GEOMETRICAL MEMORANDA. 41 The exterior angle of any triangle is the angle contained between any one side and either of the adjacent sides produced, and the adjacent angle of the exterior angle is the angle contained by the side which has been produced and the side falling on it. The exterior angle and its adjacent angle are together equal to two right angles, because one line falling on another line is equal to two right angles. The exterior angle of any triangle is equal to the sum of the two opposite angles; for let the exterior angle be called E, and the adjacent angle A, and the two other angles of the triangle B and C ; E+ A = the sum of two right angles; and A + B + C = to the sum of two right angles; if from these two equals the angle A, common to both, be subtracted, the remainders will be equal; since any exterior angle of a triangle is equal to the two interior opposite angles, therefore any such exterior angle is greater than either of the interior opposite angles. In any triangle the greater angle is subtended by the greater side, and the smaller angle by the smaller side, and therefore the greater side subtends the greater angle, and the smaller side subtends the smaller angle; and the equal angles of a triangle are subtended by equal sides, and the equal sides subtend equal angles. In any two adjacent triangles, with equal sides containing unequal angles, the greater angle will be subtended by the greater side, and the lesser angle by the lesser side. Any two triangles having the three sides of the one equal to the three sides of the other, are equal, equilateral, and equiangular. Any two triangles having each an equal angle contained by equal sides, are equal. Any two triangles having each two equal angles on equal bases, are equal. Any two triangles of equal altitude, and on equal bases, are equal. Equal triangles upon the same base, and on the same side of it, are between the same parallels. If any two sides of any triangle are intersected by a line parallel to the third side, the intercepted segments will be proportionals; and the one side will be to either of its segments as the other side is to the segment opposite to the segment of the other side; and the parallel sides will be to each other as the segments of either of the other sides are to each other; and the three sides of the larger triangle will be proportional to the three sides of the lesser triangle. It is to be observed that if the parallel to the base, instead of being drawn within the tri angle, had been drawn without, so as to intercept the two sides produced, the three sides of the two triangles thus formed would be proportionals, because they would be similar triangles having two opposite angles equal, and two sides parallel. If any angle of a triangle be divided into equal parts by a straight line intersecting the opposite base, then of the two sides containing the angle, the one side will be to its adjacent segment of the base as the other side is to the other segment, and the segments of the base shall have the same ratio which the other sides of the triangle have to one another. Any two triangles having an equal angle in each, contained by proportional sides, are similar to each other. Any two triangles having two angles of the one equal to two angles of the other, are similar triangles, because the three angles of the one triangle are equal to the three angles of the other; therefore two rectangular triangles, having, besides the two right angles, two other equal angles, are similar triangles. Two triangles having three sides of the one perpendicular to the three sides of the other are similar to each other; the equal angles will be contained by the homologous sides, and any homologous side in the one triangle will be perpendicular to its homologous side in the other triangle. In order to study the nature of similar triangles, and the proportions of homologous sides, the equal angles should be designated by the same letters in different characters. In triangles, the sides containing equal angles, being proportionals; to divide any line into parts proportional to those of another line, it is only requisite to make the two lines contain an angle between them, and to join their other extremities by a third line; then parallel to this third line draw lines through the points of division of the line already divided, to intersect that which it is required to divide. Of some of the Properties of the Circle.-In a circle, any ordinate to the diameter, that is, any perpendicular from the diameter to the circumference, is a mean proportional to the segments into which the diameter is divided by the perpendicular; let the perpendicular be O, and the segments into which the diameter is divided be A and B; then 02 A:0:0: B; and B; or B: 0 :: 0 : A, and In any circle let a diameter be drawn; at right angles to it, and from one of its extremities let a straight line be drawn, and from the centre of the circle let a third line be drawn to intersect the perpendicular to the diameter; such perpendicular will GEOMETRICAL MEMORANDA.-THE CIRCLE. 