Prop. XXXII. The three angles of a triangle may be shewn to be equal to two right angles without producing a side of the triangle, by drawing through any angle of the triangle a line parallel to the opposite side, as Proclus has remarked in his Commentary on this proposition. It is manifest from this proposition, that the third angle of a triangle is not independent of the sum of the other two; but is known if the sum of any two is known. Cor. 1 may be also proved by drawing lines from any one of the angles of the figure to the other angles. If any of the sides of the figure bend inwards and form what are called re-entering angles, the enunciation of these two corollaries will require some modification. From this proposition, it is obvious that each of the angles of an equilateral triangle, is equal to two thirds of a right angle, as it is shewn in Prop. 15, Book IV. Also, if one angle of an isosceles triangle be a right angle, then each of the equal angles is half a right angle, as in Prop. 9, Book II. Prop. xxxiv. If the other diameter be drawn, it may be shewn that the diameters of a parallelogram bisect each other, as well as bisect the area of the parallelogram. The converse of this Prop. namely, "If the opposite sides or opposite angles of a quadrilateral figure be equal, the opposite sides shall also be parallel; that is, the figure shall be a parallelogram," is not proved by Euclid. Prop. xxxv. The latter part of the demonstration is not expressed very intelligibly. Simson, who altered the demonstration, seems in fact to consider two trapeziums of the same form and magnitude, and from one of them, to take the triangle ABE; and from the other, the triangle DCF; and then the remainders are equal by the third axiom: that is, the parallelogram ABCD is equal to the parallelogram EBCF. Other wise, the triangle, whose base is DE, (fig. 2.) is taken twice from the trapezium, which would appear to be impossible, if the sense in which Euclid applies the third axiom, is to be retained here. It may be observed, that the two parallelograms exhibited in fig. 2 partially lie on one another, and that the triangle whose base is BC is a common part of them, but that the triangle whose base is DE is entirely without both the parallelograms. After having proved the triangle ABE equal to the triangle DCF, if we take from these equals the triangle whose base is DE, and to each of the remainders add the triangle whose base is BC; perhaps the proof may appear somewhat more satisfactory. Prop. XXXVIII. In this proposition, it is to be understood that the bases of the two triangles are in the same straight line. Prop. XXXIX. If the vertices of all the equal triangles which can be described upon the same base, or upon the equal bases as in Prop. 40, be joined, the line thus formed will be a straight line, and is called the locus of the vertices of equal triangles upon the same base, or upon equal bases. A locus in plane Geometry is a straight line or a plane curve, every point of which and none else satisfies a certain condition. With the exception of the straight line and the circle, the two most simple loci; all other loci, perhaps including also the Conic Sections, may be more readily and effectually investigated algebraically by means of their rectangular or polar equations. Prop. XLI. The converse of this proposition is not proved by Euclid; viz. If a parallelogram is double of a triangle, and they have the same base, or equal bases upon the same straight line, and towards the same parts, they shall be between the same parallels. Also, it may easily be shewn that if two equal triangles are between the same parallels; they are either upon the same base, or upon equal bases. Prop. XLIV. to that line. Prop. XLVII. A parallelogram described on a straight line is said to be applied In a right-angled triangle, the side opposite to the right angle is called the hypothenuse, and the other two sides, the base and perpendicular, according to their position. It is not indifferent on which sides of the lines which form the sides of the triangle the squares are described. If they were described upon the inner, instead of the outer sides of the lines, the construction would be found to fail. By this proposition may be found a square equal to the sum of any given squares, or equal to any multiple of a given square: or equal to the difference of two given squares. The truth of this proposition may be exhibited to the eye in some particular instances. As in the case of that right-angled triangle whose three sides are 3, 4, and 5 units respectively. If through the points of division of two contiguous sides of each of the squares upon the sides, lines be drawn parallel to the sides (see the notes on Book 11. p. 68), it will be obvious, that the squares will be divided into 9, 16 and 25 small squares, each of the same magnitude; and that the number of the small squares into which the squares on the perpendicular and base are divided is equal to the number into which the square on the hypothenuse is divided. Prop. XLVIII is the converse of Prop. XLVII. In this Prop. is assumed the Corollary that "the squares described upon two equal lines are equal," and the converse, which properly ought to have been appended to Prop. XLVI. The first book of Euclid's Elements, it has been seen, is conversant with the construction and properties of rectilineal figures. It first lays down the definitions which limit the subjects of discussion in the first book, next the three postulates, which restrict the instruments by which the constructions in plane geometry are effected; and thirdly, the twelve axioms, which express the principles by which a comparison is made between the ideas of the things defined. This Book may be divided into three parts. The first part treats of the origin and properties of triangles, both with respect to their sides and angles; and the comparison of these mutually, both with regard to equality and inequality. The second part treats of the generation and properties of parallelograms. The third part exhibits the connexion of the properties of triangles and parallelograms, and the equality of the squares on the base and perpendicular of a right-angled triangle to the square on the hypothenuse. When the propositions of the first book have been read, the student is recommended to use different letters in the diagrams, and where it is possible, diagrams of a form somewhat different from those exhibited in the text, for the purpose of testing the accuracy of his knowledge of the demonstrations. And further, when he is become sufficiently familiar with the method of geometrical reasoning, he may dispense with the aid of letters altogether, and acquire the power of expressing in general terms the process of reasoning in the demonstration of any proposition. Also, should he be not satisfied with the bare knowledge of the principles of the first book which have been exhibited synthetically, but be also desirous of knowing how these principles may be applied to the solution of Problems analytically and the demonstration of Theorems; he may refer to the Geometrical Exercises on the first book, which will be found at the end of the Elements, together with some brief account of the Ancient Geometrical Analysis. BOOK II. DEFINITIONS. I. EVERY right-angled parallelogram is called a rectangle, and 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 AGK, or EHC, which are at the opposite angles of the parallelograms which make the gnomon." PROPOSITION I. THEOREM. 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 BC be two straight lines; and let BC be divided into any parts in the points D, E. Then the rectangle contained by the straight lines A and BC, shall be equal to the rectangle contained by A and BD, together with that contained by A and DE, and that contained by A and EC. From the point B, draw BF at right angles to BC, (1. 11.) through G draw GH parallel to BC, (1. 31.) and through D, E, C, draw DK, EL, CH parallel to BG. Then the rectangle BH is equal to the rectangles BK, DL, EH. But BH is contained by A and BC, for it is contained by GB, BC, and GB is equal to A: because DK, that is, BG, (1. 34.) is equal to A; and in like manner the rectangle EH is contained by A, EC: therefore the rectangle contained by A, BC, is equal to the several rectangles contained by A, BD, and by A, DE, and by A, EC. 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. Then the rectangle contained by AB, BC, together with that contained by AB, AC, shall be equal to the square of AB. Upon AB describe the square ADEB,_ (1. 46.) Then AE is equal to the rectangles AF, CE. and AF is the rectangle contained by BA, AC; for BE is equal to AB: therefore the rectangle contained by AB, AC, together with the rectangle AB, BC is equal to the square of AB. If therefore a straight 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. Then the rectangle AB, BC, shall be equal to the rectangle AC, CB, together with the square of BC. Upon BC describe the square CDEB, (1. 46.) and produce ED to F, for it is contained by AB, BE, of which BE is equal to BC: 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 be divided, &c. PROPOSITION IV. THEOREM. Q. E. D. If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. Let the straight line AB be divided into any two parts in C. Then the square of AB shall be equal to the squares of AC, and CB, together with twice the rectangle contained by AC, CB. |