THE ELEMENTS OF EUCLID. BOOK I. Definitions. 1. A Point is that which hath no parts, or which hath no magnitude. 2. A line is length without breadth. 4. A straight line is that which lies evenly between its extreme points. 5. A superficies is that which hath only length and breadth. 6. The extremities of a superficies are lines. 7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. 8.“ A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction." 9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. N. B.' When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of those straight lines, and the other upon the other line : Thus the angle which is contained by the straight lines AB, CB, is named the angle ABC, СВА; that which is contained by AB, BD, is named the angle ABD, or DBA ; and that which is contained by DB, CB, is called the angle DBC, 0. CBD; but if there be only one angle at a point, it may be expressed by a letter placed at that point; as the angle at E.' 10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. 11. An obtuse angle is that which is greater than a right angle. 12. An acute angle is that which is less than a right angle. 13. “ A term or boundary is the extremity of any thing." ! 14. A figure is that which is inclosed by one or more boundaries. 15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. 16. And this point is called the centre of the circle. 17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 18. A semicircle is the figure contained by a diameter and the part of a circumference cut off by the diameter. 19.“ A segment of a circle is the figure contained by a straight line, and the circumference it cuts off." 20. Rectilineal figures are those which are contained by straight lines. 21. Trilateral figures, or triangles, by three straight lines. 22. Quadrilateral, by four straight lines. 23. Multilateral figures, or polygons, by more than four straight lines. 24. Of three-sided figures, an equilateral triangle is that which has three equal sides. 25. An isosceles triangle is that which has only two sides equal. a 26. A scalene triangle, is that which has three unequal sides. 27. A right-angled triangle, is that which has a right angle. 28. An obstuse-angled triangle, is that which has an obstuse angle Δ Δ Δ A 29. An acute-angled triangle is that which has three acute angles. 30. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles. 31. An oblong, is that which has all its angles right angles, but has not all its sides equal. 32. A rhombus, is that which has its sides equal, but its angles are not right angles. 33. 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. 34. All other four-sided figures besides these, are called Trapeziums. 35. Parallel straight lines, are such as are in the same plane, and which, being produced ever so far both ways, do not meet. POSTULATES. !. Let it be granted that a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced to any length in a straight line. 3. That a circle may be described from any centre, at any distance from that centre. AXIOMS. 1. Things which are equal to the same are equal to one ano. ther. 2. If equals be added to equals the wholes are equals 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Things which are double of the same, are equal to one another, 7. Things which are halves of the same, are equal to one another. 8. Magnitudes which coincide with one another, that is, which exactly fill the space, are equal to one another. 9. The whole is greater its part. M 10. Two lines cannot inclose a space. 11. All right angles are equal to one another. 12.“ 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." Proposition 1. 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 it. From the centre A, at the distance AB, describe (3 Postulate) the circle BCD, and from the centre B, at the distance BA, describe the circle ACE ; and from the point C, in which the circles cut one D B another, draw the straight lines, CA,CB (1 Post.) to the points A, B ; ABC shall be an equilateral triangle. Because the point A is the centre of the circle BCD, AC is equal (15 Definition) 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 CA, CB, are each of them equal to AB ; but things which are equal to the same are equal to one another (1st Axiom.) therefore CA is equal to CB ; wherefore CA, AB, BC, are equal to one another ; and the triangle ABC is therefore equilateral, and it is described upon the given straight line AB. Which was required to be done. Proposition II. Problem. 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, 't is required to draw from the point A a straight line equal to BC. From the point A to B draw (1 Post.) the straight line AB; and upon it describe (1. 1.) the equilateral triangle DAB, and produce (2 Post.) the straight lines DA, DB, to E and F; from the centre B, at the distance BC, describe (3 Post.) the circle CGH, and from the centre D, at the distance DG, describe the circle GKL. AL shall be equal to BC. Because the point B is the centre of the circle CGH, BC is equal (15 Def.) to BĠ; and because D is the centre of the circle GKL DL is equal to DG, and DA, DB, parts of them, are equal; therefore the remainder AL is equal to the remainder (3 Ar.) BG: But it has been shown, that BC is equal to BG ; wherefore AL and BC are each of them equal to BG; and things that are equal to the same are equal to one another ; therefore the straight line AL is equal to BC. Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC. · Which was to be done. Proposition III. Problem. 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, whereof 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 (2. 1.) the E B straight line AD equal 10 C; and from the centre A, and at the distance AD, describe (3 Post.) the circle DEF. And because A is the centre of the circle DEF, AE shall be equa. to AD; but the straight line C is likewise equal to AD; whence AE and C are each of them equal to AD; wherefore the straight line AE is equal to (1 Ar.) C, and from AB, the greater of two straight lines, a part AE has been cut off equal to C the less. Which was to be done. Proposition IV. Theorem. 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 those ; sides equal to one another ; they shall likewise have their bases, or third sides, equal ; and the two triangles shall be equal ; and their other angles shall be equal ; each to each, viz. those to wbich the equal sides are opposite. Let ABC, DEF be two triangles, which have the two sides AB, AC, 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, the base BC shall be A equal to the base EF; and the triangle * ABC to the triangle DEF; and the other angles to which the equal sides are opposite, shall be equal each to each, viz the angle ABC to the angle DEF, and the angle ACB to DFE. B For, if the triangle ABC be applied to DEF, so that the point A may be on D, and the stright line AB upon DE ; the point B shall coincide with the point E, becau e AB is equal :0 DE; and AB coinciding with DE, AC shall coincide with DF, because the angle BAC is equal to the angle EDF; wherefore also the point C shall coincide with the point F, because the straight line AC is equal to DF: Bu point B coincides with the point E; wherefore the base BC shall C : |