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UPon a Right Line given, to make a Parallelogram at a Right Lined Angle given, equal to a Triangle given.
It is proposed to make a Parallelo∣gram having one of its Sides equal to the Line D, and one of its Angles equal to the Right Lined Angle E, which must be equal to the Triangle
ABC. Make by the preceding Problem the Parallelogram BFGH, having the Angle HBF equal to the Angle E, and which may be equal to the Triangle ABC. Continue the Sides GH, CF, and make HI equal to the Line D; then draw the Line IBN, and by the Points I and N, draw IL Parallel to GN, and NL Parallel to GI; and continue HB to M, and FB to K. The Parallelogram KLBM is the Parallelogram required.
Demonstration. The Angle HBF equal to the given Angle E, is also equal to the Angle KBM (by the 15th.) and the Side KB is equal to the Line HI or D. Lastly, the Parallelogram MK is equal (by the foregoing) to the Pa∣rallelogram GFBH; and this was made equal to the Triangle ABC. There∣fore the Parallelogram MK is equal to the Triangle ABC.
IN this Proposition is contained a sort of Geometrical Division: For in Arithmetical Division there is proposed a number which may be imagined to be like a Rectangle: For example, the Rectangle
AB containing Twelve square Feet, which is to be Divided by another Number, as by two; that is to say, that there must be made another Rectangle equal to the Rectangle AB, which may have BD too, for one of its Sides, and to find how many Feet the other Side ought to be, that is to say the Quotient. One may attain it Geometrically by a Rule and Compass. Take BD of the Length of Two Feet, and draw the Diagonal DEF; the Line AF is that which you seek for. For having made the Rectangle DCFG, the Complements EG, EC, are equal (by the 43d.) and EG hath for one of its Sides EH equal to BD two Foot, and EI equal to AF. This way of Di∣vision is called Application, because the Rectangle AB is applyed to the Line BD or EH; and this is the reason why Division is called Application; for the Ancient Geometricians did choose rather to make use of a Ruler and Compass, than Arithmetick.
Use 44.