ART. 26. Triangles upon equal bases and between the same parallels are equal, ibid.

ART. 27. Equal triangles upon the same base, and upon the same side of it, are between the same parallels, 25.

ART. 28. Parallelograms which have equal bases and equal altitudes are equal, 26.

ART. 29. Triangles which have equal bases and equal altitudes, are equal, ibid.

ART. 30. Equal triangles on equal bases have equal altitudes, 27.

ART. 31. If a parallelogram and a triangle be upon the same base, and between the same parallels, the parallelogram is double of the triangle, ibid.

ART. 32. All the sides of a square are equal, and all its angles right angles, 28.

ART. 33. Squares described upon equal right lines are equal, ibid. ART. 34. If two squares be equal, their sides are equal, 29. ART. 35. If any side of a triangle be produced, the external angle is equal to the two farther internal angles taken together, ibid.

ART. 36. The external angle of any triangle is greater than either of the two farther internal angles, ibid.

ART. 37. The three internal angles of any triangle taken together are equal to two right angles, ibid.

ART. 38. Any two angles of a triangle are together less than two right angles; and if any angle of a triangle be obtuse or right, the other two are acute also, if two angles of a triangle be equal, they are both acute, ibid.

ART. 39. If two triangles have two angles in the one equal respectively to two angles in the other, the third angle of the one is also equal to the third angle of the other, 30.

ART. 40. The four internal angles of any four-sided rectilineal figure, taken together, are equal to four right angles, ibid.

ART. 41. In any triangle, if one side be greater than another, the angle opposite to that greater side is greater than the angle opposite to the lesser, ibid.

ART. 42. In any triangle, if one angle be greater than another, the side opposite to that greater side is greater than the side opposite to the lesser, ibid.

ART. 43. Any two sides of a triangle are together greater than the third side, 31.

ART. 44. If two triangles have two sides of the one equal respectively to two sides of the other, but the angle contained by each pair of these sides unequal the base of that triangle whose given sides contain the greater angle is greater than the base of the other triangle, ibid.

ART. 45. If two triangles have two sides of the one equal respectively to two sides of the other, but their bases unequal, the vertical angle of that triangle which has the greater base is greater than the vertical angle of the other triangle, 32.

ART. 46. If two triangles have two angles of the one equal respectively to two angles of the other, and a side of the one triangle equal to a corresponding side of the other-these triangles are in every respect equal to each other, ibid.

ART. 47. In any right-angled triangle the square described on the side

opposite the right angle is equal to the squares described on the sides containing the right angle, taken together, 33.

ART. 48. The centre of a circle falls within the circumference, 41.
ART. 49. A circle cannot have more than one centre, ibid.

ART. 50. A right line perpendicular to a chord through its middle point, will pass, if produced, through the centre of the circle, 42.

ART. 51. In a circle, a right line from the centre perpendicular to a chord divides it into two equal parts, 43.

ART. 52. In a circle, a right line, through the centre dividing a chord which does not pass through the centre into equal parts, is perpendicular to it, ibid.

ART. 53. A right line cannot meet the circumference of a circle in more than two points, ibid

ART. 54. If a right line meet a circle in two points, that part of it between the points lies wholly within, and those parts of it not between the points lie wholly without the circle, 44.

ART. 55. In a circle, a perpendicular to a diameter, at its extremity, meets the circle in but one point, ibid.

ART. 56. If a right line be a tangent to a circle, the radius drawn to the point of contact is perpendicular to the tangent, 45.

ART. 57. If a right line be a tangent to a circle, the perpendicular to it at the point of contact will, if produced sufficiently, pass through the centre, 46.

ART. 58. The diameter of a circle is the greatest chord which can be drawn in it, ibid.

ART. 59. Chords equally distant from the centre of a circle are equal, 47.

ART. 60. Equal chords in a circle are equally distant from the centre, ibid.

ART. 61. In a circle the chord which is nearer to the centre is greater than that which is farther off, 48.

ART. 62. From any point which is not the centre of a circle, the greatest right line that can be drawn to the circumference is that which actually passes through the centre, ibid.

