Semicircle

“ The semicircle test requires that a bay must comprise at least as much water area within its closing line as would be contained in a semicircle with a diameter equal to the length of the closing line. Unquestionably the 24-mile closing line together with the semicircle test now represents the position of the United States. ”

“ In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Let ABCD be a circle, of which BC is a diameter and E the centre; and draw CA, dividing the circle into the segments ABC, ADC, and join BA, AD, DC: the angle in the semicircle BAC shall be a right angle; but the angle in the segment ABC, which is greater than a semicircle, shall be less than a right angle ”

“ After having drawn a semicircle and its diameter, draw a perpendicular radius. This radius forms a right angle with the diameter, and cutting the semicircle in two equal parts or quarters of circles, leaves 90 degrees for each of them. If 90 degrees of a circle make a right angle, 45 degrees will make half a right angle, &c.
Or by another method. A radius, if made a cord of the semicircle, will allow three cords, each of which will contain 60 degrees; halve these arcs, and you have arcs of 30 degrees; halve the arcs of 30 degrees, and you have 15 degrees ”

“ A radius drawn from the centre to the number 30 on the graduated semicircle, will form an angle of 30 degrees with the diameter of the semicircle. And so for any other number of degrees. It will be seen that any number of degrees less than 90 will make an acute angle, and more than 90 degrees will form an obtuse angle thus, in fig. 4, 30 degrees form an acute angle, and the remaining 150 degrees of the half circle form an obtuse angle. ”

“ From the divisions on the circle GOA raise verticals to semicircle FOB, which will divide it in the same way. Divide the smaller semicircle EOC into the same number of parts as the others, 165 which divisions will serve as guiding points in drawing the curves of the dome that are drawn towards D, but the shading must assist greatly in giving the effect of the recess. ”

“ The first excellence of a bridge is now allowed to be strength; and it has been asserted, that a semi-ellipsis has less strength than a semicircle. To this he first answers, that granting this position for a moment, the semi-ellipsis may yet have strength sufficient for the purposes of commerce. This grant, which was made but for a moment, needed not to have been made at all; for, before he concludes his letter, he undertakes to prove, that the elliptical arch must, in all respects, be superiour in strength to the semicircle. ”

“ Find and mark the centre line A to B in Fig. 1 of drawings No. 18. Set the compasses to a radius from A to B, and describe, with A for the centre, the semicircle shown at the top of the Fig. Fit a 1-in. Centre-bit to the Brace, and drill a hole with A for the centre, taking care that it goes vertically through at right angles, and that there is no splitting when the bit is nearly through. With the small turning saw, cut round the semicircle at right angles, and, with the chisel, shave off and round the two corners that spring from the semicircle, also at right angles. ”

“ With a radius of one inch describe an inner semicircle (5-6) parallel to the outer one. Again, with a radius of one inch describe a third semicircle, parallel to the other two. Set the compass at half the radius and divide each semicircle into six equal parts. Connect these points of intersection by straight lines (9-10) . Make [Pg 45] a stencil that will fit in one of these sections. Using the stencil, draw the same figure in each section. Carefully cut out the stenciled space. Next lay the construction paper on the Japanese rice paper and trace on it the stencil design. ”

“ The Circles are all fixed in a plain and movable flat compasses (or better, a movable semicircle) are used for fixing any two positions; (2) There is a “double projection” of each logarithmic line “inlarged on a Plaine,” one fixed, the other movable, as shown in his first figure on the title-page, a single index only being used ”

“ And in lyke sorte shall you iudg of the ãgle made in a semicircle, whiche is distinct frõ the angles of the semicircle. For in this figure, the angles of the semicircle are those angles which be by A. and D, and be made of the right line A.D, beeyng the diameter, and of the halfe circumference of the circle, but by the angle made in the semicircle is that angle by B, whiche is made of the righte line A.B, and that other right line B.D, whiche as they mete in the circumference, and make an angle, so they ende with their other extremities at the endes of the diameter. ”

