AB and BC are together equal to the square of AC (hyp.), therefore the squares of A D and AC are equal, and therefore the lines themselves are equal; but also D B and BC are equal, and the side AB is common to both triangles, therefore the triangles A B C and ABD are mutually equilateral, and therefore also mutually equiangular, and therefore the angle ABC is equal to the angle ABD; but ABD is a right angle, therefore A B C is also a right angle. This proposition may be extended thus: The vertical angle of a triangle is less than, equal to, or greater than a right angle, according as the square of the base is less than, equal to, or greater than the sum of the squares of the sides. For from B draw B D perpendicular to A B and equal to B C, and join A D. D B The square of A D is equal to the squares of A B and BD or B C. The line A C is less than, equal to, or greater than A D, according as the square of the line A C is less A than, equal to, or greater than the squares of the sides A B and B C. But the angle B is less than, equal to, or greater than a right angle, according as the side A C is less than, equal to, or greater than A D (XXV, VIII); therefore, &c. BOOK. II. DEFINITIONS. (217) I. Every rectangle or right angled parallelogram is said to be contained by two right lines which contain one of its right angles. (218) II. In any parallelogram either of the pa- or O F) with the two complements (A G K D G L (219) Next to the triangle, the most important rectilinear figure is the rectangle or right angled parallelogram. The areas of all figures whatever, whether bounded by straight lines or curves, are expressed by those of equivalent rectangles. To determine a rectangle it is only necessary to know two sides which are conterminous, for the other sides being opposed to these are equal to them, and the angles are all right. It is usual, therefore, to express a rectangle by its two conterminous sides, and it is said to be contained by these. Thus, if A and B express two lines which are the conterminous sides of a rectangle, the rectangle itself is called 'the rectangle under A and B.' (220) It was proved in (186) that the area of a parallelogram can be expressed in numbers by multiplying the number which expresses the length of its base by that which expresses the length of its altitude. Hence, the area of a rectangle is expressed by multiplying the numbers representing its sides. The product then expresses the area. In arithmetic and algebra the product of two numbers is expressed by placing the sign x between them. Hence, we derive a shorter way of expressing a rectangle whose sides are A and B, scil. A x B. By what has been established in (186), it appears that the area of every parallelogram is expressed by the product of its base and altitude, and that every triangle is expressed by the product of any side and the perpendicular on it from the opposite angle. The entire of the second book is appropriated to the investigation of the relations between the rectangles under the segments of right lines divided into two or more parts. (221) If there be two right lines (A and B C), one of which is divided into any number of parts F (BD, DE, EC), the rectangle under the two lines is equal to the sum of the rectangles under the undivided line (A) and the several parts of the divided line (B C). From the point B draw B H perpendicular to B C, take on it B F equal to A, and through F draw FL parallel to B C, and draw DG, E K, and CL parallel to B F. H F A B GK DE C It is evident that the rectangle BL is equal to the rectangles BG, DK, and E L; but the rectangle BL is the rectangle under A and B C, for B F is equal to A: and the rectangles B G, DK, and E L are the rectangles under A and B D, A and D E, and A and E C, for each of the lines B F, DG, and EK is equal to A (XXXIV, Book I.). If the line B C be considered as the sum of the several lines B D, DE, &c. this proposition may be thus announced : The rectangles under one line and several others is equal to the rectangle under that line and the sum of the others.' (222) COR.-The rectangle under any two lines is equal to twice the rectangle under either of them and half the other, to three times the rectangle under either of them and a third of the other, &c. &c. (223) If a right line (A B) be divided into any two parts (in C), the square of the whole line is equal to the sum of the rectangles under and each of the parts D E P On A B describe the square ADFB (XLV, Book I.), and through C draw CE parallel to AD. The square AF is p equal to the rectangles A E and CF. But the rectangle AE is the rectangle under A B and A C, because AD is equal to A B (const.), and the rectangle C F is the rectangle under A B and C B, because C E is equal to A B (XXXIV, Book I. and const.). A C (224) In this and the succeeding propositions there is no necessity for the absolute construction of the rectangles to establish the relations they express. We shall, therefore, subjoin