ART. 63. From any point which is not the centre of a circle, the least right line that can be drawn to the circumference is that which does not, but which would, if produced, pass through the centre, 49.

ART. 64. If from any point not the centre of a circle two right lines be drawn to the circumference, which make with that drawn through the centre, equal angles opening towards the same parts, these two lines are equal, ibid.

ART. 65. If from any point not the centre of a circle, but either within or on the circumference two right lines be drawn to the circumference, which make with the right line drawn actually through the centre, unequal angles opening towards the same parts, that which makes the smaller angle is greater than the other, 50.

ART. 66. If from a point outside a circle two right lines be drawn to the concave part of the circumference, which make with the right line through the centre unequal angles, that which makes the smaller angle is greater than the other, 51.

ART. 67. If from a point outside a circle two right lines be drawn to the convex part of the circumference, which make with the right line through the centre unequal angles, that which makes the smaller angle is less than the other, 52.

ART. 68. More than two equal right lines cannot be drawn to the circumference of a circle from any one point but the centre, ibid.

ART. 69. The opposite angles of a four-sided rectilineal figure inscribed in a circle are together equal to two right angles, 53.

ART. 70. The angles in the same segment of a circle are equal, 55.

ART. 71. The angle in the segment of a circle is half of the external angle at the centre whose sides are terminated in the extremities of the same segment of the circumference, ibid.

ART. 72. The angle in a semicircle is a right angle, 56.

ART. 73. The angle in a segment greater than the semicircle is less than a right angle, ibid.

ART. 74. The angle in a segment less than a semicircle is greater than a right angle, ibid.

ART. 75. If a tangent and a chord of a circle be drawn from the same point, the angle between them is equal to the angle in the alternate segment, 57.

ART. 76. If two different circles meet one another, they cannot have the same centre, 58.

ART. 77. One circle cannot meet another in more than two points, ibid. ART. 78. If one circle meet another in two points, one portion of the former will be wholly within and the other wholly without the latter circle, ibid.

ART. 79. If two circles having their centres at the two extremities of a given finite right line pass through the same point on that finite line, they meet in that point, but in no other, ibid.

ART. 80. If two circles touch, the right line joining the centres, if produced, will pass through the point of contact, 59.

ART. 81. Equal circles have equal diameters, 60.

ART. 82. In equal circles equal chords cut off equal arches, ibid.

ART. 83. In equal circles equal arches have equal bases, ibid.

ART. 84. In equal circles, equal angles, whether they be at the centres

or the circumferences, stand upon equal arches, 61.

ART. 85. In equal circles, the angles which stand upon equal arches are equal, whether they be at the centres or the circumferences, 62.

ART. 86. These latter four articles, it is evident, are true for the same circle as well as equal ones, 63.

ART. 87. In a circle parallel chords intercept equal arches, ibid.

ART. 88. In a circle, the chords joining the extremities of equal arches, and not intersecting, are parallel, 64.

ART. 89. If there be two right lines, one of which is divided into any number of parts, the rectangle under the two lines is equal to the sum of the rectangles under the undivided line and the several parts of the divided line, 65.

ART. 90. If a right line be divided into any two parts, the square of the whole line is equal to the sum of the rectangles under the whole line and each of the parts, ibid.

ART. 91. If a right line be divided into any two parts, the rectangle under the whole line and either part is equal to the square of this part together with the rectangle under the parts themselves, 66.

ART. 92. If a right line be divided into any two parts, the square of the whole line is equal to the sum of the squares of the parts together with twice the rectangle under the parts, ibid.

ART. 93. The square of a right line is equal to four times the square of its half, 67.

ART. 94. Parallelograms which have equal altitudes, have to each other the same ratio as their bases, 75.

ART. 95. Parallelograms which have equal bases, have to each other the same ratio as their altitudes, 76.

ART. 96. Two equal parallelograms, which are also equi-angular, have the sides about their equal angles reciprocally proportional, 77.

ART. 97. Two equi-angular parallelograms which have their sides about their equal angles reciprocally proportional are equal, 78.

