XXVI. A scalene [that is, unequal] triangle is that which has three unequal sides. XXVII. A right-angled triangle is that which has a right angle. XXVIII. An obtuse-angled triangle is that which has an obtuse angle. XXIX. An acute-angled triangle is that which has three acuteangles. The acute angled triangle must have three acute angles, because the two preceding species of triangles have each two acute angles, as will be shown in the sequel. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles. This definition is redundant. If the general definition annexed to the 34th Prop. of this book be considered, the square is only a species of parallelogram, viz. that which has one angle a right angle and the sides which contain it equal to one another. XXXI. An oblong is that which has all its angles right angles, but has not all its sides equal. This definition is also redundant; for an oblong [or rectangle, that is, a right-angled parallelogram] is that which has one angle a right angle, and the sides which contain it equal. XXXII. A rhombus is that which has all its sides equal, but its angles are not right angles. This is also redundant; for a rhombus is that which has one angle oblique, and the sides which contain it unequal. XXXIII. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. This definition may be replaced by that of a parallelogram above mentioned. 0 XXXIV. All other four-sided figures besides these, are called Trapeziums. Quadrilateral figures whose opposite sides are not parallel, are called trapeziums; but if one opposite pair be parallel and the other pair not, the figure is called a trapezoid. XXXV. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet. The meaning of this definition is, that the space between the lines is always of the same breadth. POSTULATES. Let it be granted, that a straight line may be drawn from any one point to any other point. When a straight line is drawn from one point to another point, the points are said to be joined. The points are understood to be in the same plane. II. That a terminated straight line may be produced to any length in a straight line. By terminated here, is meant of a definite length; and by produced is mean-t lengthened or extended indefinitely, in the same plane. III. And that a circle may be described from any centre, at any distance from that centre. By describing a circle at any distance, is meant drawing a circle in a plane with any given radius. Various other postulates (that is, demands of common sense,) are tacitly assumed by Euclid; as, that one figure or angle may be applied to another, for the purpose of comparison. See Prop. IV. of this book. AXIOMS. Things which are equal to the same thing, are equal to one another. II. If equals be added to equals, the wholes are equal. III. If equals be taken from equals, the remainders are equal. IV. If equals be added to unequals, the wholes are unequal. V. If equals be taken from unequals, the remainders are unequal. VI. Things which are double of the same, are equal to one another. VII. Things which are halves of the same, are equal to one another. VIII. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. IX. The whole is greater than its part. Dr. Thomson, in his edition of Euclid, has added to this axiom, another, viz., that "the whole is equal to all its parts taken together." X. Two straight lines cannot enclose a space. XI. All right angles are equal to one another. XII. If a straight line meets two straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines, being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles. It is admitted by Dr. Simson that this axiom is not self-evident, which all axioms ought to be. Accordingly he demonstrates the truth of it as a proposition in his notes, by help of five different propositions! Though not considered free from objection, the substitute for this axiom, given in Playfair's edition of Euclid, is to be preferred: viz., "If two straight lines intersect each other, they cannot be both parallel to the same straight line." The number of axioms is in this book limited to twelve; but Euclid has tacitly assumed the truth of various other axioms, which will be noticed in the sequel. PROP. I. PROBLEM. To describe an equilateral triangle upon a given finite straight line. Let AB be the given straight line. It is required to describe an equilateral triangle upon AB. From the centre A, at the distance AB, describe (Post. 3) the circle BCD. From the centre B, at the distance BA, describe the circle ACE. And from the point C, in which the circles cut one another, draw the straight lines (Post. 1) CA, CB, to the points A, B. Then ABC is an equilateral triangle. D B E Because the point A is the centre of the circle B CD, AC is equal (Def. 15) to AB. And because the point B is the centre of the circle ACE, BC is equal to BA. But it has been proved that CA is equal to AB. Therefore the two straight lines CA, CB, are each of them equal to AB. But things which are equal to the same thing are equal (Ax. 1) to one another. Therefore CA is equal to CB. Wherefore the three sides CA, AB, BC, are equal to one another. The triangle ABC is, therefore, equilateral. And it is described upon the given straight line AB. Q. E. F. From the construction of this problem, it is plain that, upon any given straight line, two equilateral triangles may be described, viz. one on each side. Exercise. To describe an isosceles triangle upon a given finite straight line, that shall have each of its sides double the base. From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line. It is required to draw from the point A a straight line equal to BC. From the point A to B draw (Post. 1) the straight line AB. Upon AB describe (I. 1) the equilateral triangle DAB. And produce (Post. 2) the straight lines DA, DB, to the points E and F. From the centre B, at the distance BC, describe (Post. 3) the circle CGH. And from the centre D, at the distance D G, describe the circle G K L. Then, the straight line AL is equal to BC. Because the point B is the centre of the circle CGH, BC is equal (Def. 15) to B G. And because the point D is the centre of the circle G KL, DL is equal to D G. But (Const.) DA, DB, parts of these equals, are equal. Therefore the remainder AL is equal to the remainder (Ax. 3) B G. But it has been shown that BC is equal to B G. Wherefore AL and BC are each of them equal to BG. And things that are equal to the same thing are equal (Ax. 1) to one another. Therefore the straight line AL is straight line BC. Wherefore from the given point A, AL has been drawn equal to the given straight line B C. K FI D. B G equal to the a straight line Q. E. F. The construction of this problem might be improved thus:- Join AB. Upon A B describe the equilateral triangle ABD. From the centre B, at the distance BC, describe the circle CG H. Produce DB to meet the circumference in G. The demon Exercise. Draw the figures, and show the application of the construction and demonstration to different positions of the point and the straight line; such as, when the given point is situated above the straight line or below the straight line; also, when in the straight line itself, at the extremities, or at any point between them. From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, of which AB is the greater. It is required to cut off from AB, the greater, a part equal to C, the less. From the point A draw (I. 2) the straight line AD equal to C. And from the centre A, at the distance AD, describe (Post. 3) the circle DEF. Then the part AE shall be equal to C. D C A. E B Because A is the centre of the circle DEF, AE is equal (Def. 15) to AD. But the straight line C is likewise equal (Const.) to AD. Therefore AE and C are each of them equal to AD. Wherefore the straight line AE is equal (Ax. 1) to C. Therefore from AB the greater of two given straight lines, a part AE has been cut off equal to C the less. Q. E. F. Exercise. To produce the smaller of two given straight lines, so that with the part produced, it shall be equal to the greater. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by these sides equal to one another, their bases, or third sides, are equal; the two triangles are equal; and their other angles are equal, each to each; viz., those to which the equal sides are opposite. D Let A B C, DEF be two triangles, which have the two sides AB, A C equal to the two sides DE, DF, each to each; viz., AB to DE, and AC to DF; and the angle BAC equal to the angle EDF. Then the base BC is equal to the base EF; the triangle ABC is equal to the triangle DEF; and the remaining angles of the one are equal to the remaining angles of the other, each to each; viz., those to which the equal sides are opposite; that is, the angle ABC is equal to the angle DEF, and the angle A CB to the angle DFE. B For, if the triangle ABC be applied to the triangle DEF, so that the point A may be on the point D, and the straight line AB upon the straight line DE. The point B shall coincide (that is, fall upon, so as to agree) with the point E, because AB is equal (Hyp.) to DE. And AB coinciding with DE, AC shall coincide with D F, because the angle |