ART. 129. If a parallelogram and triangle be upon equal bases and between the same parallels, the parallelogram is double of the triangle, 100. ART. 130. If a parallelogram and triangle be between the parallels, and the base of the triangle double the base of the parallelogram, then the parallelogram and triangle are equal, ibid.

same

ART. 131. In a given parallelogram the complements of the parallelograms about the diagonal are equal to each other, 101.

ART. 132. If the square described upon one side of a triangle be equal to the squares described on the other sides of the triangle, taken together, the angle opposite to the first-mentioned side is a right angle, ibid.

ART. 133. Two perpendiculars cannot be drawn from the same point to the same right line, 102.

ART. 134. If two right lines be drawn from the same point to the same given right line, and if one of them be perpendicular, the other not, the perpendicular will fall at the side of the acute angle, ibid.

ART. 135. If two right lines be drawn from the same point to the same right line, and if one of them be perpendicular, the other not, the perpendicular is less than the other, ibid.

ART. 136. Each angle of an equal-sided triangle is one-third of two right angles, 103.

ART. 137. In a right-angled triangle whose sides about the right angle are equal, the remaining angles are each equal to half a right angle, ibid.

ART. 138. If from a point within a triangle right lines be drawn to the extremities of any side, these are together less than the other two sides of the triangle, but contain a greater angle, ibid.

ART. 139. In a triangle which has two equal sides, the right line drawn from the vertex of the angle between them, perpendicular to the third side, divides that angle, and also the third side into two equal parts, respectively, 104.

ART. 140. In a triangle which has two equal sides, a right line dividing the angle between them into two equal parts, if drawn to the third side, will divide it into two equal parts, and also be perpendicular to it, ibid.

ART. 141. In a triangle which has two equal sides, a right line drawn from the vertex of the angle between them to the middle point of the third side, divides the opposite angles into two equal parts, and is also perpendicular to the third side, ibid.

ART. 142. In a circle, if two chords intersect, which are not both diameters, they do not divide each other into equal parts, 105.

ART. 143. If from a point within a circle there can be drawn more than two equal right lines to the circumference, this point is the centre of the circle, ibid.

ART. 144. If a right line be drawn from the point of contact nearer the centre than the tangent, it cuts the circle, 106.

ART. 145. At the same point of the same circle but one right line can be drawn touching the circle, ibid.

ART. 146. If one side of a quadrilateral figure inscribed in a circle be produced, the external angle thus formed is equal to the internal remote angle of the quadrilateral, ibid.

ART. 147. The difference between the squares of any two unequal

right lines is equal to the rectangle under the sum of the lines and their difference, 107.

ART. 148. If a right line be equally divided, and produced to any point, then, the rectangle under the whole line and the produced part, is equal to the difference between the square of half the original line, and the square of the line made up of that half and the produced part, ibid.

ART. 149. If a right line be divided into two equal parts, and into two unequal parts, the rectangle under the unequal parts is equal to the difference between the square of half the line, and the square of the in

termediate part, 108.

ART. 150. If a right line be divided equally and unequally, the rectangle under the equal parts is greater than the rectangle under the unequal parts, ibid.

ART. 151. If a right line be cut equally and unequally, the sum of the squares of the unequal parts is greater than the sum of the squares of the equal parts,—or, in other words, greater than twice the square of half the line, ibid.

ART. 152. If two equal triangles have an angle in the one which together with the angle in the other is equal to two right angles, the sides about these angles are reciprocally proportioned, 109.

ART. 153. If two triangles have an angle in the one which together with the angle in the other is equal to two right angles, and if the sides about the given angles are reciprocally proportional, then these two triangles are equal, ibid.

ART. 154. If four right lines be proportionals, the parallelogram under the extremes is equal to an equi-angular parallelogram under the means, ibid.

ART. 155. If two chords of a circle intersect, the rectangle under the segments of one is equal to the rectangle under the segments of the other, 110.

ART. 156. If from a point without a circle a secant and a tangent be drawn to the circle, the square of the tangent is equal to the rectangle under the whole secant and its external part, ibid.

