From the point o draw a right line perpendicular to the line C B, and from it cut off a part CD, equal to C A.* Join the points D and B by the right line D B. Because the lines C A and C D are equal, the squares on them are equal. If to each of these squares we add the square on CB, the sum of the squares on A C and CB will be equal to the sum of the squares on D C and C B. But since B C D is a right Z, the square on the side B D in the ABCD is equal to the sum of the squares on the sides B C and C D. Therefore the square on the side B D is also equal to the sum of the squares on AC and C B. But we know, to start with, that the square on AB is equal to the sum of the squares on AC and C B. Therefore the square on A B is equal to the square on D B. Consequently, the line A B must be equal to the line D B. Hence, the three sides A C, C B, and B A in the A A CB, are equal respectively to the three sides DC, CB, and B D in the A DOB. It follows therefore (according to the 8th proposition) that these As are also equal in every other respect. Among these respects is, that the < ACB is equal to the ZDOB. But D C B is a right 2 (by construction). . It would do equally well if a line were drawn perpendicular to C A, and a part cut off, equal to C B. THE SECOND BOOK OF EUCLID EXPLAINED TO BEGINNERS. DEFINITIONS. 1. A rectangle is a parallelogram, the angles of which are right angles. To show that a parallelogram is a rectangle, it is enough to show that one _ is a right Z; because the opposite to this must be equal to it (I. 34), and any two of the Z8 which are not opposite must be together equal to two right Zs. (I. 29.) 2. A rectangle is said to be contained by any two of its sides which form one of its <s, or (in other words) a rectangle is said to be contained by any two conterminous sides. It is also described as the rectangle under two conterminous sides. Thus the rectangle ABCD is said to be D contained by AB and BC, by BC and CD, by AB and AD, or by AD and DC; or is described as the rectangle under AB and BC, &c. A rectangle is also said to be contained by two right lines which are respectively equal to two of its conterminous sides. 3. A square is a rectangle whose conterminous sides are equal. A 4. In any parallelogram either of the parallelograms about the diagonal, taken together with the two complements, forms a figure which is called a gnomon. In what follows the abbreviation rect. is frequently used for the word rectangle, and the symbol ofor the word square. Thus “rect. A B, BC," means “the rectangle contained by A B and BC;" on A B” means “ the square on A B.” Of course, “ rects.” means "rectangles," and "Os” means “ squares.” A rectangle is often named by means of two letters standing at opposite Zs. Thus the rectangle given in p. 134 may be named as the rectangle A C,” or “the rectangle DB.” If lines are named by single letters, “rect. P,Q" means “ the rectangle contained by the lines P and Q." The operations of addition and subtraction are as applicable to magnitudes as to numbers. Two lines or two areas may be added together, or one may be taken from another. Consequently, the signs +, and used with relation to magnitudes, as well as with relation to numbers. If A and B are two magnitudes (lines, or angles, or areas), A + B denotes their sum, A B denotes what is left when B has been taken from A. Such an equation as A + B = C + D + E means that the sum of the two magnitudes, A and B, is equal to the sum of the three magnitudes, A, B, and C. But multiplication is a purely arithmetical operation : to speak of multiplying a line by a line is absolutely meaningless. Consequently, the signs x and • cannot be used to connect symbols which denote magnitudes. It is utterly wrong and misleading to denote the rectangle contained by two lines, A B and CD by the symbols AB x CD. For a similar reason, it is objectionable to speak of “the square of the line A B,” because the expression “square of” is used with reference to numbers. The “square of five" is the product of 5 multiplied by 5. It is the more important to adhere to these distinctions, because (as will be shown further on), there is a very real and close connection between a rectangle may be and a product which can never be properly understood if the two are confounded from the beginning. The reasoning in several of the propositions in the Second Book of Euclid are rendered much more intelligible by the use of the signs t, and = They will accordingly be occasionally employed. PROPOSITION I, If there are 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 sum of the rectangles contained by the undivided line and the several parts of the divided line. For the construction made use of in this proposi tion we are supposed to be able :1. Through a given point in a given straight line to draw a line at right angles to the given line. (I. 11.) 2. From a given right line to cut off a part equal to another given right line. (I. 3.) 3. Through a given point to draw a right line parallel to a giver right line. (I. 31.) We are also supposed to know that: 1. Magnitudes which are equal to the same are equal to each other. (Ax. I.) 2. If a right line intersect two parallel right lines, the exterior angle is equal to the interior opposite angle on the same side of the intersecting line. (I. 29.) 3. The opposite sides of a parallelogram are equal to each other. (I. 34.) By equal is of course meant equal in area. R R G L A B D E Let A and B C be two straight lines, of which BO is divided into the parts BD, DE, and EO. We have to show that the rectangle under A and B C is equal to the sum of the rectangles, contained by A and BD, A and DE, and A and EO. From the point B draw BF at right angles to B C. From BF cut off BG equal to A. Through the point G draw G H parallel to B C. Through the points I), E, and C draw DK, EL, and CH parallel to BG, and meeting the line G H in the points K, L, and H. Since the parallel lines B G and D K are intersected by the right line BC, the < XDE is equal to the 2. GBC, and is therefore a right Z. For a similar reason the _ LE C is a right Z. Therefore (Def. 1), the parallelograms BH, BK, DL, and E H are all rectangles. Now it is self-evident that the rectangle BH is equal to the sum of the rectangles BK, DL, and E H. But the rectangle B H is the rect. under A and B 0; for it is contained by B G and BC, and B G is equal to A. The rectangle BK is the rectangle contained by A and B D, for it is contained by B G and BD, and BG is equal to A. The rectangle D L is the rectangle contained by A and D Е, for it is contained by D K and D E, and DK is equal to BG, which is equal to A. Also the rectangle E H is the rectangle under A and EC, for it is contained by EL and EC, and EL is equal to BG, which is equal to A, Therefore the rectangle contained by the lines A and |