PROP. XIV. PROB. To describe a square that shall be equal to a given rectilineal figure. Let A be the given rectilineal figure; it is required to describe a square that shall be equal to A. Describe the rectangular parallelogram BCDE equal to the rectilineal figure A. If then the sides of it BE, ED are equal to one another, BOOK II. a xlv. 1. to F, and make EF equal to ED and bisect BF in G; and from the centre G, at the distance GB, or GF, describe the semicircle BHF, and produce DE to H, and join GH: Therefore because the straight line BF is divided into two equal parts in the point G, and into two unequal at E, the rectangle BE, EF, together with the square of EG, is equal to the square of GF: But GF is equal to GH: therefore the rectangle BE, EF, together with the square of EG, is equal to the square of GH: b v.2. But the squares of HE, EG are equal to the square of vii. 1. GH: Therefore the rectangle BE, EF, I BOOK II. together with the square of EG, is equal to the squares of HE, EG: which is common to both; and the remaining rectangle BE, EF is equal to the square of EH: But the rectangle contained by BE, EF is the parallelogram BD, because EF is equal to ED; therefore BD is equal to the square of EH; but BD is equal to the rectilineal figure A; therefore the rectilineal figure A is equal to the square of EH. viz. the square described upon EH. [In constructing this figure it will be necessary to divide the rectilineal figure A into triangles, and construct a rectangle equal to one of them, and then apply to one of the sides of that rectangle another rectangle which shall be equal to another of the triangles, and so on till a rectangle has been made equal to the whole rectilineal figure. Then proceed as directed in the Proposition. The learner should here draw correctly the figure which he was directed to omit at the first reading of i. 45.] DEDUCTIONS FROM BOOK II. THERE is not much of interest or importance in the Deductions usually appended to this book. The learner may exercise himself with the following, or proceed immediately to the Third Book. (1.) To divide a straight line so that the rectangle of the parts shall be equal to A (2.) If a straight line be drawn from the vertical angle of a the rectangle AD, DE (II. 12.). E B Or, in other words, is equal to the squares of AD, DC, together with twice the rectangle AD, DE; that is, is equal to the squares of AD, DC, together with twice the rectangle BD, DE, for AD is BOOK II. The square of BC is equal to the squares of BD, DC, less twice the rectangle BD, DE (II. 13.). Add these equals together, and the Proposition is proved. (4.) The squares described upon the sides of a parallelogram are together equal to the squares upon its diameters. Remember that the diago nals of a paral- A lelogram bisect each other. B D Then apply Ex. 2. to both the triangles ABC, ADC. C (5.) In any triangle, if a perpendicular be drawn from one The square on AC is equal to A B The square on BC is equal to the squares on BO, OC. (6.) Prove that if a straight line be divided into two parts, the rhombus described upon the whole line shall be equal to those which are described upon the parts, and have their angles equal to its angles, together with twice the parallelogram whose sides are equal to the parts, and whose angles are equal to E B K H BOOK II. In other respects construct the figure as in Prop V., and the proof is nearly the same with that in Prop. V. |