廿二。

18. A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.

第十九界
有平理之錯者。此數幾何。彼數幾何。此之前與後。若彼之前與後。而此之後與他率。若彼之他率與其前。
甲、乙、丙、數幾何。丁、戊、己、數幾何。其甲與乙。若戊與己。又此之後乙、與他率丙。若彼之他率丁、與前戊。是錯也。今平推甲與丙、若丁與己也。十八、十九、界推法。於十七界　\\　中通論之。故兩題中不再著也。(p. 二三一)
證見本卷廿三。
增。一幾何。有一幾何、相與為比例。卽此幾何。必有彼幾何、相與為比例。而兩比例等。一幾何。有一幾何、相與為比例。卽必有彼幾何、與此幾何為比例。而兩比例等。此例同理。省　\\　曰比例等。
甲幾何。與乙幾何、為比例。卽此幾何丙。亦必有彼幾何、如丁。相與為比例。若甲與乙也。丙幾何。與丁幾何、為比例。卽必有彼幾何、如戊。與此幾何丙、為比例。若丙與丁也。此理推廣無礙。於理有之。不必舉其率也。舉率之理。備見後卷。

PROPOSITION I.

幾何原本第五卷本篇論比例　計三十四題
第一題

If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all. ma+mb+mc...=m(a+b+c...).

此數幾何。彼數幾何。此之各率。同幾倍於彼之各率。則此之幷率。亦幾倍於彼之幷率。

Let any number of magnitudes whatever AB, CD be respectively equimultiples of any magnitudes E, F equal in multitude; I say that, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.

For, since AB is the same multiple of E that CD is of F, as many magnitudes as there are in AB equal to E, so many also are there in CD equal to F. Let AB be divided into the magnitudes AG, GB equal to E, and CD into CH, HD equal to F; then the multitude of the magnitudes AG, GB will be equal to the multitude of the magnitudes CH, HD. Now, since AG is equal to E, and CH to F, therefore AG is equal to E, and AG, CH to E, F. For the same reason GB is equal to E, and GB, HD to E, F; therefore, as many magnitudes as there are in AB equal to E, so many also are there in AB, CD equal to E, F; therefore, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.

Therefore etc. Q. E. D.

PROPOSITION 2.

第二題

If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth. (ma+na) is the same multiple of as that (mb+nb) is of n.

六幾何。其第一倍第二之數。等於第三倍第四之數。而第五倍第二之數。等於第六倍第四之數。則第一、第五、幷、倍第二之數。等於第三、第六、幷、倍第四之數。

Let a first magnitude, AB, be the same multiple of a second, C, that a third, DE, is of a fourth, F, and let a fifth, BG, also be the same multiple of the second, C, that a sixth, EH, is of the fourth F; I say that the sum of the first and fifth, AG, will be the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F.

For, since AB is the same multiple of C that DE is of F, therefore, as many magnitudes as there are in AB equal to C, so many also are there in DE equal to F. For the same reason also, as many as there are in BG equal to C, so many are there also in EH equal to F; therefore, as many as there are in the whole AG equal to C, so many also are there in the whole DH equal to F. Therefore, whatever multiple AG is of C, that multiple also is DH of F. Therefore the sum of the first and fifth, AG, is the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F.

Therefore etc. Q.E.D.

PROPOSITION 3.

第三題

If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth.

四幾何。其第一之倍於第二。若第三之倍於第四。次倍第一。又倍第三。其數等。則第一所倍之與第二。若第三所倍之與第四。

Let a first magnitude A be the same multiple of a second B that a third C is of a fourth D, and let equimultiples EF, GH be taken of A, C; I say that EF is the same multiple of B that GH is of D.

For, since EF is the same multiple of A that GH is of C, therefore, as many magnitudes as there are in EF equal to A, so many also are there in GH equal to C. Let EF be divided into the magnitudes EK, KF equal to A, and GH into the magnitudes GL, LH equal to C; then the multitude of the magnitudes EK, KF will be equal to the multitude of the magnitudes GL, LH. And, since A is the same multiple of B that C is of D, while EK is equal to A, and GL to C, therefore EK is the same multiple of B that GL is of D. For the same reason KF is the same multiple of B that LH is of D. Since, then, a first magnitude EK is the same multiple of a second B that a third GL is of a fourth D, and a fifth KF is also the same multiple of the second B that a sixth LH is of the fourth D, therefore the sum of the first and fifth, EF, is also the same multiple of the second B that the sum of the third and sixth, GH, is of the fourth D. [V. 2]

Therefore etc. Q. E. D.

PROPOSITION 4.

第四題其系為反理

If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order. If A:B=C:D, then mA:nB=mC:nD.

四幾何。其第一與二。偕第三與四。比例等。第一、第三、同任為若干倍。第二、第四、同任為若干倍。則第一所倍、與第二所倍。第三所倍、與第四所倍。比例亦等。

For let a first magnitude A have to a second B the same ratio as a third C to a fourth D; and let equimultiples E, F be taken of A, C, and G, H other, chance, equimultiples of B, D; I say that, as E is to G, so is F to H.

For let equimultiples K, L be taken of E, F, and other, chance, equimultiples M, N of G, H.

Since E is the same multiple of A that F is of C, and equimultiples K, L of E, F have been taken, therefore K is the same multiple of A that L is of C. [V. 3] For the same reason M is the same multiple of B that N is of D. And, since, as A is to B, so is C to D, and of A, C equimultiples K, L have been taken, and of B, D other, chance, equimultiples M, N, therefore, if K is in excess of M, L also is in excess of N, if it is equal, equal, and if less, less. [V. Def. 5] And K, L are equimultiples of E, F, and M, N other, chance, equimultiples of G, H; therefore, as E is to G, so is F to H. [V. Def. 5]

Therefore etc. Q. E. D.

本卷界　\\　說六
一系。凡四幾何。第一與二。偕第三與四。比例等。卽可反推第二與一。偕第四與三。比例亦等。何者。如上倍甲之壬、與倍乙之子。偕倍丙之癸、與倍丁之丑。等、大、小、俱同類。而顯甲與乙、若丙與丁。卽可反說。倍乙之子、與倍甲之壬。偕倍丁之丑、與倍丙之癸。等、大、小、俱同類。而乙與甲。亦若丁與丙。本卷界　\\　說六
二系。別有一論。亦本書中所恆用也。曰。若甲與乙、偕丙與丁。比例等。則甲之或二或三倍、與乙之或二、或三倍。偕丙之或二、或三倍、與丁之或二、或三倍。比例俱等。倣此以至無窮。

PROPOSITION 5.

THEOR. 5 PROPOS. 5

第五題

If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.

