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 1
... ( Eucl . i . 32. ) and consequently less than ACD . But the greater angle is subtended by the greater side ( Eucl . i . 19. ) ; therefore AD is greater than AC . In the same manner every other line drawn from A to BD may be shewn to be ...
... ( Eucl . i . 32. ) and consequently less than ACD . But the greater angle is subtended by the greater side ( Eucl . i . 19. ) ; therefore AD is greater than AC . In the same manner every other line drawn from A to BD may be shewn to be ...
Page 2
... ( Eucl . i . 20. ) ; therefore , & c . ( 3. ) Through a given point to draw a straight line which shall make equal angles with two straight lines given in position . Let P be the given point , and BE , CF the lines given in position ...
... ( Eucl . i . 20. ) ; therefore , & c . ( 3. ) Through a given point to draw a straight line which shall make equal angles with two straight lines given in position . Let P be the given point , and BE , CF the lines given in position ...
Page 5
... ( Eucl . i . 16. ) , and therefore greater than AGF , hence ( Eucl . i . 19. ) AG is greater than AF . In the same manner it may be shewn that AH is greater than AG . And from A there can only be drawn to BC two straight lines equal to ...
... ( Eucl . i . 16. ) , and therefore greater than AGF , hence ( Eucl . i . 19. ) AG is greater than AF . In the same manner it may be shewn that AH is greater than AG . And from A there can only be drawn to BC two straight lines equal to ...
Page 6
... ( Eucl . vi . 2. ) EP : PF :: ED : DA , i . e . in a ratio of equality . COR . If it be required to draw a line through P which shall be terminated by the given lines , and divided in any given ratio in P , draw PD parallel to AC and take ...
... ( Eucl . vi . 2. ) EP : PF :: ED : DA , i . e . in a ratio of equality . COR . If it be required to draw a line through P which shall be terminated by the given lines , and divided in any given ratio in P , draw PD parallel to AC and take ...
Page 7
... ( Eucl . vi . 2. ) DB : BE :: AB : BC , i . e . in the given ratio . ( 12. ) If from a given point any number of straight lines be drawn to a straight line given in position , to determine the locus of the points of section which divide ...
... ( Eucl . vi . 2. ) DB : BE :: AB : BC , i . e . in the given ratio . ( 12. ) If from a given point any number of straight lines be drawn to a straight line given in position , to determine the locus of the points of section which divide ...
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Common terms and phrases
ABCD angle ABC angles at F base centre chord circle ABC circles cut circles touch circumference describe a circle divided draw a line draw the diameter drawn parallel duplicate ratio equal angles equiangular Eucl extremities G draw given angle given circle given in position given line given point given ratio given square given straight line given triangle 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 point of bisection point of contact point of intersection point required radius rectangle right angles segments semicircle shewn tangent touching the circle trapezium triangle ABC
Popular passages
Page 14 - IF a straight line be divided into two equal, and also into two unequal parts ; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section.
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 230 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.
Page 327 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds.
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 212 - 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 123 - If from a point, without a parallelogram, there be drawn two straight lines to the extremities of the two opposite sides, between which, when produced, the point does not lie, the difference of the triangles thus formed is equal to half the parallelogram. Ex. 2. The two triangles, formed by drawing straight lines from any point within a parallelogram to the extremities of its opposite sides, are together half of the parallelogram.
Page 305 - Given the vertical angle, the difference of the two sides containing it, and the difference of the segments of the base made by a perpendicular from the vertex ; construct the triangle.
Page 247 - The perpendicular from the vertex on the base of an equilateral triangle is equal to the side of an equilateral triangle inscribed in a circle, whose diameter is the base.
Page 299 - AB be equal to the given bisecting line ; and upon it describe a segment of a circle containing an angle equal to the given angle.