angle, repeated twice, are equivalent to twice the squares of BD, CE, and AF, with half the squares of the sides of the triangle. Consequently four times the squares of AB, BC, and AC are equivalent to four times the squares of BD, CE, and AF, with once the squares of AB, BC, and AC; wherefore thrice the squares of the sides AB, BC, and AC are together equivalent to four times the squares of the bisecting lines BD, CE, and AF. PROP. IV. THEOR. The squares of the sides of a quadrilateral figure are together equivalent to the squares of its diagonals, together with four times the square of the straight line joining their middle points. Let ABCD be a quadrilateral figure, in which the straight lines AC, BD, drawn to the opposite corners, are bisected at the points E, F: the squares of AB, BC, CD, and DE, are together equivalent to the squares of AC, BD, together with four times the square of EF. For join EF. And because AC is bisected in F, the squares of AB and BC are equivalent to twice the square of AF and twice the square of BF (II. 22.); and, for the same reason, the squares of CD and DA are equivalent to twice the square of AF and twice the square of DF. Consequently the squares of all the sides AB, BC, CD, and DA, are equivalent to four times the square of AF—or the square of AC–with twice the squares of BF and of DF. But twice these squares of BF and DF is equivalent (II. 22.) to four times the square of BE, or the square of BD, with four times the square of EF; whence the squares of all the sides of the quadrilateral figure are together equivalent to the squares of its diagonals AC, BD, with four times the square of the straight line EF which joins their points of equal section. This general theorem seems to have been first given by the illustrious Leonard Euler in the Petersburg Memoirs. It evidently comprehends the twenty-fourth Proposition of this Book ; for when the quadrilateral figure becomes a rhomboid, the diagonals bisect each other, and the middle . points E and F coincide; whence the squares of all the sides are equivalent simply to the squares of those diagonals.-If this rhomboid again becomes a rectangle, it will have equal diagonals, and consequently, as in the 10th Proposition of the Second Book, the squares of the sides of a right-angled triangle are equivalent to the square of the hypotenuse. BOOK III. 1. Proposition fifteenth. Hence angles are sometimes measured by a circular instrument, from a point in the circumference, as well as from the centre. 2. Proposition eighteenth. On this proposition depends the construction of amphitheatres; for the visual magnitude of an object is measured by the angle which it subtends at the eye, and consequently the whole extent of the stage, the intermediate objects being purposely darkened or obscured, will be seen with equal advantage by every spectator seated in the same are of a circle. 3. Proposition twenty-second. To erect a perpendicular, any point D is taken, as in Prop. 34. Book I., and from it a circle is described passing through C and B; the diameter CDF, by its intersection at the point B, determines the position of the perpendicular BF. To let fall a perpendicular, draw to AB any straight line FC, which bisect in D, and from this point as a centre describe a circle through the points C, B and F; FB is the perpendicular required. 4. To this Book may be subjoined some useful propositions. The inclination of two straight lines is equal to the angle terminated at the circumference by the sum or difference of the arcs which they intercept, according as their vertex is within or without the circle. If the two straight lines AB and CD intersect each other in the point E within a circle; the angle AED which they form, is equal to an angle at the circumference and standing on the sum of the intercepted arcs AD and BC. For draw the chord BF parallel to CD. Because ED and BF are parallel, the angle AED (I. 22.) is equal to the interior angle ABF, which stands on the arc AF; but since the chords BF and CD are parallel, the arc BC is equal to DF (III. 18.) and consequently the arc AF, which termimates at the circumference an angle equal to AED, is the sum of the two intercepted arcs AD and BC. Again, if the straight lines AB and CD meet at E, without the circle, their inclination AED is equal to an angle at the circumference, having for its base the excess of the arc AD above BC. For BF being drawn parallel to CD, the arc BC is equal to FD, and consequently the arc AF is the excess of AD above BC; but the angle ABF which stands on AF, is equal to the interior angle AED. Cor. Hence if two chords intersect each other at right angles within a circle, the opposite intercepted arcs are equal to the semicircumference. This proposition is of some utility in practice, for an angle may be hence measured by help of a circular protractor, without the trouble of applying the centre to its vertex or the point of concourse of the sides. The same principle is likewise applicable to the construction of some optical instruments, adapted to measure lateral angles by the intersection of micrometer wires. PROP. II. THEOR. If a circle be described on the radius of another circle, any straight line drawn from the point where they meet to the outer circumference, is bisected by the interior one. Let AEC be a circle described on the radius AC of the circle ADB, and AD a straight line drawn from A to terminate in the ex- T) terior circumference; the part AE in the smaller circle is equal to the part H5 ED intercepted between the two cir- A. NJ a semicircle, the angle contained in it is a right angle (III. 19.); consequently the straight line CE, drawn from the centre C, is perpendicular to the chord AD, and therefore (III. 4.) bisects it, a- PROP. III. THEOR. If, on each side of any point in the circumference of a circle, equal arcs be repeated; the chords which join the opposite points of section will be together equal to the last chord extended till it meets a straight line drawn through the middle point and either extremity of the first chord. Let DAG be the circumference of a circle, in which the arcs AB, BC, CD on the one side of a point A, and the corresponding arcs AE, EF, FG on the other side, are all assumed equal; the chords BE, CF, and DG, are together equal to the line GH, formed by extending GD till it meets the production of AB. For join FD and CE, and produce this to meet GH in the point I. Because the arcs BC A and CD are equal to R2- T. same reason, since the Ti arcs BC and CD are equal to AE and EF, the chords BA, CE and DF are likewise parallel. Hence the figures HBEI and ICFD are rhomboids, and therefore the extended chord GH, being composed of the segments HI, ID, and DG, is equal to the sum of their opposite chords BE, CF and DG.-It is obvious that the same train of reasoning may be pursued to any number of equal al’CS. PROP. Iv. THEOR. If from any point in the diameter of a circle or its extension, straight lines be drawn to the ends of a parallel chord ; the squares of these lines are together equivalent to the squares of the segments into which the diameter is divided, Let BEFD be a circle, and from the point A in its extended diameter the straight lines AE and AF be drawn to the ends of the parallel chord EF, the squares of AE and AF are together equivalent to the squares of AB and AD. For, from the centre C, let fall the perpendicular CG upon EF (I. 6.), and join AG and CE. y |