Bibliotheca Polyglotta

Let BA, AC be the two given straight lines, and let them be placed so as to contain any angle; thus it is required to find a third proportional to BA, AC. For let them be produced to the points D, E, and let BD be made equal to AC; [I. 3] let BC be joined, and through D let DE be drawn parallel to it. [I. 31]

SINT duae rectae AB, AC, ita dispositae, ut efficiant angulum A, quemcunque, fitque inuenienda illis tertia proportionalis, sicut quidem AB, ad AC, ita AC, ad tertiam. Producatur AB, quam volumus esse antecedentem, et capiatur BD, aequalis ipsi AC, quae consequens esse debet, sive media. Deinde ducta recta BC, agatur illi ex D, parallela DE, occurrens ipsi AC, productae in E.

法曰。甲乙、甲丙、兩線。求別作一線。相與為連比例者。合兩線。任作甲角。而甲乙與甲戊之比例。若甲丙與他線也。先于甲乙引長之、為乙丁。與甲丙等。次作丙乙線相聯。次從丁作丁戊線。與丙乙平行。末于甲丙引長之、遇于戊。卽丙戊為所求線。如以甲丙為前率。倣此。

Since, then, BC has been drawn parallel to DE, one of the sides of the triangle ADE, proportionally, as AB is to BD, so is AC to CE. [VI. 2] But BD is equal to AC; therefore, as AB is to AC, so is AC to CE.

Dico CE, esse tertiam proportionalem: hoc est, esse ut AB, ad AC, ita AC, ad CE. Cum enim in triangulo ADE, lateri DE, parallela sit recta BC;¹ erit ut AB, ad BD, ita AC, ad CE: ² Sed ut AB, ad BD, ita eadem AB, ad AC, aequalem ipsi BD. Ut igitur AB, ad AC, ita AC, ad CE. quod est propositum.

1. 4. sexti 2. 7. quinti.

論曰。甲丁戊角形內之丙乙線。旣與戊丁邊平行。卽甲乙與乙丁之比例。若甲丙與丙戊也。本篇二而乙丁、甲丙、元等。卽甲乙與甲丙。若甲丙與丙戊也。五卷七

Therefore to two given straight lines AB, AC a third proportional to them, CE, has been found. Q. E. F.

Duabus ergo datis rectis lineis, tertiam proportionalem adinvenimus. Quod erat faciendum.

SCHOLIUM. ALITER idem demonstrabimus, hoc modo. Duae rectae datae AB, BC, constituantur ad angulum rectum ABC, et coniungatur recta AC. Producta autem AB, antecedente, ducatur ex C, ad AC, perpendicularis CD, occurrens ipsi AB, productae in D. Dico BD, esse tertiam proportionalem.

注曰。別有一法。以甲乙、乙丙、兩線。列作甲乙丙直角。次以甲丙線聯之。而甲乙引長之。末從丙作丙丁。為甲丙之垂線。遇引長線于丁。卽乙丁為所求線。

Cum enim in triangulo ACD, angulus ACD, sit rectus, et ab eo ad basin AD, deducta perpendicularis CB; erit por corollarium propositio 8 huius liber BC, media proportionalis inter AB, et BD, hoc est, ut AB, ad BC, ita erit BC, ad BD. Quod est propositum.

論曰。甲丙丁角形之甲丙丁。旣為直角。而從直角至甲。丁底。有丙乙垂線。卽丙乙為甲乙、乙丁、比例之中率。本篇八之系則甲乙與乙丙。若乙丙與乙丁也。

INVENTA autem tertia linea continue proportionali, si primam omiseris, et alijs duabus tertiam inueneris, habebis quatuor lineas continue proportionales. Ut si lineis A, et B, adinueniatur tertia proportionalis C, et duabus B, et C, tertia proportionalis D, erunt quatuor lineae A, B, C, D, continue proportionales. Eadem artereperietur quinta proportionalis, sexta, septima, octaua; et sic in infinitum.INVENTA autem tertia linea continue proportionali, si primam omiseris, et alijs duabus tertiam inueneris, habebis quatuor lineas continue proportionales. Ut si lineis A, et B, adinueniatur tertia proportionalis C, et duabus B, et C, tertia proportionalis D, erunt quatuor lineae A, B, C, D, continue proportionales. Eadem artereperietur quinta proportionalis, sexta, septima, octaua; et sic in infinitum.

旣從一二得三。卽從二、三、求四、以上、至于無窮。俱倣此。

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