1 XLI. A triangle having two equal sides may be called an isosceles triangle.. XLII. A triangle having the three sides unequal may be called a scalene triangle. XLIII. If a straight line move on two straight lines which make an angle, the surface thus described may be called a plane. XLIV. A plane may be supposed to be described on any two straight lines which make an angle, and enlarged to any extent. XLV. To enlarge a plane on the original lines may be called extending the plane. XLVI. Because a plane may be described on any two sides of a rectilineal triangle, and extended to the remaining side; therefore a rectilineal triangle is a plane figure. XLVII. A straight line may be applied to any point in a plane, and moved in the plane till it coincides with any other point therein. Therefore, if any two points be taken in a plane, the straight line between them lies wholly in the plane. XLVIII. If a part of a straight line is in a plane, the remainder is also in the plane; and if the line is produced, and the plane extended, the Îine remains in the plane. XLIX. If one plane be applied to another they will coincide, except so far as one extends beyond the other; and if they are extended, they will still coincide. L. A straight line which revolves in a plane about one of its own extremities till it returns to its original position, may be called a radius. LI. The point of revolution may be called a centre. LII. The boundary described by the opposite extremity of the radius may be called a circumference. LIII. And the figure thus described may be called a circle. LIV. Because the radius measures all straight lines from the centre to the circumference of the circle, therefore they are all equal. LV. A straight line passing through the centre of a circle, and terminated at either extremity by the circumference, may be called a diameter. LVI. A part of the circumference of a circle may be called an arc. LVII. If a line or figure is divided into two equal parts, it may be said to be bisected. DEFINITIONS Of terms which are used in Geometry. I. A request that cannot reasonably be denied, may be called a postulate. II. That which is proposed to be done or proved, may be called a proposition. III. A proposition to be done may be called a problem. IV. A proposition to be proved may be called a theorem. V. A theorem, whose evidence arises immediately from the consideration of the terms in which it is stated, may be called an axiom. VI. A proposition to assist in doing or proving another, may be called a lemma. VII. A supposition made in stating or performing a proposition, may be called an hypothesis. VIII. An inference which follows from one or more propositions, may be called a corollary. IX. A remark upon one or more propositions, tending to show their connection, their restriction, their extension, or the manner of their application, may be called a scholium. PROPOSITION I. THEOREM. If from two points in two straight lines, two other straight lines be drawn, the sum of the angles made by one of the lines is equal to the sum of the angles made by the other. From the points C and G in the straight lines AB and EF, let the straight lines CD and GH be drawn, the sum of the angles ACD, BCD is equal to the sum of the angles EGH, FGH. Let AB be applied to EF, so that C coincides with G; AB will coincide with EF (Def. 20.), and let CD fall as GD, then the angles ACD, BCD will coincide with the angles EGD and DGF. Because the angles EGD and DGF are composed of the three angles EGD, DGH, FGH, and the angles EGH, FGH also composed of the same three E D C B D H G F COROLLARY 1. As the straight lines CD and GH may be drawn on either side of the straight lines AB and EF; therefore the sum of the angles on one side of a straight line is equal to the sum of the angles on the other side. COR. II. Because a right angle is equal to half the sum of all the angles on one side of a straight line (Def. 27.); therefore all right angles are equal. COR. III. The angles which one straight line makes with another on the same side of it, are together equal to two right angles. PROP. II. THEOR. If, at a point in a straight line, two other straight lines on the opposite sides of it make the adjacent angles together equal to two right angles, these two straight lines shall be in the same straight line. At the point B in the straight line BD, let the straight lines AB, BC make the angles ABD, CBD together equal to two right angles; AB equal to ABD and CBD. Take away the common angle ABD, and the remaining angle DBE is equal to the remaining angle CBD; and therefore BE coincides with BC. Therefore, if from a point in a straight line, two other straight lines on the opposite sides of it make the adjacent angles together equal to two right angles, these two straight lines shall be in the same straight line. Which was to be proved. PROP. III. THEOR. If two straight lines cut one another, the vertical or opposite angles shall be equal. Let the two straight lines AB, CD cut one another in the point E, the angle AEC shall be equal to the angle BED, and the angle AED to the angle CEB. Because the straight line AE makes with CD the angles AEC, AED; these angles -B E are together equal to two right angles (3 Cor. I. Prop. I.). Again, because the straight line DE makes with AB the angles AED, BED; these also are to- A gether equal to two right angles (3 Cor. I. Prop. I.): and AEC, AED have been shewn to be equal to two right angles; therefore the angles AEC, AED are equal to the angles AED, BED. Take away the common angle AED, and the remaining angle AEC is equal to the remaining angle BED. In the same manner it can be shewn, that the angles AED, BEC are equal. Therefore, if two straight lines cut one another, the vertical or opposite angles shall be equal. Which was to be proved. COR. I. If two straight lines cut one another, the four angles which they make at the point where they cut, are together equal to four right angles. COR. II. All the angles made by any number of straight lines, which meet in one point, are together equal to four right angles. PROP. IV. THEOR. 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 which 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 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 the angle DFE. For, if the triangle ABC be applied to DEF, so that the point A may be on D, and the straight line AB upon DE, the point B shall coincide with the point E, because AB is equal to 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; but the point B coincides with the point E; wherefore, the base BC shall coincide with the base EF (Def. 21.). Wherefore, the whole triangle ABC shall coincide with the whole triangle DEF, and be equal to it; and the other angles of the one shall coincide with the remaining angles of the other, and be equal to them, viz. the angle ABC to the angle DEF, and the angle ACB to DFE. Therefore, 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; their bases shall likewise be equal, and the triangles be equal, and their other angles to which the equal sides are opposite shall be equal, each to each. Which was to be proved. PROP. V. THEOR. The angles at the base of an isosceles triangle are equal to one another; and, if the equal sides be produced, the angles upon the other side of the base shall be equal. Let ABC be an isosceles triangle, of which the side AB is equal to |