SI magnitudo magnitudinis aeque fuerit multiplex, atque ablata ablatae : Etiam reliqua reliquae ita multiplex erit, ut tota totius.

大小兩幾何。此全所倍於彼全。若此全截取之分、所倍於彼全截取之分。則此全之分餘、所倍於彼全之分餘。亦如之。

For let the magnitude AB be the same multiple of the magnitude CD that the part AE subtracted is of the part CF subtracted; I say that the remainder EB is also the same multiple of the remainder FD that the whole AB is of the whole CD.

Ita multiplex fit tota AB, totius CD, ut est multiplex AE, ablata ablate CF : Dico reliquam EB, ita esse multipticem reliquae CD, ut est tota AB, totius CD.

解曰。甲乙大㡬何。丙丁小㡬何。甲乙所倍于丙丁。若甲乙之截分甲戊、所倍于丙丁之截分、丙己。題言甲戊之分餘、戊乙、所倍于丙己之分餘，己丁。亦如其數。

For, whatever multiple AE is of CF, let EB be made that multiple of CG.

Ponatur enim EB, ita multiplex cuiuspiam magnitudinis ut delicet ipsius GC, ut est AE, ipsus CF.

論曰。試作一他幾何。為庚丙。令戊乙之倍庚丙。若甲戊之倍丙己也。本卷界說增

Then, since AE is the same multiple of CF that EB is of GC, therefore AE is the same multiple of CF that AB is of GF. [V. 1] But, by the assumption, AE is the same multiple of CF that AB is of CD. Therefore AB is the same multiple of each of the magnitudes GF, CD; therefore GF is equal to CD. Let CF be subtracted from each; therefore the remainder GC is equal to the remainder FD. And, since AE is the same multiple of CF that EB is of GC, and GC is equal to DF, therefore AE is the same multiple of CF that EB is of FD. But, by hypothesis, AE is the same multiple of CF that AB is of CD; therefore EB is the same multiple of FD that AB is of CD. That is, the remainder EB will be the same multiple of the remainder FD that the whole AB is of the whole CD.

Quoniam igitur AE, EB, aeque sunt multiplices ipsarum CF, GC,¹ erit tota AB, totius GF, ita multiplex, ut AE, ipsus CF, hoc est, omnes omnium, ut una unius. Sed tam multiplex etiam ponitur AB, ipsius CD, quam est multiplex AE, ipsius CF. Igitur AB, tam est multiplex ipsius GF, quam multiplex est ipsius CD;² atque idcirco aequales sunt GF, CD. Ablata igitur communi CF, aequales erunt GC, FD. Tam multiplex igitur erit EB, ipsius FD, quam multiplex est ipsius GC. Sed ita multiplex posita fuit EB, ipsius GC, ut AE, ipsius CF, hoc est, ut tota AB, totius CD. Quare tam multiplex est reliqua EB, reliquae FD, quam est tota AB, totius CD: quod est propositum.

1. 1. quinti. 2. 6. pron.

甲戊、戊乙、之倍丙己、庚丙。其數等。卽其兩幷、甲乙之倍庚己。亦若甲戊之倍丙己也。本篇一而甲乙之倍丙丁。元若甲戊之倍丙己。則丙丁與庚己等也。次每減同用之丙己。卽庚丙與己丁、亦等。而戊乙之倍己丁。亦若戊乙之倍庚丙矣。夫戊乙之倍庚丙。旣若甲戊之倍丙己。則戊乙、為甲戊之分餘。所倍於己丁、為丙己之分餘者。亦若甲乙之倍丙丁也。

Therefore etc. Q. E. D.

ALITER. Sit ita multiplex tota AB, totius CD, ut ablata AE, ablatae CF. Dico reliquam EB, reliquae FD, esse sic multiplicem, ut est tota totius. Posita enim GA, ita multiplici ipsius FD, ut est AE, ipsius CF, vel ut tota AB, totius CD: quoniam AE, GA, aeque multiplices sunt ipsarum CE, FD,³ erit tota GE, sic multiplex totius CD, ut AE, ipsius CF: Sed ita quoque multiplex est AB, eiusdem CD, ut AE, ipsius C F, ex hypothesi. Aeque multiplices sunt igitur GE, AB, ipsius CD;⁴ atque adeo inter se aequales. Quare dempta communi AE, aequalet erunt GA, EB: Ideoque aequemultiplices ipsius FD; cum GA, sit multiplex posita ipsius FD: Atqui ita est multiplex posita GA, ipsius FD, ut AB, ipsius CD. Igitur et EB, reliqua sic erit multiplex ipsius FD, reliquae, ut AB, tota totius CD; quod est propositum. Si magnitudo itaque magnitudinis aeque fuerit multiplex, etc. Quod erat demonstrandum.

3. 1. quinti. 4. 6. pron.

又論曰。試作一他幾何、為庚甲。令庚甲之倍己丁。若甲戊之倍丙己。本卷界說二十卽其兩幷、庚戊之倍丙丁。亦若甲戊之倍丙己也。本篇一而甲乙之倍丙丁。元若甲戊之倍丙己。是庚戊與甲乙等矣。次每減同用之甲戊。卽庚甲與戊乙等也。而庚甲之倍己丁。若甲乙之倍丙丁也。則戊乙之倍己丁。亦若甲乙之倍丙丁也。

PROPOSITION 6.

第六題

If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them. mA-nA (n<m) is the same multiple of A that mB-nB is of B.

兩幾何名倍于彼兩幾何其數等于此兩幾何每滅一分其一分之各倍于所當彼幾何其數等則其餘或各與彼幾何等。或尚各倍於彼幾何。其數亦等。

For let two magnitudes AB, CD be equimultiples of two magnitudes E, F, and let AG, CH subtracted from them be equimultiples of the same two E, F; I say that the remainders also, GB, HD, are either equal to E, F or equimultiples of them.

For, first, let GB be equal to E; I say that HD is also equal to F.

For let CK be made equal to F. Since AG is the same multiple of E that CH is of F, while GB is equal to E and KC to F, therefore AB is the same multiple of E that KH is of F. [V. 2] But, by hypothesis, AB is the same multiple of E that CD is of F; therefore KH is the same multiple of F that CD is of F. Since then each of the magnitudes KH, CD is the same multiple of F, therefore KH is equal to CD. Let CH be subtracted from each; therefore the remainder KC is equal to the remainder HD. But F is equal to KC; therefore HD is also equal to F. Hence, if GB is equal to E, HD is also equal to F.

Similarly we can prove that, even if GB be a multiple of E, HD is also the same multiple of F.

Therefore etc. Q. E. D.

PROPOSITION 7.

第七題二支

Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.

