Geometrical Problems Deducible from the First Six Books of Euclid, Arranged and Solved: To which is Added an Appendix Containing the Elements of Plane Trigonometry ... |
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Page ii
... proportion , and from that point also a line equal to the mean proportional be inclined at any angle ; the lines joining the extremity of this line and of the proportionals will contain equal angles . 16. To trisect a right angle . € 4 ...
... proportion , and from that point also a line equal to the mean proportional be inclined at any angle ; the lines joining the extremity of this line and of the proportionals will contain equal angles . 16. To trisect a right angle . € 4 ...
Page viii
... mean proportional between the diameters of the circles . 37. If two circles touch each other externally , and the line join- ing their centres be produced to the circumferences ; and from its middle point as a centre with any radius ...
... mean proportional between the diameters of the circles . 37. If two circles touch each other externally , and the line join- ing their centres be produced to the circumferences ; and from its middle point as a centre with any radius ...
Page ix
... proportional . 40. If three circles , whose diameters are in continued proportion , touch each other internally ... mean proportional between the lines drawn from the point of intersection with the circumference to the extremities of the ...
... proportional . 40. If three circles , whose diameters are in continued proportion , touch each other internally ... mean proportional between the lines drawn from the point of intersection with the circumference to the extremities of the ...
Page x
... mean proportional between its segments . 54. Two circles being given in magnitude and position ; to find a point in the circumference of one of them , to which if a tangent be drawn cutting the circumference of the other , the part of ...
... mean proportional between its segments . 54. Two circles being given in magnitude and position ; to find a point in the circumference of one of them , to which if a tangent be drawn cutting the circumference of the other , the part of ...
Page xii
... mean proportional to two dis- tances from the centre in the same straight line ; the lines drawn from their extremities to any point in the circumference will have the same ratio that the distances of these points from the circum ...
... mean proportional to two dis- tances from the centre in the same straight line ; the lines drawn from their extremities to any point in the circumference will have the same ratio that the distances of these points from the circum ...
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Common terms and phrases
ABCD angle ABC base bisects the angle centre chord circle ABC circles cut circles touch circumference describe a circle divided draw any line drawn parallel duplicate ratio equal angles equiangular Eucl extremities given angle given circle given in position given line given point given ratio given straight line given triangle inscribed intercepted isosceles triangle Join AE Join BD Let AB Let ABC let fall line given line joining line required lines be drawn lines drawn mean proportional opposite side parallel to BC parallelogram pendicular perpendicular be drawn point of bisection point of contact point of intersection quadrant radius rectangle contained right angles right-angled triangle segments semicircle shewn tangent touches the circle trapezium triangle ABC vertex vertical angle
Popular passages
Page 124 - If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally; and if the sides, or the sides produced, be cut proportionally, the straight '.line which joint the points of section, shall be parallel to the remaining side of the triangle.
Page xiii - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle,. shall be equal to the square of the line which touches it.
Page 158 - Iff 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...
Page 160 - Upon a given straight line, to describe a segment of a circle, containing an angle equal to a given angle. Let AB be the given straight line, and C the given angle ; it is required to.
Page 230 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.
Page 157 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.
Page 33 - FC ; (ax. 1.) and FA, FB, FC are equal to one another : wherefore the circle described from the centre F, at the distance of one of them, will pass through the extremities of the other two, and be described about the triangle ABC.
Page xxv - ... the squares of the diagonals, is equal to the sum of the squares of the bisected sides together with four times the square of the line joining those points of bisection.
Page 248 - If an equilateral triangle be inscribed in a circle, and the adjacent arcs cut off by two of its sides be bisected, the line joining the points of bisection shall be trisected by the sides.
Page 355 - The sine of an angle is equal to the sine of its supplement. The sine rule Consider fig.