Bibliotheca Polyglotta

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.¹ 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.² 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.³ 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.⁴ 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.⁵ 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.⁶ 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.

1. Translation of the Chinese by Engelfriet, p. 257. 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. 258. 5. Translation of the Chinese by Engelfriet, p. 258. 6. 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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