ART. 98. Equal parallelograms have their bases and altitudes reciprocally proportional, ibid.

ART. 99. Parallelograms which have their bases and altitudes reciprocally proportional are equal, 79.

ART. 100. If four right lines be proportionals, the rectangle under the extremes is equal to the rectangle under the means, ibid.

ART. 101. If there be four right lines, and the rectangle under any two of them equal to the rectangle under the remaining ones, these right lines are four proportionals, ibid.

ART. 102. If three right lines be proportionals, the rectangle under the extremes is equal to the square of the mean, 80.

ART. 103. If there be three right lines, and the rectangle under any two of them equal to the square of the third, these three right lines are proportionals, ibid.

ART. 104. Triangles which have equal altitudes are to each other as their bases, 81.

ART. 105. Triangles which have equal bases are to each other as their altitudes, ibid.

ART. 106. Equal triangles which have also an angle in the one equal to an angle in the other, have the sides about these equal angles reciprocally proportional, ibid.

ART. 107. Two triangles which have an angle in one equal to an angle in the other, and have also the sides about these equal angles reciprocally proportional, are equal, 82.

ART. 108. Equal triangles have their bases and altitudes reciprocally proportional, ibid.

ART. 109. Triangles which have their bases and altitudes reciprocally proportional are equal, ibid.

ART. 110. If a right line be drawn parallel to any side of a triangle, and meeting the other sides, the segments of one of these sides have the same ratio as the segments of the other, 83.

ART. 111. If a right line meets the sides of a triangle, so that the segments of one side shall have the same ratio as the segments of the other, and if the corresponding segments be at the same side of the

intersecter, this right line is parallel to the remaining side of the triangle, 83.

ART. 112. If the angles of one triangle be respectively equal to the angles of another, the three sides of one triangle have to the corresponding sides of the other, respectively, the same ratio, ibid.

ART. 113. If the three sides of any triangle have respectively to the three sides of another the same ratio, the angles of one triangle are respectively equal to the angles of the other, 87.

ART. 114. If two sides of any triangle have respectively to two sides of another the same ratio, and likewise the angles contained by each pair of sides equal, the other angles of the triangles will be also respectively equal, ibid.

ART. 115. If two triangles have an angle in the one equal to an angle in the other; and if the sides containing a second angle in the former have, respectively, the same ratio to the sides containing a second anglę in the latter; and if likewise the third angles of the triangles are either both acute or both obtuse, or both right; all the angles of these triangles are respectively equal to each other, 88.

ART. 116. Equi-angular parallelograms have to each other the ratio compounded of the ratios of the sides about the equal angles, 91.

ART. 117. Triangles which have an angle in the one equal to an angle in the other, have to one another a ratio compounded of the ratios of the sides about the equal angles, ibid.

ART. 118. If the angles of one triangle be respectively equal to those of another, these triangles have to each other a ratio duplicate of that which their corresponding sides have to each other, 92.

ART. 119. In equal circles, angles at the centres have to each other the same ratio as the arches on which they stand, 93.

ART. 120. In equal circles, angles at the circumferences have to each other the same ratio as the arches on which they stand, 94.

ART. 121. Every triangle which has its three sides equal has also its three angles equal, 95.

ART. 122. Every triangle which has its three angles equal has also its three sides equal, ibid.

ART. 123. Where several right lines meeting another right line at the same point, make angles with it, these angles are altogether equal to two right angles, ibid.

ART. 124. Two right lines intersecting each other, make angles which taken together are equal to four right angles, ibid.

ART. 125. If several right lines intersect one another in the same point,

all the angles taken together are equal to four right angles, 96.

ART. 126. A triangle which has two of its sides equal, if these equal sides be produced, will have the angles beneath the third side equal to each other, ibid.

ART. 127. If two right lines be parallel to the same right line, they are parallel to one another, 98.

ART. 128. If a right line intersect two right lines, and make the two internal angles at the same side of the intersecting line together less than two right angles, these two latter right lines will meet if produced sufficiently, 99.