Bibliotheca Polyglotta

Let A be the given point, and BC the given straight line. Thus it is required to place at the point A (as an extremity) a straight line equal to the given straight line BC.

SIT punctum datum A, & data recta linea B C, cui aliam rectam æqualem ponere oportet ad punctum A. Facto alterutro extremo lineæ B C, nempe C, centro ¹ describatur circulus B E, interuallo rectæ B C.

1. 3. petit

法曰。有甲點。及乙丙線。求以甲為界。作一線。與乙丙等。先以丙為心。乙為界。乙為心丙為界亦可作。作丙乙圜。第三求

From the point A to the point B let the straight line AB be joined; [Post. 1] and on it let the equilateral triangle DAB be constructed. [I. 1] Let the straight lines AE, BF be produced in a straight line with DA, DB; [Post. 2] with centre B and distance BC let the circle CGH be described; [Post. 3] and again, with centre D and distance DG let the circle GKL be described. [Post. 3]

Et ex A, ad centrum C, ² recta ducatur A C; (nisi punctum A, intra rectam B C, fuerit: Tunc enim pro linea ducta sumetur A C, vt secunda figura indicat.) Super recta verò A C, ³ construatur triangulum æquilaterum A C D, sursum, aut deorsum versus, vt libuerit; cuius duo latera modò constituta D A, D C, versus rectam A C, ⁴ extendantur; D C, quidem oppositum puncto dato A, vsque ad circumferentiam in E; D A, vero oppositum centro C, quantumlibet in F. Clavius made this circle already in the third sentence of the proposition. Deinde centro D, interuallo vero rectæ D E, per C, centrum transeuntis, ⁵ alter circulus describatur E G, secans rectam D F, in G.

2. 1. petit. 3. 1. primi. 4. 2. pet. 5. 3. pet.

次觀甲點、若在丙乙之外。則自甲至丙。作甲丙線。第一求如上前圖。或甲在丙乙之內。則截取甲至丙一分線。如上後圖。兩法俱以甲丙線為底。任於上下作甲丁丙平邊三角形。本篇一。次自三角形兩腰線引長之第二求其丁丙、引至丙乙圜界而止。為丙戊線。其丁甲、引之出丙乙圜外、稍長。為甲己線。 As did Clavius 末以丁為心。戊為界。作丁戊圜。其甲己線、與丁戊圜、相交於庚。

Then, since the point B is the centre of the circle CGH,BC is equal to BG. Again, since the point D is the centre of the circle GKL, DL is equal to DG. And in these DA is equal to DB; therefore the remainder AL is equal to the remainder BG. [C.N. 3] But BC was also proved equal to BG; therefore each of the straight lines AL, BC is equal to BG. And things which are equal to the same thing are also equal to one another; [C.N. 1] therefore AL is also equal to BC.

Dico rectam A G, quæ posita est ad punctum datum A, æqualem esse datæ rectæ B C. Quoniam D E, D G, ductæ sunt ex centro D, ad circumferentiam E G, ⁶ ipsæ inter se æquales erunt: Ablatis igitur D A, D C, æqualibus lateribus trianguli æquilateri A C D, ⁷ remanebit A G, æqualis rectæ C E. Sed eidem C E, ⁸ æqualis est recta B C. (cùm ambæ rectæ C B, C E, cadant ex centro C, ad circumferentiam B E.) Igitur rectæ A G, B C, quandoquidem vtraque æqualis est ostensa rectæ C E, inter se ⁹ æquales erunt. See above

6. 15. def. 7. 3. pron. 8. 15. def. 9. 1. pron.

卽甲庚線、與乙丙線等。論曰。丁戊、丁庚線。同以丁為心。戊、庚、為界。故等。界說十五於丁戊線減丁丙。丁庚線減丁甲。其所減兩腰線等。則所存亦等。公論三夫丙戊、與丙乙。同以丙為心。戊、乙、為界。亦等。界說十五卽甲庚、與丙乙等。公論一。 (p. 二四） See previous Chinese sentence See above

Therefore at the given point A the straight line AL is placed equal to the given straight line BC. (Being) what it was required to do.

Ad datum igitur punctum. &c. quod erat faciendum.
QVOD si punctum datum fuerit in extremo datæ lineæ, quale est C, facilè absoluetur problema. Si enim centro C, & interuallo C B, ¹⁰ describatur circulus, ad cuius circumferentiam recta ¹¹ ducatur vtcunque C E, erit hæc posita ad punctum datum C, ¹² æqualis datæ rectæ B C, cum vtraque & B C, & C E, ex eodem centro egrediatur ad circumferentiam B E.

10. 3. petit 11. 1. pet. 12. 15. def.

若所設甲點。卽在丙乙線之一界。其法尤易。假如點在丙。卽以丙為心。作乙戊圜。從丙至戊、卽所求。

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