R. On Figures contained in Circles. HYPOTHESES. CONSEQUENCES. If a four-sided figure is con-Its opposite angles are to tained within a circle. If one. side of a four-sided III. 22. schol. 2. If a four-sided figure has its III. 36, cor. 2. opposite angles together area. gether equal to two right angles. The external angle is equal to the angle opposite to the internal adjacent angle. A circle may be described about it. PROBLEMS. A. Relating to straight Lines. From a given point to draw a straight line equal to a given finite straight line. Through a given point to draw a straight line parallel to a given straight line. From the greater of two given straight lines to cut off a part equal to the less. To bisect a given finite straight line. To divide a given finite straight line into two parts, so that the rectangle under the whole line and one segment shall be equal in area to the square on the other segment. To divide a given finite straight line into any given number of equal parts. B. Relating to rectilineal Angles. At a given point in a given straight line to form a rectilineal angle equal to a given rectilineal angle. To bisect a given rectilineal angle. From a given point in a given straight line to draw a perpendicular to that line. To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it. To draw a straight line through the extremity of a given straight line, perpendicular to the same. To bisect a given right angle. C. Relating to Triangles. Given three finite straight lines, of which any two together are greater than the third, to construct a triangle whose sides shall be respectively equal to the given lines. To construct an equilateral triangle upon a given finite straight line. Any two sides of a right-angled triangle being given, to find the third side. D. Relating to Parallelograms. To construct a parallelogram equal in area to a given triangle, and having an angle equal to a given rectilineal angle. Upon a given finite straight line to construct a parallelogram equal in area to a given triangle, and having an angle equal to a given rectilineal angle. To construct a parallelogram equal in area to a given rectilineal figure, and having an angle equal to a given rectilineal angle. Upon a given finite straight line to construct a parallelogram equal in area to a given rectilineal figure, and having an angle equal to a given rectilineal angle. To construct a rectangle under two given finite straight Upon a given finite straight line to construct a square. To construct a square equal in area to the difference of two To construct a square equal in area to a given rectilineal figure. To find geometrical values of 1, √2, √3, &c. E. Relating to Circles. To find the center of a given circle. From a given point, either without a given circle or in its circumference, to draw a straight line touching the cir cumference. To draw a tangent to a given circle, from a given point without it. To bisect a given arc. A segment of a circle being given, to describe the circle of which it is a segment. On a given finite straight line to describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle. To cut off from a given circle a segment which shall contain an angle equal to a given rectilineal angle. THE ELEMENTS OF EUCLID, WITH MANY ADDITIONAL PROPOSITIONS, AND EXPLANATORY NOTES; TO WHICH IS PREFIXED AN INTRODUCTORY ESSAY ON LOGIC. BY HENRY LAW, CIVIL ENGINEER. PART II. CONTAINING THE 4TH, 5TH, 6TH, 11TH, & 12TH BOOKS. London: JOHN WEALE, 59, HIGH HOLBORN. 1855. PREFACE. IN presenting to the public the Second Volume of Euclid's Elements, the Author feels that some explanation is required for the interval which has elapsed since the publication of the former volume. He feels that it will be only necessary for him to state that the work has been written in the intervals snatched from his professional duties, which have occupied so considerable a share of his time as to leave him no choice between delaying the publication, or hurrying it forward in an imperfect form; and he felt that he would best promote the interests of the public, as well as his own reputation, by the adoption of the former alternative. 15, ESSEX STREET, STRAND. 15th May, 1855. H. L. |