Bibliotheca Polyglotta

Let ABCD be the given rectilineal figure and E the given rectilineal angle; thus it is required to construct, in the given angle E, a parallelogram equal to the rectilineal figure ABCD.

DATVM rectilineum sit A B C, & datus angulus D: oportet igitur construere parallelogrammum æquale rectilineo A B C, quod habeat angulum æqualem angulo D.

. .

Let DB be joined, and let the parallelogram FH be constructed equal to the triangle ABD, in the angle HKF which is equal to E; [I. 42] let the parallelogram GM equal to the triangle DBC be applied to the straight line GH, in the angle GHM which is equal to E. [I. 44] Then, since the angle E is equal to each of the angles HKF, GHM, the angle HKF is also equal to the angle GHM. [C.N. 1] Let the angle KHG be added to each; therefore the angles FKH, KHG are equal to the angles KHG, GHM. But the angles FKH, KHG are equal to two right angles; [I. 29] therefore the angles KHG, GHM are also equal to two right angles. Thus, with a straight line GH, and at the point H on it, two straight lines KH, HM not lying on the same side make the adjacent angles equal to two right angles; therefore KH is in a straight line with HM. [I. 14] And, since the straight line HG falls upon the parallels KM, FG, the alternate angles MHG, HGF are equal to one another. [I. 29] Let the angle HGL be added to each; therefore the angles MHG, HGL are equal to the angles HGF, HGL. [C.N. 2] But the angles MHG, HGL are equal to two right angles; [I. 29] therefore the angles HGF, HGL are also equal to two right angles. [C.N. 1] Therefore FG is in a straight line with GL. [I. 14] And, since FK is equal and parallel to HG, [I. 34] and HG to ML also, KF is also equal and parallel to ML; [C.N. 1; I. 30] and the straight lines KM, FL join them (at their extremities); therefore KM, FL are also equal and parallel. [I. 33] Therefore KFLM is a parallelogram. And, since the triangle ABD is equal to the parallelogram FH, and DBC to GM, the whole rectilineal figure ABCD is equal to the whole parallelogram KFLM.

Resoluat rectilineum in triangula A, B, & C. Deinde triangulo A, æquale parallelogrammum constituatur E F G H, habens angulum F, angulo D, æqalem. Item super rectam G H, parallelogrammum G H I K, æquale triangulo B, habens angulum G, æqualem angulo D. Item super rectam I K, parallelogrammum I K L M, æquale triangulo C, habens angulum K, æqualem anglo D; Et sic deinceps procedatur, si plura fuerint triangula in dato rectilineo, factumque erit, quod iubetur. Nam tria parallelogramma constructa, quæ quidem æqualia sunt rectilineo dato A B C, conficiunt totum unum parallelogrammum, quod sic demonstratur. Duo anguli E F G, H G K, inter se sunt æquales, cum uterque æqualis sit angulo D. Addto igitur communi angulo F G H, erunt duo anguli E F G, F G H, qui duobus rectis æquivalent, æquales duobus angulis H G K, F G H, ideoque hi anguli duobus rectis æquales sunt. Quare F G, G K, unam rectam lineam efficient. Eadem ratione ostendemus, E H, H I, unam rectam lineam efficere, propterea quod duo anguli E H G, H I K, æquales inter se sint, (cum sint æquales oppositis angulis æqualibus E F G, H G K.) & duo anguli H I K, I H G, duobus sint rectis æquales, &c. Cum igitur E I, F K, sint parallelæ; Itemque E F, I K, quod utraque parallela sit rectæ H G; Parallelogrammum erit E F K I. Eodemque modo demonstrabitur, parallelogrammum I K L M, adiunctum parallelogrammo E F K I, constituere totum unum parallelogrammum E F L M.

. . . . . . . . . . . . . . . . . . . . .

Therefore the parallelogram KFLM has been constructed equal to the given rectilineal figure ABCD, in the angle FKM which is equal to the given angle E. Q. E. F.

Dato ergo rectilineo A B C, constituimus æquale parallelogrammum E F L M, habens angulum F, æqualem angulo D, dato. Quod erat efficiendum.

SCHOLION
QVAMVIS in hoc problemate Euclides absolute, & simpliciter docuerit, quanam arte parallelogrammum constituatur æquale rectilineæ figuræ datæ non astringendo nos ad certam aliquam rectam lineam datam, ut in propos. 44. fecerat; Tamen eodem modo, quod iubetur, efficiemus, si recta aliqua lineanobis fuerit assignata. Nam si detur recta linea E F, super ipsam construemus parallelogrammum E F G H, æquale triangulo A. Et eodem modo super G H, constituemus aliud G H I K, æquale triangulo B, &c. Quibus peractis, constitutum erit super datam rectam E F, parallelogrammum E F L M, æquale rectilineo dato, in dato angulo F, qui æqualis est angulo D, proposio.
PARI ratione, propositis quotcunque rectilineis, constituemus illis parallelogrammum æquale, si omnia resoluantur in triangula, quibus æqualia parallelogramma exhibeantur, singulis singula, per propos. 44. veluti factum est in hoc problemate. Nam cum omnia hæc parallelogramma efficiant unum parallelogrammum, uti hic demonstratum fuit, constitutum erit parallelogrammum æquale rectilineis propositis. Vt si quis intelligat duo rectilinea proposita A B, & C; Atque A B, resoluatur in triangula A, & B, singulisque triangulis, A, B, C, singula parallelogramma E G, G I, I L, super rectas E F, H G, I K, iuxta artem huius problematis, æqualia constituantur, ex propos. 42. & 44. erit constructum parallelogrammum totum E F L M, æquale duobus rectilineis A B, & C. Et sic de pluribus.
HVC referri poterit problema utilissimum ex Peletario, quod nos tamen alia ratione, et breuiori demonstrabimus in hunc modum.

DATIS duobus rectilineis inæqualibus, excessum maioris supra minus inquirere.
SINT data rectilinea A, & B, sitque A, maius. Oportet igitur indagare, qua magnitudine rectilineum A, superet rectilineum B. Fiat parallelogrammum C D E F, in quocunque angulo D, æquale maiori rectilineo A. Et super rectum C D, parallelogrammum C D G H, in eodem angulo D, æquale rectilineo minori B. Quoniam igitur parallelogrammum C D E F, superat parallelogrammum C D G H, parallelogrammo E F H G, superabit quoque figura A, figuram B, eodem parallelogrammo E F H G. Quod est propositum.

. .

Permanent link

http://www2.hf.uio.no/common/apps/permlink/permlink.php?app=polyglotta&context=ctext&uid=c8280f78-28d7-11e0-8f33-001cc4df1abe

Choose languages

Choose images, etc.

Choose languages

Choose display