此兩幾何等。則與彼幾何各為比例、必等。而彼幾何、與此相等之兩幾何。各為比例、亦等。

Let A, B be equal magnitudes and C any other, chance, magnitude; I say that each of the magnitudes A, B has the same ratio to C, and C has the same ratio to each of the magnitudes A, B.

For let equimultiples D, E of A, B be taken, and of C another, chance, multiple F.

Then, since D is the same multiple of A that E is of B, while A is equal to B, therefore D is equal to E. But F is another, chance, magnitude. If therefore D is in excess of F, E is also in excess of F, if equal to it, equal; and, if less, less. And D, E are equimultiples of A, B, while F is another, chance, multiple of C; therefore, as A is to C, so is B to C. [V. Def. 5]

I say next that C also has the same ratio to each of the magnitudes A, B.

For, with the same construction, we can prove similarly that D is equal to E; and F is some other magnitude. If therefore F is in excess of D, it is also in excess of E, if equal, equal; and, if less, less. And F is a multiple of C, while D, E are other, chance, equimultiples of A, B; therefore, as C is to A, so is C to B. [V. Def. 5]

Therefore etc.

PORISM.
From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely. Q. E. D.

後論與本篇第四題之系。同用反理。如甲與丙。若乙與丙。反推之。丙與甲。亦若丙與乙也。

PROPOSITION 8.

第八題

Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.

大小兩幾何。各與他幾何為比例。則大與他之比例。大於小與他之比例。而他與小之比例。大於他與大之比例

Let AB, C be unequal magnitudes, and let AB be greater; let D be another, chance, magnitude; I say that AB has to D a greater ratio than C has to D, and D has to C a greater ratio than it has to AB.

For, since AB is greater than C, let BE be made equal to C; then the less of the magnitudes AE, EB, if multiplied, will sometime be greater than D. [V. Def. 4]

[Case I.] First, let AE be less than EB; let AE be multiplied, and let FG be a multiple of it which is greater than D; then, whatever multiple FG is of AE, let GH be made the same multiple of EB and K of C; and let L be taken double of D, M triple of it, and successive multiples increasing by one, until what is taken is a multiple of D and the first that is greater than K. Let it be taken, and let it be N which is quadruple of D and the first multiple of it that is greather than K.

Then, since K is less than N first, therefore K is not less than M. And, since FG is the same multiple of AE that GH is of EB, therefore FG is the same multiple of AE that FH is of AB. [V. 1] But FG is the same multiple of AE that K is of C; therefore FH is the same multiple of AB that K is of C; therefore FH, K are equimultiples of AB, C. Again, since GH is the same multiple of EB that K is of C, and EB is equal to C, therefore GH is equal to K. But K is not less than M; therefore neither is GH less than M. And FG is greater than D; therefore the whole FH is greater than D, M together. But D, M together are equal to N, inasmuch as M is triple of D, and M, D together are quadruple of D, while N is also quadruple of D; whence M, D together are equal to N. But FH is greater than M, D; therefore FH is in excess of N, while K is not in excess of N. And FH, K are equimultiples of AB, C, while N is another, chance, multiple of D; therefore AB has to D a greater ratio than C has to D. [V. Def. 7]

I say next, that D also has to C a greater ratio than D has to AB.

For, with the same construction, we can prove similarly that N is in excess of K, while N is not in excess of FH. And N is a multiple of D, while FH, K are other, chance, equimultiples of AB, C; therefore D has to C a greater ratio than D has to AB. [V. Def. 7]

[Case 2.] Again, let AE be greater than EB. Then the less, EB, if multiplied, will sometime be greater than D. [V. Def. 4] Let it be multiplied, and let GH be a multiple of EB and greater than D; and, whatever multiple GH is of EB, let FG be made the same multiple of AE, and K of C. Then we can prove similarly that FH, K are equimultiples of AB, C; and, similarly, let N be taken a multiple of D but the first that is greater than FG, so that FG is again not less than M. But GH is greater than D; therefore the whole FH is in excess of D, M, that is, of N. Now K is not in excess of N, inasmuch as FG also, which is greater than GH, that is, than K, is not in excess of N. And in the same manner, by following the above argument, we complete the demonstration.

Therefore etc. Q. E. D.

PROPOSITION 9.

第九題二支

Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the same ratio are equal.

兩幾何、與一幾何。各為比例、而等。則兩幾何必等。一幾何、與兩幾何。各為比例、而等。則兩幾何亦等。
先解曰。甲、乙、兩幾何。各與丙為比例、等。題言甲與乙等。

For let each of the magnitudes A, B have the same ratio to C; I say that A is equal to B.

For, otherwise, each of the magnitudes A, B would not have had the same ratio to C; [V. 8] but it has; therefore A is equal to B.

Again, let C have the same ratio to each of the magnitudes A, B; I say that A is equal to B.

For, otherwise, C would not have had the same ratio to each of the magnitudes A, B; [V. 8] but it has; therefore A is equal to B.

Therefore etc. Q. E. D.

PROPOSITION 10.

第十題二支

Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.

彼此兩幾何。此幾何、與他幾何之比例。大於彼與他之比例。則此幾何、大於彼。他幾何、與彼幾何之比例。大於他與此之比例。則彼幾何、小於此。

For let A have to C a greater ratio than B has to C; I say that A is greater than B.

For, if not, A is either equal to B or less. Now A is not equal to B; for in that case each of the magnitudes A, B would have had the same ratio to C; [V. 7] but they have not; therefore A is not equal to B. Nor again is A less than B; for in that case A would have had to C a less ratio than B has to C; [V. 8] but it has not; therefore A is not less than B. But it was proved not to be equal either; therefore A is greater than B.

Again, let C have to B a greater ratio than C has to A; I say that B is less than A.

For, if not, it is either equal or greater. Now B is not equal to A; for in that case C would have had the same ratio to each of the magnitudes A, B; [V. 7] but it has not; therefore A is not equal to B. Nor again is B greater than A; for in that case C would have had to B a less ratio than it has to A; [V. 8] but it has not; therefore B is not greater than A. But it was proved that it is not equal either; therefore B is less than A.

Therefore etc. Q. E. D.

PROPOSITION 11.

第十一題

Ratios which are the same with the same ratio are also the same with one another.

此兩幾何之比例。與他兩幾何之比例、等。而彼兩幾何之比例。與他兩幾何之比例、亦等。則彼兩幾何之比例。與此兩幾何之比例、亦等。
解曰。甲乙偕丙丁之比例。各與戊己之比例等。題言甲乙與丙丁之比例、亦等。

For, as A is to B, so let C be to D, and, as C is to D, so let E be to F; I say that, as A is to B, so is E to F.

For of A, C, E let equimultiples G, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.