to each a second demonstration independent of any construction. Let A be the right line divided into the parts x and y. We are to prove that the square of A is equal to the rectangles Axx and A × y taken together. BA Let B be drawn equal to A. By (I*) the rectangle B x A is equal to the rectangles B x x and B x y taken together; that is, (since В is equal to A) to the rectangles Axx and Axy taken together. As we shall frequently have occasion to express the equality of quantities, the language will be abridged by the use of the sign =. Thus, 'A B' means 'the line A is equal to the line B.' = (225) The second book is generally found to be one of the greatest difficulties which the student has to encounter in plane geometry. One of the causes of this (if not the only cause) is, the great variety of forms under which the same proposition may present itself. We cannot do any thing more calculated to remove this difficulty, than to show from whence this variety of forms arises. We have already stated that the object of most of the propositions of this book is, to determine the relations between the rectangles under the parts of divided lines. We shall first confine our attention to a finite right line divided into two parts. In this case there are three lines to be considered; 1st, the whole line; 2nd, its greater part; 3rd, its lesser part; and in the present proposition the square of the first is compared with the rectangles under it and the second and third. If, however, the two parts be considered as two independent lines, the whole line must be considered as their sum. Under this view the second proposition becomes, The square of the sum of any two lines is equal to the rectangles under the sum and each of them.' Again, if the whole line A be considered as the greater of two given right lines and one of the parts x as the less, the other part y must be their difference. Thus the greater line is, in fact, supposed to be divided into two parts equal to the less and difference. Under this view, the second proposition assumes the form, ' The square of the greater of two lines is equal to the rectangle under those lines together with the rectangle under the greater and difference.' These, though apparently different from the second proposition, as announced in the text, are really the same, no other change being made than in the names given to the line and its parts. should not, therefore, be denominated corollaries, as is sometimes the case. They If W express the whole line, and P,p its parts, the proposition as announced in the text may be expressed thus: The square of W= WXP+ W × p. (The sign interposed between two magnitudes signifies their sum.) If L, express any two lines and S express their sum, the second method of announcing the proposition may be expressed thus: The square of SS x L + S x l. *When a reference is made to a proposition without any mention of a 'Book,' the present book is to be understood. And if D represent the difference between L and 1, the third method is : The square of L = LX1+LX D In the study of the second book considerable facility may be derived from the use of these symbols. PROPOSITION III. THEOREM. (226) If a right line (A B) be divided into any two parts (in C), the rectangle under the whole line (A B) and either part (AC) is equal to the square of that part (A C) together with the rectangle under the parts (A C and CB). On A C describe the square AD FC, and through B draw BE parallel to A D, until it meet D F produced to E. The rectangle A E is equal to the square ADFC together with the rectangle CE. But the rectangle A E is the rectangle under AC and AB, for A D is equal to AC (const.), D F E A and the square A D FC is the square of A C (const.), and the rectangle ĈE is the rectangle under AC and C B, for CF is equal to AC (const.). Otherwise thus: Let A be the right line divided into the parts x and y, and let B be another line equal to x. By (I) the rectangle Ax B =Bxx + Bxy. But since B = x, * the rectangle A Bxx is the square of x, and the rectangle B x y is equal to the rectangle x x y. Hence, &c. (227) Conformably to the observations on the last proposition, this may be announced in two other ways. 1. If the two parts of the divided line be considered as two independent lines, the whole line being their sum, the proposition becomes, The rectangle under the sum of two lines, and one of them, is equal to the square of that one together with the rectangle under the lines.' 6 2. If the whole line be considered as the greater, one part as the less, and the other as the difference, the proposition becomes, The rectangle under two lines is equal to the square of the less together with the rectangle under the less and difference.' (228) COR. 1.-From this and the last proposition combined it follows, that the difference of the squares of two lines is equal to the rectangle under their sum and difference. For by the second, the *This sign expresses the word 'therefore.' |