“ Now, from zero along the curve to the end of the straight edge, on both sides of zero, are ninety little points marking ninety degrees, making two halves of a semicircle, or in other words, one-half of a whole circle of 360 degrees. ”

“ The circle was undoubtedly known to the early civilizations, its simplicity specially recommending it as an object for study. Euclid defines it (Book I. def. 15) as a “plane figure enclosed by one line, all the straight lines drawn to which from one point within the figure are equal to one another.” In the succeeding three definitions the centre, diameter and the semicircle are defined, while the third postulate of the same book demands the possibility of describing a circle for every “centre” and “distance.” ”

“ If one axis of our ellipse vanish, while the other remain of finite length, then the ellipse is reduced to a straight line, and its roulette will appear as a succession of semicircles touching one another upon the axis (C) ; the solid of revolution will be a series of equal spheres. If as before one axis of the ellipse vanish, but the other be infinitely long, then the curve described by the rotation {219} of this latter will be a circle of infinite radius, i.e. a straight line infinitely distant from the axis; and the surface of rotation is now a plane. ”

“ In a perfect circle of conductors, where either one of the second class is placed between two different from each other of the first class, or, contrariwise, one of the first class is placed between two of the second class different from each other, an electric stream is occasioned by the predominating force either to the right or to the left—a circulation of this fluid, which ceases only when the circle is broken, and which is renewed when the circle is again rendered complete. ”

“ Suppose the circle whose centre is O and one of the points of intersection A to remain fixed, while the second circle turns round that point in such a manner that the second point of intersection B becomes ultimately consecutive to A; then, since the line OO′ always bisects AB, we see that when B ultimately becomes consecutive to A, the line OO′ passes through A. In consequence of the motion, the common chord will become in the limit a tangent to each circle, as in the second diagram.—Comberousse, Géométrie Plane, page 57. ”

“ Centrifugal force, it must be explained, is the tendency which bodies have to fly off in a straight line from motion round a centre; and the power which prevents bodies from flying off, and draws them towards a centre, is called centripetal force. All bodies moving in a circle are constantly acted upon by these opposing forces, as may be seen by attaching one end of a piece of string to a ball, and the other to a stick driven into the ground. ”

“ In projecting, the lines in one view are used to mark those in other views, and to find their shapes or curvature as they will appear in other views. Thus, in Figure 225a we have a spiral, wound around a cylinder whose end is cut off at an angle. The pitch of the spiral is the distance A B, and we may delineate the curve of the spiral looking at the cylinder from two positions (one at a right-angle to the other, as is shown in the figure) , by means of a circle having a circumference equal to that of the cylinder. ”

“ Is an arc of a hundred and seventy-nine degrees greater or less than a semi-circumference?
Can there be more than three hundred and sixty degrees in a circumference?
If the circumference of circle 1 were divided into degrees, each degree would be so small an arc that it would look like a dot.
If a degree were divided into sixty equal parts, each part would be called a minute.
If a minute were divided into sixty equal parts, each part would be called a second.
How many degrees in the large circle of Fig. 2?
How many in the smaller one?
Has a large circle any more degrees than a small circle? ”

“ When the instrument is in motion a red circle is seen at C when θ is positive, the radius of which indicates roughly the value of θ. When is negative, a green circle is seen at C.
By means of nuts working on screws attached to the electromagnet, the axis CC' is adjusted to be a principal axis having its moment of inertia just exceeding that round the axis A, so as to make the instrument very sensible to the action of the force if it exists. ”

“ It is a very important proposition, theoretically and practically, for it enables us to find the center of a circle if we know any part of its arc. A civil engineer, for example, who wishes to find the center of the circle of which some curve (like that on a running track, on a railroad, or in a park) is an arc, takes two chords, say of one hundred feet each, and erects perpendicular bisectors. It is well to ask a class why, in practice, it is better to take these chords some distance apart. ”