Then since, as A is to B, so is C to D, and of A, C equimultiples G, H have been taken, and of B, D other, chance, equimultiples L, M, therefore, if G is in excess of L, H is also in excess of M, if equal, equal, and if less, less. Again, since, as C is to D, so is E to F, and of C, E equimultiples H, K have been taken, and of D, F other, chance, equimultiples M, N, therefore, if H is in excess of M, K is also in excess of N, if equal, equal, and if less, less. But we saw that, if H was in excess of M, G was also in excess of L; if equal, equal; and if less, less; so that, in addition, if G is in excess of L, K is also in excess of N, if equal, equal, and if less, less. And G, K are equimultiples of A, E, while L, N are other, chance, equimultiples of B, F; therefore, as A is to B, so is E to F.

Therefore etc. Q. E. D.

PROPOSITION 12.

第十二題

If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. If A:a=B:b=C:c etc., each ratio is equal to the ratio (A+B+C...):(a+b+c...)

數幾何。所為比例皆等。則幷前率、與幷後率、之比例。若各前率、與各後率、之比例。

Let any number of magnitudes A, B, C, D, E, F be proportional, so that, as A is to B, so is C to D and E to F; I say that, as A is to B, so are A, C, E to B, D, F.

For of A, C, E let equimultiples G, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.

Then since, as A is to B, so is C to D, and E to F, and of A, C, E equimultiples G, H, K have been taken, and of B, D, F other, chance, equimultiples L, M, N, therefore, if G is in excess of L, H is also in excess of M, and K of N, if equal, equal, and if less, less; so that, in addition, if G is in excess of L, then G, H, K are in excess of L, M, N, if equal, equal, and if less, less. Now G and G, H, K are equimultiples of A and A, C, E, since, if any number of magnitudes whatever are respectively equimultiples of any magnitudes equal in multitude, whatever multiple one of the magnitudes is of one, that multiple also will all be of all. [V. 1] For the same reason L and L, M, N are also equimultiples of B and B, D, F; therefore, as A is to B, so are A, C, E to B, D, F. [V. Def. 5]

Therefore etc. Q. E. D.

PROPOSITION 13.

第十三題

If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth. If a:b=c:d and c:d>e:f, then a:b>e:f.

數幾何。第一與二之比例。若第三與四之比例。而第三與四之比例。大於第五與六之比例。則第一與二之比例亦大於第五與六之比例。

For let a first magnitude A have to a second B the same ratio as a third C has to a fourth D, and let the third C have to the fourth D a greater ratio than a fifth E has to a sixth F; I say that the first A will also have to the second B a greater ratio than the fifth E to the sixth F.

For, since there are some equimultiples of C, E, and of D, F other, chance, equimultiples, such that the multiple of C is in excess of the multiple of D, while the multiple of E is not in excess of the multiple of F, [V. Def. 7] let them be taken, and let G, H be equimultiples of C, E, and K, L other, chance, equimultiples of D, F, so that G is in excess of K, but H is not in excess of L; and, whatever multiple G is of C, let M be also that multiple of A, and, whatever multiple K is of D, let N be also that multiple of B.

Now, since, as A is to B, so is C to D, and of A, C equimultiples M, G have been taken, and of B, D other, chance, equimultiples N, K, therefore, if M is in excess of N, G is also in excess of K, if equal, equal, and if less, less. [V. Def. 5] But G is in excess of K; therefore M is also in excess of N. But H is not in excess of L; and M, H are equimultiples of A, E, and N, L other, chance, equimultiples of B, F; therefore A has to B a greater ratio than E has to F. [V. Def. 7]

Therefore etc. Q. E. D.

PROPOSITION 14.

第十四題

If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less. If a:b=c:d, then, accordingly as a>=<b, also c>=<d.

四幾何。第一與二之比例。若第三與四之比例。而第一幾何大於第三。則第二幾何亦大於第四。第一或等、或小、於第三。則第二亦等、亦小、於第四。

For let a first magnitude A have the same ratio to a second B as a third C has to a fourth D; and let A be greater than C; I say that B is also greater than D.

For, since A is greater than C, and B is another, chance, magnitude, therefore A has to B a greater ratio than C has to B. [V. 8] But, as A is to B, so is C to D; therefore C has also to D a greater ratio than C has to B. [V. 13] But that to which the same has a greater ratio is less; [V. 10] therefore D is less than B; so that B is greater than D.

Similarly we can prove that, if A be equal to C, B will also be equal to D; and, if A be less than C, B will also be less than D.

Therefore etc. Q. E. D.

PROPOSITION 15.

Parts have the same ratio as the same multiples of them taken in corresponding order. a:b=ma:mb.

兩分之比例。與兩多分幷之比例、等。

For let AB be the same multiple of C that DE is of F; I say that, as C is to F, so is AB to DE.

For, since AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, so many are there also in DE equal to F. Let AB be divided into the magnitudes AG, GH, HB equal to C, and DE into the magnitudes DK, KL, LE equal to F; then the multitude of the magnitudes AG, GH, HB will be equal to the multitude of the magnitudes DK, KL, LE. And, since AG, GH, HB are equal to one another, and DK, KL, LE are also equal to one another, therefore, as AG is to DK, so is GH to KL, and HB to LE. [V. 7] Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [V. 12] therefore, as AG is to DK, so is AB to DE. But AG is equal to C and DK to F; therefore, as C is to F, so is AB to DE.

Therefore etc. Q. E. D.

PROPOSITION 16.

THEOR. 16. PROPOS. 16.

第十六題更理

If four magnitudes be proportional, they will also be proportional alternately. If a:b=c:d, then a:c=b:d.

SI Quatuor magnitudines proportionales fuerint, et vicissim proportionales erunt.

四幾何、為兩比例、等。卽更推前與前、後與後為比例。亦等。

Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D; I say that they will also be so alternately, that is, as A is to C, so is B to D.

HIC demonstratur Alterna, sive Permutata proportio, seu ratio, quae definitione 12. explicata est. Sit enim A, ad B, ut C, ad D. Dico vicissim, seu permutando, esse quoque A, ad C, ut B, ad D.

解曰。甲、乙、丙、丁、四幾何。甲與乙之比例。若丙與丁。題言更推之。甲與丙之比例。亦若乙與丁。

For of A, B let equimultiples E, F be taken, and of C, D other, chance, equimultiples G, H.