43 be a mean proportional between the segments of the line last drawn, one of which is inside the circle and the other outside. If any straight line touch a circle, without cutting it, it shall be perpendicular to the radius. In a circle, a perpendicular bisecting a chord and produced through the circle will be the diameter, and the middle of the diameter will be the centre of the circle. Equal straight lines in a circle are equally distant from the centre; and those equally distant are equal to one another; the greatest line is the diameter, and of all others, that which is nearer the centre is always greater than that more remote. The angle at the centre of a circle is double the angle at the circumference, upon the same base, or upon the same arc of the circle. The angles in the same segment of a circle are all equal to one another. The opposite angles of any quadrilateral inscribed in a circle are together equal to two right angles. In equal circles, equal arcs are subtended by equal chords; and equal chords cut off equal arcs;. and equal angles stand upon equal chords, whether they be at the circumferences or at the centres, and any arc is equally bisected by equal chords. In a circle, the angle contained by a semicircle is a right angle; and an angle in a segment greater than the semicircle is less than a right angle; and an angle in a segment less than a semicircle is greater than a right angle. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle. On a given line to describe a segment which shall contain an angle equal to a given angle. If the given angle is a right angle, the required segment will be a semicircle; and it is only requisite to bisect the given line, and from the point of bisection to describe a semicircle with half the line as radius. But if the given angle be less or greater than a right angle at either end of the given line, which we will call A, make an angle equal to the given angle, and call the second line B; from the point where A and B meet, draw a line perpendicular to B, which call C; bisect A, and from the point of bisection raise a perpendicular to intersect C; take this point as a centre, and take for radius the distance on C between this centre and B, and describe a circle; the circle will be divided into two segments by the line A ; if the given angle be less than a right angle, it will be contained by the greater segment, and if less than a right angle, it will be contained by the lesser segment. From a given circle to cut off a segment which shall contain an angle equal to a given angle. Draw any line tangent to the given circle; at the point of contact describe an angle equal to a given angle, making the tangent one side of it; produce the other side to cut through the circle, which will thus be divided into two segments, the greater of which will contain the given angle if less than a right angle, and the lesser segment if the given angle is greater than a right angle. If two straight lines cut one another within a circle, the rectangle contained by the segments of one line will be equal to the rectangle of the segments of the other line. In any circle the square of the diameter is equal to the square of the chords drawn from the two extremities of the diameter to the circumference. In any quadrilateral inscribed in a circle, the rectangle contained by the diagonals is equal to the rectangles contained by the opposite sides of the quadrilateral. If from any point outside of a circle two straight lines be drawn, one of which cuts the circle and the other touches it, then the rectangle under the line which cuts the circle, and that part of it outside the circle, is equal to the rectangle under the line which touches the circle. Of Rectangles. In any right angled triangle, the square of the side subtending the right angle is equal to the square under the sides containing the right angle. If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line. If a straight line be divided into any two parts, the rectangle contained by the whole and one part is equal to the rectangle contained by the two parts, together with the square of the aforesaid part. If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line, which is made up of the whole line. and that part. In obtuse angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than GEOMETRICAL MEMORANDA,-PERPENDICULARS. 45 the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse angle. In every triangle the square of the side subtending an acute angle is less than the squares of the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the acute angle and the perpendicular let fall upon it from the opposite angle. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals. If three straight lines be proportionals, the rectangle contained by the extremes is equal to the square of the mean, and if the rectangle contained by the extremes be equal to the square of the mean, the three straight lines are proportionals. PROBLEM 2. Fig. 16. On the ground from a given point on a given line to set off a perpendicular. Double a long, light cord, fasten the ends at D and F, the middle of the cord held towards G with an equal strain on both ends of the line, will give the perpendicular GC; the chain may be used for doing this, or in the following manner: make DC equal to 30 links, fix one end of the chain at C, and the 90 at D, and fix 40 (not 60) towards G; GC will be the perpendicular required; G D will be equal to 50, as will also GF, if the work is correctly done. PROBLEM 3. Fig. 17. To raise a perpendicular from a point near the end of a line. Let A B be the given line, and P the given point. From any convenient point C, with the distance CP, describe a circle; through A and E draw AE; EP will be the perpendicular required. To do this on the ground, from the point P set off 50 feet or links at D, just as you may happen to be using Gunter's chain, or a chain or tape of 100 feet, and 50 more to E; stick an arrow at E and P, pass the loops of a line one 100 long over each of them, as E and P, and take the middle of the line towards G, which fix exactly with another arrow, holding the line quite tight; EGP will be an equilateral triangle. Now set |