whole fbc; and because the two sides a b, b d, are equal to the two fb, bc, f each to each, and the angle dba equal to the angle fbc; therefore the base ad is equal (i. 4) to the base fc, and the triangle abd to the triangle fbc: now the parallelogram bl is double (i. 41) of the triangle abd, because they are upon the same base bd, and between the same parallels bd, al; and the square gb is double of the triangle fbc, because these also are upon the same base fb, and between the same parallels fb, gc. But the doubles of equals are equal (6 ax.) to one another therefore the parallelogram bl is equal to the square gb: and, in the same manner, by joining a e, bk, it is demonstrated, that the parallelogram cl is equal to the square hc; therefore the whole square bdec is equal to the two squares gb, hc; and the square bdec is described upon the straight line bc, and the squares gb, hc upon ba, ac: wherefore the square upon the side bc is equal to the squares upon the sides ba, a c. Therefore, in any rightangled triangle, &c. Q. E. D. d 1 e PROPOSITION XLVIII.-THEOREM. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle. IF the square described upon bc, one of the sides of the triangle a bc, be equal to the squares upon the other sides ba, a c, the angle bac is a right angle. a d From the point a draw (i. 11) a d at right angles to a c, and make ad equal to ba, and join dc: Then, because da is equal to a b, the square of d a is equal to the square of a b. To each of these add the square of a c; therefore the squares of d a, a c are equal to the squares of ba, a c. But the square of dc is equal (i. 47) to the squares of d a, a c, because dac is a right angle; and the square of bc, by hypothesis, is equal to the squares of ba, ac; therefore the square of dc is equal to the square of bc; and therefore also the side dc is equal to the side bc. And because the side da is equal to cab, and ac common to the two triangles dac, bac, the two da, ac are equal to the two ba, a c; and the base dc is equal to the base bc; therefore the angle dac is equal (i. 8) to the angle bac; but dac is a right angle; therefore also bac is a right angle. Therefore, if the square, &c. Q. E. D. b 1. Given a straight line and two points, to find a point in the line equidistant from each of the given points. 2. Given a straight line, a point, and an angle, to draw from the point to the line another straight line that shall make with it an angle equal to the given angle. 3. To divide a given finite straight line into any given number of equal parts. 4. To trisect a right angle, that is, to divide it into three equal parts. 5. Given an angle divided into any number of equal angles, to divide one half of the angle into the same number of equal angles. 6. Given the base, one of the angles at the base, and the sum of the other two sides, to construct the triangle. 7. To find a point in a triangle equidistant from the three angular points of the triangle. 8. Given the three angles of a triangle and the sum of its three sides to construct the triangle. 9. To find a square which shall be equal to any number of given squares. 10. To bisect a parallelogram by a straight line drawn through a given point in one of its sides. 11. To inscribe a square in a parallelogram.. 12. Given a right-angled isosceles triangle to inscribe in it a square. 13. Given a finite straight line to describe a square of which that line Ishall be the diameter. 14. To construct a triangle that shall be equal to any given rectilineal figure. D SECT. II.-THEOREMS. 15. There cannot be drawn more than two equal straight lines to another straight line from a given point without it. 16. The straight line drawn from a given point perpendicular to a given line is the shortest that can be drawn from the point to the line. 17. If a triangle have its exterior angle and one of its opposite interior angles double of those in another triangle, its remaining opposite interior angle is also double of the corresponding angle in the other. 18. If two sides of a triangle be bisected, the straight line joining the two points of bisection will be parallel to the other side. 19. The difference between the two sides of a triangle is less than the third side. 20. The sum of two sides of a triangle is greater than twice the line joining the vertex and the middle of the base.. 21. Each angle at the base of an isosceles triangle is equal to, or is less, or is greater than the half of the vertical angle, according as the triangle is right, obtuse, or acute-angled. 22. If the vertical angle of an isosceles triangle be bisected, the bisecting line will bisect the base, and be perpendicular to it. 23. In an isosceles triangle, if either of the equal sides be produced beyond the vertex, the line that bisects the exterior angle will be parallel to the base. 24. The diameters of a rectangle are equal to one another. 25. If any number of parallelograms be inscribed in a given parallelogram, the diameters of all the figures shall cut one another in the same point. 26. In a parallelogram the diameters bisect each other. BOOK II. DEFINITIONS. I. EVERY right-angled parallelogram is said to be contained by any two of the straight lines which contain one of the right angles. II. In every parallelogram, any of the parallelograms about a diameter, together with the two complements, is called a gnomon. Thus the parallelogram hg, together with the complements af, fc, is the gnomon, which is more briefly expressed by the letters a gk, or eh c, which h are at the opposite angles of the parallelograms which make the gnomon. a b g e k If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line. LET a and be be two straight lines; and let bc be divided into any parts in the points d, e; the rectangle contained by the straight lines a, bc is equal to the rectangle contained by a, bd, together with that contained by a, b de, and that contained by a, e c. d e k 1 From the point b draw (i. 11) bf at right angles to bc, and make bg equal (i. 3) to a; and through g draw (i. 31) gh parallel to bc; and through d, e, c, draw (i. 31) dk, el ch parallel to bg; then 8 the rectangle bh is equal to the rectangles bk, dl, eh; and bh is contained by a, bc, for it is contained by gb, bc, and gb is equal to a; and bk is contained by a, bd, for it is contained by gb, bd, of which gb is equal to a; and dl is contained by a, de, because dk, that is (i. 34) bg is equal to a; and in like manner the rectangle eh is contained by a, e c. Therefore the rectangle contained fl by a, bc, is equal to the several rectangles contained by a, bd, and by a, de, and also by a, e c. Wherefore, if there be two straight lines, &c. Q. E. D. PROPOSITION II. THEOREM. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square of the whole line. LET the straight line ab be divided into any two parts in the point c; the rectangle contained by a b, bc, together with the rectangle* a b, bac, shall be equal to the square of a b a d C f therefore a straight Upon ab describe (i. 46) the square a deb, and through c draw (i. 31) cf, parallel to ad or be. Then a e is equal to the rectangles af, ce; and ae is the square of a b; and af is the rectangle contained by ba, a c; for it is contained by da, a c, of which ad is equal to a b; and ce is contained by ab, bc, for be is equal to ab; therefore the rectangle contained by ab, a c, together with the rectangle ab, bc, is equal to the square of a b. If line, &c. Q. E. D. PROPOSITION III.—THEOREM. If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts is equal to the rectangle contained by the two parts, together with the square of the aforesaid part. LET the straight line ab be divided into any two parts in the point c; the rectangle ab, bc is equal to the rectangle ac, cb, together with the square of bc. a C Upon bc describe (i. 46) the square cdeb, and produce ed to f, and through a draw (i. 31) af parallel to cd or be; then the rectangle ae is equal to the rectangles ad, ce; and ae is the rectangle contained by a b, bc, for it is contained by a b, be, of which be is equal to bc; and ad is contained by a c, cb, for cd is equal to cb; and db is the square of bc; therefore the rectangle ab, bc, is equal to the rectangle ac, cb, together with the square of bc. If therefore a straight line, &c. Q. E. D. f d e * To avoid repeating the word contained too frequently, the rectangle contained by two straight lines ab, a c is sometimes simply called the rectangle ab, a c. |