Sumantur enim ipsarum A, B, primae ac secundae, aequemultiplices E, F; Item ipsarum C, D, tertiae et quartae aequemultiplices G, H;

論曰。試以甲與乙。同任倍之為戊、為己。別以丙與丁。同任倍之為庚、為辛。

Then, since E is the same multiple of A that F is of B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is E to F. But as A is to B, so is C to D; therefore also, as C is to D, so is E to F. [V. 11] Again, since G, H are equimultiples of C, D, therefore, as C is to D, so is G to H. [V. 15] But, as C is to D, so is E to F; therefore also, as E is to F, so is G to H. [V. 11] But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less. [V. 14] Therefore, if E is in excess of G, F is also in excess of H, if equal, equal, and if less, less. Now E, F are equimultiples of A, B, and G, H other, chance, equimultiples of C, D; therefore, as A is to C, so is B to D. [V. Def. 5]

⁵ eritque E, ad F, ut A, ad B; cum E, et F, sint pariter multiplices partium A, et B. Eadem ratione erit G, ad H, ut C, ad D. Cum igitur proportiones E, ad F, et C, ad D, sint eaedem proportioni A, ad B; ⁶ erunt et ipsae inter se eaedem. Rursus quia proportiones E, ad F, et G, ad H, eaedem sunt proportioni C, ad D; ⁷ erunt et ipsae eaedem inter se; hoc est, ut est E, prima ad F, secundam, ita erit G, tertia ad H, quartam, ⁸ Quare si E, prima maior est quam G, tertia, vel aequalis, vel minor, erit quoque F, secunda maior quam H, quarta, vel aequalis, vel minor, in quacunque multiplicatione accepta sint aeque multiplicia E, F, et aeque multiplicia G, H. ⁹ Est igitur A, prima ad C, secundam, ut B, tertia ad D, quartam (cum E, et F, sint aeque multiplices primae A, ac tertiae B; At G, et H, aeque multiplices C, secundae, et D, quartae, et illae ab his una deficiant, vel una aequales sint, vel una excedant, etc.) quod est propositum.

5. 15. quinti. 6. 11. quinti. 7. 11. quinti. 8. 14. quinti. 9. 6. defin. quinti.

卽戊與己。若甲與乙也。本篇十五庚與辛。若丙與丁也。夫甲與乙。若丙與丁。而戊與己。亦若甲與乙。卽戊與己。亦若丙與丁矣。依顯庚與辛。若丙與丁。卽戊與己。亦若庚與辛也。本篇十一次三試之。若戊大於庚則己亦大於辛也。若等、亦等。若小、亦小。任作幾許倍。恆如是也。本篇十四則倍一甲之戊。倍三乙之己。與倍二丙之庚。倍四丁之辛。其等、大、小、必同類也。而甲與丙。若乙與丁矣。

Therefore etc. Q. E. D.

Si quatuorigitur magnitudines proportionales fuerint, et vicissim proportionales erunt. Quod ostendendum erat.

PROPOSITION 17.

第十七題　分理

If magnitudes be proportional componendo, they will also be proportional separando. If a:b=c:d, then (a-b):b=(c-d):d.

相合之兩幾何、為比例等。則分之為比例、亦等。

Let AB, BE, CD, DF be magnitudes proportional componendo, so that, as AB is to BE, so is CD to DF; I say that they will also be proportional separando, that is, as AE is to EB, so is CF to DF.

For of AE, EB, CF, FD let equimultiples GH, HK, LM, MN be taken, and of EB, FD other, chance, equimultiples, KO, NP.

Then, since GH is the same multiple of AE that HK is of EB, therefore GH is the same multiple of AE that GK is of AB. [V. 1] But GH is the same multiple of AE that LM is of CF; therefore GK is the same multiple of AB that LM is of CF. Again, since LM is the same multiple of CF that MN is of FD, therefore LM is the same multiple of CF that LN is of CD. [V. 1] But LM was the same multiple of CF that GK is of AB; therefore GK is the same multiple of AB that LN is of CD. Therefore GK, LN are equimultiples of AB, CD. Again, since HK is the same multiple of EB that MN is of FD, and KO is also the same multiple of EB that NP is of FD, therefore the sum HO is also the same multiple of EB that MP is of FD. [V. 2] And, since, as AB is to BE, so is CD to DF, and of AB, CD equimultiples GK, LN have been taken, and of EB, FD equimultiples HO, MP, therefore, if GK is in excess of HO, LN is also in excess of MP, if equal, equal, and if less, less. Let GK be in excess of HO; then, if HK be subtracted from each, GH is also in excess of KO. But we saw that, if GK was in excess of HO, LN was also in excess of MP; therefore LN is also in excess of MP, and, if MN be subtracted from each, LM is also in excess of NP; so that, if GH is in excess of KO, LM is also in excess of NP. Similarly we can prove that, if GH be equal to KO, LM will also be equal to NP, and if less, less. And GH, LM are equimultiples of AE, CF, while KO, NP are other, chance, equimultiples of EB, FD; therefore, as AE is to EB, so is CF to FD.

Therefore etc. Q. E. D.

PROPOSITION 18.

第十八題　合理

If magnitudes be proportional separando, they will also be proportional componendo. If a:b=c:d, then (a+b):b=(c+d):d

兩幾何。分之為比例、等。則合之為比例、亦等。

Let AE, EB, CF, FD be magnitudes proportional separando, so that, as AE is to EB, so is CF to FD; I say that they will also be proportional componendo, that is, as AB is to BE, so is CD to FD.

For, if CD be not to DF as AB to BE, then, as AB is to BE, so will CD be either to some magnitude less than DF or to a greater.

First, let it be in that ratio to a less magnitude DG. Then, since, as AB is to BE, so is CD to DG, they are magnitudes proportional componendo; so that they will also be proportional separando. [V. 17] Therefore, as AE is to EB, so is CG to GD. But also, by hypothesis, as AE is to EB, so is CF to FD. Therefore also, as CG is to GD, so is CF to FD. [V. 11] But the first CG is greater than the third CF; therefore the second GD is also greater than the fourth FD. [V. 14] But it is also less: which is impossible. Therefore, as AB is to BE, so is not CD to a less magnitude than FD. Similarly we can prove that neither is it in that ratio to a greater; it is therefore in that ratio to FD itself.

Therefore etc. Q. E. D.

PROPOSITION 19

第十九題其系為轉理

If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole. If a:b=c:d (c<a and d<b), then (a-c)=(b-d).

兩幾何。各截取一分。其所截取之比例。與兩全之比例等。則分餘之比例。與兩全之比例亦等。

For, as the whole AB is to the whole CD, so let the part AE subtracted be to the part CF subtracted; I say that the remainder EB will also be to the remainder FD as the whole AB to the whole CD.

For since, as AB is to CD, so is AE to CF, alternately also, as BA is to AE, so is DC to CF. [V. 16] And, since the magnitudes are proportional componendo, they will also be proportional separando, [V. 17] that is, as BE is to EA, so is DF to CF, and, alternately, as BE is to DF, so is EA to FC. [V. 16] But, as AE is to CF, so by hypothesis is the whole AB to the whole CD. Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD. [V. 11]

Therefore etc.

