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Euclid: Elementa
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Proposition 3. 
PROBL. 3. PROPOS. 3. 
第三題 
Given two unequal straight lines, to cut off from the greater a straight line equal to the less. 
DVABVS datis rectis lineis inæqualibus, de maiore æqualem minori rectam lineam detrahere. 
兩直線。一長一短。求於長線、減去短線之度。 
Let AB, C be the two given unequal straight lines, and let AB be the greater of them.  Thus it is required to cut off from AB the greater a straight line equal to C the less. 
SINT duæ rectæ inæquales A, minor, & B C, maior,  oporteat que ex maiore B C, detrahere lineam æqualem minori A. 
法曰。甲短線。乙丙長線。  求於乙丙、減甲。 
At the point A let AD be placed equal to the straight line C; [I. 2]  and with centre A and distance AD let the circle DEF be described. [Post. 3] 
Ad alterutrum extremorum lineæ maioris B C, nempe ad punctum B, 1 ponatur aliqua linea, quæ sit B D, æqualis minori A.  Deinde centro B, interuallo autem B D, circulus 2 describatur secans B C, in E. 
先以甲為度。從乙引至別界。作乙丁線。本篇二  次以乙為心。丁為界。作圜。第三求圜界與乙丙、交於戊。 
Now, since the point A is the centre of the circle DEF, AE is equal to AD. [Def. 15]  But C is also equal to AD.  Therefore each of the straight lines AE, C is equal to AD;  so that AE is also equal to C. [C.N. 1] 
Dico B E, detractam esse æqualem ipsi A. Quoniam B E, 3 . æqualis est rectæ B D, & eidem B D, æqualis est recta A, per constructionem;  4 erunt A, & B E, inter se æquales.    See previous 
卽乙戊、與等甲之乙丁等。蓋乙丁、乙戊。同心、同圜故。界說十五。  See previous  See previous  See previous 
Therefore, given the two straight lines AB, C, from AB the greater AE has been cut off equal to C the less.  (Being) what it was required to do. 
Duabus igitur datis rectis, &c.  quod erat faciendum.
QVOD si duæ rectæ datæ coniungantur in vno extremo, quales sunt B D, & B C, coniunctæ in extremo vtriusque B; describendus erit circulus ex B, ad interuallum minoris B D. Hic enim auferet B E, æqualem ipsi B D, vt constat ex definitione circuli.
SCHOLIUM
VARIOS etiam posse casus esse in hoc problemate, nemo ignorat, cum duæ lineæ inæquales datæ vel inter se distent, ita vt neutra alteram contingat, vel non; sed vel coniungantur ad vnum extremum, vel se mutuo secent, vel certè altera alteram suo extremo tangat duntaxat, &c. de quare lege Proclum hoc in loco. 
  (p. 二五) 
 
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