ART. 157. If from a point without a circle two right lines be drawn, one cutting the circle, the other meeting it at any point, and if the rectangle under the whole secant and its exterior part be equal to the square of the line which meets the circle, then, this line is a tangent, ibid.

ART. 158. A right line parallel to any side of a triangle, and meeting the other sides produced, cuts them so that the segments of the one have the same ratio as the segments of the other, 111.

ART. 159. A right line cutting any two produced sides of a triangle, so as to make the segments of the one proportional to the corresponding segments of the other, is parallel to the third side, 112.

ART. 160. If a right line parallel to any side of a triangle divides either of the other sides equally, it will divide both equally, ibid.

ART. 161. If a right line divide two sides of a triangle equally, it is parallel to the third, ibid.

ART. 162. If several right lines be drawn parallel to a side of any triangle, and meeting the other sides, the segments of one of these sides have the same ratio as the corresponding segments of the other, ibid.

ART. 163. A right line dividing any angle of a triangle into two equal parts, divides the opposite sides into segments which have the same ratio as the sides which contain the divided angle, 113.

ART. 164. If a right line drawn from any angle of a triangle to the opposite side divide that side into parts, which have the same ratio as the corresponding sides about the given angle, this angle is divided into two equal parts, 114.

ART. 165. In a right-angled triangle, a perpendicular drawn from the vertex of the right angle to the opposite side, is a mean proportional between the segments of this side, ibid.

ART. 166. In such a triangle as above described, each side about the right angle is a mean proportional between the corresponding segment and the side opposite the right angle, 115.

ART. 167. In such a triangle as above described, the three sides and the perpendicular are proportionals, ibid.

ART. 168. In the same circle, the angles, whether at the centre or circumference, have the same ratio to each other as the arches they stand on, ibid.

ART. 169. In a circle, any angle is to four right angles as the arch on which it stands to the whole circumference, ibid.

ART. 170. In a series of four proportional right lines, the second is to the first as the fourth to the third, 116.

ART. 171. In a series of four proportional right lines, the first is to the third as the second to the fourth, ibid.

ART. 172. In a series of four proportional right lines, the sum of the first and second is to the second as the sum of the third and fourth is to the fourth, ibid.

ART. 173. In a series of four proportional right lines, the difference between the first and second is to the second as the difference between the third and fourth is to the fourth, 117.

ART. 174. In a series of four proportional right lines, the first is to the sum of the first and second as the third is to the sum of the third and fourth, ibid.

ART. 175. In a series of four proportional right lines, the first is to the difference between the first and the second as the third is to the difference between the third and the fourth, 118.

POPULAR SYSTEM OF GEOMETRY.

A SPACE may have length, breadth, and depth;, or length and breadth only; or length alone. These are called its dimensions.

SOLIDS are magnitudes which have the three dimensions of space: length, breadth, and depth. Thus a die, or any other magnitude having these three dimensions, is a solid.

SURFACES are magnitudes which have but two dimensions length and breadth. Thus the face of a die is a surface; having length and breadth, but not depth,-else it would be, not the face of the die, but part of the solid.

LINES are magnitudes which have but one dimension : length. Thus the edge of a die is a line; having length, but not breadth,-else it would be, not the edge of the die, but part of a surface.

A Right Line is a line which is perfectly even or straight throughout its whole extent*.

A Plane Surface is a surface which is perfectly even or· flat throughout its whole extent.

A Mathematical Solid is a solid with its surface or surfaces either plane, or, if not, perfectly smooth, throughout the whole extent of each particular surface.

GEOMETRY is that Science which treats of Right Lines, Plane Surfaces, and Mathematical Solids, with regard to magnitude.

PLANE GEOMETRY is that Science which treats of the Right Line and Circle. [See NOTE A.]

* A right line and a straight line are, in geometry, synonymous terms, but it is better to use the former, which always indicates a perfectly straight line, whilst the word "straight" is often applied in common speech to lines which are not perfectly straight, but only nearly and visibly so.

B