.

[PORISM.
From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.] Q. E. D.

PROPOSITION 20

第二十題　三支

If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; and, if less, less. If a:b=d:e and b:c=e:f, then, accordingly as a>=<c, also d>=<f.

有三幾何。又有三幾何。相為連比例。而第一幾何大於第三。則第四亦大於第六。第一或等、或小於第三。則第四亦等、亦小、於第六。

Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, so that, as A is to B, so is D to E, and as B is to C, so is E to F; and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and, if less, less.

For, since A is greater than C, and B is some other magnitude, and the greater has to the same a greater ratio than the less has, [V. 8] therefore A has to B a greater ratio than C has to B. But, as A is to B, so is D to E, and, as C is to B, inversely, so is F to E; therefore D has also to E a greater ratio than F has to E. [V. 13] But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater; [V. 10] therefore D is greater than F. Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.

Therefore etc. Q. E. D.

PROPOSITION 21.

第二十一題　三支

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less. If a:b=e:f and b:c=d:e, then, accordingly as a>=<c, also d>=<f.

有三幾何。又有三幾何。相為連比例而錯。以平理推之。若第一幾何大於第三。則第四亦大於第六。若第一或等、或小、於第三。則第四亦等、亦小、於第六。

Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that, as A is to B, so is E to F, and, as B is to C, so is D to E, and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and if less, less.

For, since A is greater than C, and B is some other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8] But, as A is to B, so is E to F, and, as C is to B, inversely, so is E to D. Therefore also E has to F a greater ratio than E has to D. [V. 13] But that to which the same has a greater ratio is less; [V. 10] therefore F is less than D; therefore D is greater than F. Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.

Therefore etc. Q. E. D.

PROPOSITION 22.

第二十二題平理之序

If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali. If a:b=d:e and b:c=e:f then a:c=d:f.

有若干幾何。又有若干幾何。其數等。相為連比例。則以平理推。

Let there be any number of magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two together are in the same ratio, so that, as A is to B, so is D to E, and, as B is to C, so is E to F; I say that they will also be in the same ratio ex aequali, .

For of A, D let equimultiples G, H be taken, and of B, E other, chance, equimultiples K, L; and, further, of C, F other, chance, equimultiples M, N.

Then, since, as A is to B, so is D to E, and of A, D equimultiples G, H have been taken, and of B, E other, chance, equimultiples K, L, therefore, as G is to K, so is H to L. [V. 4] For the same reason also, as K is to M, so is L to N. Since, then, there are three magnitudes G, K, M, and others H, L, N equal to them in multitude, which taken two and two together are in the same ratio, therefore, ex aequali, if G is in excess of M, H is also in excess of N; if equal, equal; and if less, less. [V. 20] And G, H are equimultiples of A, D, and M, N other, chance, equimultiples of C, F. Therefore, as A is to C, so is D to F. [V. Def. 5]

Therefore etc. Q. E. D.

PROPOSITION 23.

第二十三題　平理之錯

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, they will also be in the same ratio ex aequali.

若干幾何。又若干幾何。相為連比例而錯。亦以平理推。

Let there be three magnitudes A, B, C, and others equal to them in multitude, which, taken two and two together, are in the same proportion, namely D, E, F; and let the proportion of them be perturbed, so that, as A is to B, so is E to F, and, as B is to C, so is D to E; I say that, as A is to C, so is D to F.

Of A, B, D let equimultiples G, H, K be taken, and of C, E, F other, chance, equimultiples L, M, N.

Then, since G, H are equimultiples of A, B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is G to H. For the same reason also, as E is to F, so is M to N. And, as A is to B, so is E to F; therefore also, as G is to H, so is M to N. [V. 11] Next, since, as B is to C, so is D to E, alternately, also, as B is to D, so is C to E. [V. 16] And, since H, K are equimultiples of B, D, and parts have the same ratio as their equimultiples, therefore, as B is to D, so is H to K. [V. 15] But, as B is to D, so is C to E; therefore also, as H is to K, so is C to E. [V. 11] Again, since L, M are equimultiples of C, E, therefore, as C is to E, so is L to M. [V. 15] But, as C is to E, so is H to K; therefore also, as H is to K, so is L to M, [V. 11] and, alternately, as H is to L, so is K to M. [V. 16] But it was also proved that, as G is to H, so is M to N. Since, then, there are three magnitudes G, H, L, and others equal to them in multitude K, M, N, which taken two and two together are in the same ratio, and the proportion of them is perturbed, therefore, ex aequali, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less. [V. 21] And G, K are equimultiples of A, D, and L, N of C, F. Therefore, as A is to C, so is D to F.

Therefore etc. Q. E. D.

PROPOSITION 24.

第二十四題

If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth. If a:c=d:f and b:c=e:f then (a+b):c=(d+e):f.

凡第一與二幾何之比例。若第三與四幾何之比例。而第五與二之比例。若第六與四。則第一第五幷。與二之比例。若第三第六幷。與四。

Let a first magnitude AB have to a second C the same ratio as a third DE has to a fourth F; and let also a fifth BG have to the second C the same ratio as a sixth EH has to the fourth F; I say that the first and fifth added together, AG, will have to the second C the same ratio as the third and sixth, DH, has to the fourth F.

For since, as BG is to C, so is EH to F, inversely, as C is to BG, so is F to EH. Since, then, as AB is to C, so is DE to F, and, as C is to BG, so is F to EH, therefore, ex aequali, as AB is to BG, so is DE to EH. [V. 22] And, since the magnitudes are proportional separando, they will also be proportional componendo; [V. 18] therefore, as AG is to GB, so is DH to HE. But also, as BG is to C, so is EH to F; therefore, ex aequali, as AG is to C, so is DH to F. [V. 22]

Therefore etc. Q. E. D.

PROPOSITION 25.

第二十五題

If four magnitudes be proportional, the greatest and the least are greater than the remaining two. If a:b=c:d and a is the greatest and d is the least, then a+d>b+c.

四幾何、為斷比例。則最大與最小兩幾何幷。大於餘兩幾何幷。

Let the four magnitudes AB, CD, E, F be proportional so that, as AB is to CD, so is E to F, and let AB be the greatest of them and F the least; I say that AB, F are greater than CD, E.

For let AG be made equal to E, and CH equal to F.

