PROP. XXXIX. THEOR. 47. 1 Eu. In any right-angled triangle, the square which is described upon the hypothenuse, or side D LE the same lls Cor. 2. (BD, AL; * Squares and parallelograms are frequently expressed, for the sake of brevity, by the letters at their opposite angles; and the square on any line, as BC, is represented by BC%. PROP. XL. THEOR. exterior angle, also one of the interior and Let CBA be a A. Prod. BC to D, and let the ext. L ACD be bisected by the str. line CE, and the int. and opp. L ABC by the str. line BE. Then will Z BAC = 2L BEC. DEFINITIONS. Every right-angled parallelogram, or rect angle, 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 parallelo grams 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 AGK or EHC, which are at the opposite angles of the parallelograms which make the gnomon.' A Gc N.B.-The rectangle contained by any two straight lines AB, AD, is generally expressed by the words “rectangle of AB and AD;" or by “AB. AD.” PROP. XLI. THEOR. 1. 2 Eu. If there be two straight lines, one of which is divided into any number of parts; the rect |