Since, as AB is to CD, so is E to F, and E is equal to AG, and F to CH, therefore, as AB is to CD, so is AG to CH. And since, as the whole AB is to the whole CD, so is the part AG subtracted to the part CH subtracted, the remainder GB will also be to the remainder HD as the whole AB is to the whole CD. [V. 19] But AB is greater than CD; therefore GB is also greater than HD. And, since AG is equal to E, and CH to F, therefore AG, F are equal to CH, E. And if, GB, HD being unequal, and GB greater, AG, F be added to GB and CH, E be added to HD, it follows that AB, F are greater than CD, E.

Therefore etc. Q. E. F. Proposition 26. If the ratio between a first and a second magnitude is larger than that between a third and a fourth, then, invertendo, the ratio between the second and the first is smaller than that between the fourth and the third. If a:b>c:d then b:a<d:c.² Proposition 27. If the ratio between a first and a second [quantity] is greater thatn that between a third and a fourth, then, alternando, the ratio of the first and the third is also greater than that of the second and the fourth. If a:b>c:d then a:c>b:d.³ If the ratio bewteen a first and a second [quantity] is greater than that between a third and a fourth, then, componendo, the ration of the first and the second together to the second will also be greater than that of the thrid and the fourth together to the fourth. If a:b>c:d then (a+b):b>(c+d):d.⁴ Proposition 29. If the ratiio of a first and second [quantity] together to the second is greater than that of a third and a fouth, then the ratio of the first and the second is also greater than that of the third and the fourth. If (a+b):b>(c+d):d then a:b>c:d.⁵ Proposition 30. If the ratio of a first and a second [quantity] together to the second is greater than that of a third and a fourth together to the fourth, then, invertendo, the ratio of the first and the second together to the first is lesser than that of the third and the fourth together to the third. If (a+b):b>(c+d):d then (a+b):a<(c+d):c.⁶ Proposition 31. If there be these three quantities, and those three quantities, such as the ratio of the first and the second of these is greater than that between the first and the second of those, while the ratio of the second and the third of these is greater than that of the second and third of those, if arranged in order. then ex equali the ratio between the first and the third of these will also be greater than the ratio between the first and the third of those. If a:b>d:e and b:c>e:f then a:c>d:f.⁷ Proposition 32. If a:b>e:f and b:c>d:e then a:c>d:f. Proposition 33. If a:b>c:d then (a-c):(b-d)>a:b. If A:a, B:b,...K:k then (A+B...+K)>(a+b+...+k):k and also (A+B+...+K):K>(B+...+K):(b+...+k) but (A+B+...+K):K<A:a.

2. Translation of the Chinese by Engelfriet, p. 257. 3. Translation of the Chinese by Engelfriet, p. 257. 4. Translation of the Chinese by Engelfriet, p. 257. 5. Translation of the Chinese by Engelfriet, p. 258. 6. Translation of the Chinese by Engelfriet, p. 258. 7. Translation of the Chinese by Engelfriet, p. 258.

第二十六題
第一與二幾何之比例。大於第三與四之比例。反之。則第二與一之比例。小於第四與三之比例。
解曰。一甲與二乙之比例。大於三丙與四丁。題言反之。二乙與一甲之比例。小於四丁與三丙。(p. 二七○)
論曰。試作戊與乙之比例。若丙與丁。卽甲與乙之比例。大於戊與乙。而甲幾何大於戊。本篇　\\　十則乙與戊之比例。大於乙與甲也。本篇　\\　八反之。則乙與戊之比例。若丁與丙本篇　\\　四而乙與甲之比例。小於丁與丙。第二十七題
第一與二之比例。大於第三與四之比例。更之。則第一與三之比例。亦大於第二與四之比例。
解曰。一甲與二乙之比例。大於三丙與四丁。題言更之。則一甲與三丙之比例。亦大於二乙與四丁。
論曰。試作戊與乙之比例。若丙與丁。卽甲與乙之比例。大於戊與乙。而甲幾何大於戊。本篇　\\　十則甲與丙之比例。大於戊於丙也。本篇　\\　八夫戊與乙之比例。旣若丙與丁。更之。則戊與丙之比例。亦若乙與丁。本篇　\\　十六而甲與丙之比例。大於乙與丁矣。第二十八題
第一與二之比例。大於第三與四之比例。合之。則第一、第二、幷、與二之比例。亦大於第三、第四、幷、與四之比例。
解曰。一甲乙與二乙丙之比例。大於三丁戊與四戊己。題言合之。則甲丙與乙丙之比例。亦大於丁己與戊己。(p. 二七一)
論曰。試作庚乙與乙丙之比例。若丁戊與戊己。卽甲乙與乙丙之比例。大於庚乙與乙丙。而甲乙幾何大於庚乙矣。本篇　\\　十此二率者。每加一乙丙。卽甲丙亦大於庚丙。而甲丙與乙丙之比例。大於庚丙與乙丙也。本篇　\\　八夫庚乙與乙丙之比例。旣若丁戊與戊己。合之。則庚丙與乙丙之比例。亦若丁己與戊己也。本篇　\\　十八而甲丙與乙丙之比例。大於丁己與戊己矣。第二十九題
第一合第二、與二之比例。大於第三合第四、與四之比例。分之。則第一與二之比例。亦大於第三與四之比例。
解曰。甲丙與乙丙之比例。大於丁己與戊己。題言分之。則甲乙與乙丙之比例。亦大於丁戊與戊己。
論曰。試作庚丙與乙丙之比例。若丁己與戊己。卽甲丙與乙丙之比例。亦大於庚丙與乙丙。而甲丙幾何大於庚丙矣。本篇　\\　十此二率者。每減一同用之乙丙。卽甲乙亦大與庚乙。而甲乙與乙丙之比例。大於庚乙與乙丙也。本篇　\\　八夫庚丙與乙丙之比例。旣若丁己與戊己。分之。則庚乙與乙丙之比例。亦若丁戊與戊己也。本篇　\\　十七而甲乙與乙丙之比例。大於丁戊與戊己矣。第三十題
(p. 二七二)第一合第二、與二之比例。大於第三合第四、與四之比例。轉之。則第一合第二、與一之比例。小於第三合第四、與三之比例。
解曰。甲丙與乙丙之比例。大於丁己與戊己。題言轉之。則甲丙與甲乙之比例。小於丁己與丁戊。
論曰。甲丙與乙丙之比例。旣大於丁己與戊己。分之。卽甲乙與乙丙之比例。亦大於丁戊與戊己也。本篇　\\　廿九又反之。乙丙與甲乙之比例。小於戊己與丁戊矣。本　\\　篇廿　\\　六又合之。甲丙與甲乙之比例。亦小於丁己與丁戊也。本篇　\\　廿八第三十一題
此三幾何。彼三幾何。此第一與二之比例。大於彼第一與二之比例。此第二與三之比例。大於彼第二與三之比例。如是序者。以平理推。則此第一與三之比例。亦大於彼第一與三之比例。
解曰。甲、乙、丙。此三幾何。丁、戊、己。彼三幾何。而甲與乙之比例。大於丁與戊。乙與丙之比例。大於戊與己。如是序者。題言以平理推。則甲與丙之比例。亦大於丁與己。
論曰。試作庚與丙之比例。若戊與己。卽乙與丙之比例。大於庚與丙。而乙幾何大於庚本篇　\\　十是甲與小庚之比例。大於甲與大乙矣。本篇　\\　八夫甲與乙之比例。元大於丁與戊。卽(p. 二七三)甲與庚之比例。更大於丁與戊也次作辛與庚之比例若丁與戊卽甲與庚之比例。亦大於辛與庚。而甲幾何大於辛。本篇　\\　十是大甲與丙之比例。大於小辛與丙矣。本篇　\\　八夫辛與丙之比例。以平理推之。若丁與己也。本篇　\\　廿二則甲與丙之比例。大於丁與己也。第三十二題
此三幾何。彼三幾何。此第一與二之比例。大於彼第二與三之比例。此第二與三之比例。大於彼第一與二之比例。如是錯者。以平理推。用此第一與三之比例。亦大於彼第一與三之比例。
解曰。甲、乙、丙。此三幾何。丁、戊、己。彼三幾何。而甲與乙之比例。大於戊與己。乙與丙之比例。大於丁與戊。如是錯者。題言以平理推。則甲與丙之比例。亦大於丁與己。
論曰。試作庚與丙之比例。若丁與戊。卽乙與丙之比例。大於庚與丙。而乙幾何大於庚。本篇　\\　十是甲與小庚之比例。大於甲與大乙矣。本篇　\\　八夫甲與乙之比例。旣大於戊與己。卽甲與庚之比例。更大於戊與己也。次作辛與庚之比例。若戊與己。卽甲與庚之比例。亦大於辛與庚。而甲幾何大於辛。本篇　\\　十是大甲與丙之比例。大於小辛與丙矣。本篇　\\　八(p. 二七四)夫辛與丙之比例。以平理推之。若丁與己也。本篇　\\　廿三則甲與丙之比例。大於丁與己也。第三十三題
此全與彼全之比例。大於此全截分、與彼全截分之比例。則此全分餘、與彼全分餘之比例。大於此全與彼全之比例。
解曰。甲乙全與丙丁全之比例。大於兩截分、甲戊與丙己。題言兩分餘、戊乙與己丁之比例。大於甲乙與丙丁。
論曰。甲乙與丙丁之比例。旣大於甲戊與丙己。更之。卽甲乙與甲戊之比例。亦大於丙丁與丙己也。本篇　\\　廿七又轉之。甲乙與戊乙之比例。小於丙丁與己丁也。本篇　\\　三十又更之。甲乙與丙丁之比例。小於戊乙與己丁也。本篇　\\　廿七戊乙與己丁。分餘也。則分餘之比例。大於甲乙全、與丙丁全矣。依顯兩全之比例。小於截分。則分餘之比例。小於兩全。第三十四題　三支
若干幾何。又有若干幾何。其數等。而此第一與彼第一之比例。大於此第二與彼第二之比例。此第二與彼第二之比例。大於此第三與彼第三之比例。以(p. 二七五)後俱如。是則此幷與彼幷之比例。大於此末與彼末之比例。亦大於此幷減第一、與彼幷減第一之比例。而小於此第一與彼第一之比例。
解曰。如甲、乙、丙、三幾何。又有丁、戊、己、三幾何。其甲與丁之比例。大於乙與戊。乙與戊之比例。大於丙與己。題先言甲乙丙幷、與丁戊己幷之比例。大於丙與己。次言亦大於乙丙幷、與戊己幷。後言小於甲與丁。
論曰。甲與丁之比例。旣大於乙與戊。更之。卽甲與乙之比例。大於丁與戊也。本篇　\\　廿七又合之。甲乙幷與乙之比例。大於丁戊幷與戊也。本篇　\\　廿八又更之。甲乙幷與丁戊幷之比例。大於乙與戊也。本篇　\\　廿七是甲乙全與丁戊全之比例。大於減幷乙與減幷戊也。旣爾。卽減餘甲與減餘丁之比例。大於甲乙全與丁戊全也。本篇　\\　卅三依顯乙與戊之比例。亦大於乙丙全與戊己全。卽甲與丁之比例。更大於乙丙全與戊己全也。又更之。甲與乙丙幷之比例。大於丁與戊己幷也。本篇　\\　廿七又合之。甲乙丙全與乙丙幷之比例。大於丁戊己全與戊己幷也。本篇　\\　廿八又更之。甲乙丙全與丁戊己全之比例。大於乙丙幷與戊己幷也。本篇　\\　廿七則得次解也。又甲乙丙全與丁戊己全之比例。旣大於減幷乙丙與減幷戊己。卽減餘甲與減餘丁之比例。大於甲乙丙全與丁戊己全也本篇　\\　卅二則得後解也。(p. 二七六)又乙與戊之比例。旣大於丙與己。更之。卽乙與丙之比例。大於戊與己也。本篇　\\　廿七又合之。乙丙全與丙之比例。大於戊己全與己也。本篇　\\　廿八又更之。乙丙幷與戊己幷之比例。大於丙與己也。本篇　\\　廿七而甲乙丙幷與丁戊己幷之比例。旣大於乙丙幷與戊己幷。卽更大於末丙與末己也則得先解也。
若兩率各有四幾何而丙與己之比例。亦大於庚與辛。卽與前論同理。蓋依上文論、乙與戊之比例。大於乙丙庚幷與戊己辛幷。卽甲與丁之比例。更大於乙丙庚幷與戊己辛幷也。更之。卽甲與乙丙庚幷之比例。大於丁與戊己辛幷也。本篇　\\　十八又合之。甲乙丙庚全與乙丙庚幷之比例。大於丁戊己辛全與戊己辛幷也。又更之。甲乙丙庚全與丁戊己辛全之比例。大於乙丙庚幷與戊己辛幷也。本篇　\\　廿七則得次解也。又甲乙丙庚全與丁戊己辛全之比例。旣大於減幷乙丙庚與減幷戊己辛。卽減餘甲與減餘丁之比例。大於甲乙丙庚全與丁戊己辛全也。本篇　\\　卅三則得後解也。又依前論、顯乙丙庚幷與戊己辛幷之比例。旣大於庚與辛。而甲乙丙庚全與丁戊己辛全之比例。大於乙丙庚幷與戊己辛幷。卽更大於末庚與末辛也。則得先解也。自五以上。至於無窮。俱@此論。可@全題之旨。

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