Circle (geometry)

“ It is important to realize that geometry, as taught in schools since Greek times, ceases to exist as a separate science, and becomes merged in physics. The two fundamental notions in elementary geometry were the straight line and the circle. What appears to you as a straight road, whose parts all exist now, may appear to another observer to be like the flight of a rocket, some kind of curve whose parts come into existence successively. The circle depends upon measurement of distances, since it consists of all the points at a given distance from its center. ”

“ That the circle belongs to the curves of the second order is seen at once if we state in a slightly different form the theorem that in a circle all angles at the circumference standing upon the same arc are equal. If two points S1, S2 on a circle be joined to any other two points A and B on the circle, then the angle included by the rays S1A and S1B is equal to that between the rays S2A and S2B, so that as A moves along the circumference the rays S1A and S2A describe equal and therefore projective pencils. ”

“ The object still being to illustrate the mode of working with coordinates for descriptive purposes, we consider the theorem of the polar of a point in regard to a circle. Given a circle and a point O (fig. 52) , we draw through O any two lines meeting the circle in the points A, A′ and B, B′ respectively, and then taking Q as the intersection of the lines AB′ and A′B, the theorem is that the locus of the point Q is a right line depending only upon O and the circle, but independent of the particular lines OAA′ and OBB′. ”

“ This ratio, invariably denoted by π, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number. Very early in the history of geometry it was known that the circumference and area of a circle of radius r could be expressed in the forms 2πr and πr2. The exact geometrical evaluation of the second quantity, viz. πr2, which, in reality, is equivalent to determining a square equal in area to a circle, engaged the attention of mathematicians for many centuries. ”

“ 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. ”

“ The problem of finding a square equal in area to a given circle, like all problems, may be increased in difficulty by the imposition of restrictions; consequently under the designation there may be embraced quite a variety of geometrical problems. It has to be noted, however, that, when the “squaring” of the circle is especially spoken of, it is almost always tacitly assumed that the restrictions are those of the Euclidean geometry. ”

“ To prove the same proposition regarding π is to prove that a Euclidean construction for circle-quadrature is impossible. For in such a construction every point of the figure is obtained by the intersection of two straight lines, a straight line and a circle, or two circles; and as this implies that, when a unit of length is introduced, numbers employed, and the problem transformed into one of algebraic geometry, the equations to be solved can only be of the first or second degree, it follows that the equation to which we must be finally led is a rational equation of even degree. ”

“ The centre of a circle is its focus, and its directrix has gone to infinity, having no special direction. This last fact illustrates the necessity, which is also forced on plane geometry by three-dimensional considerations, of treating all points at infinity in a plane as lying on a single straight line. ”

“ Since if the cutting plane is perpendicular to the axis the section is a circle, and if oblique it is an ellipse, a parabola, or an hyperbola, it follows that if light proceeds from a point, the shadow of a circle is a circle, an ellipse, a parabola, or an hyperbola, depending on the position of the plane on which the shadow falls. ”

“ The general problem of inscribing in a given circle a regular polygon of n sides depends upon the problem of dividing the circumference of a circle into n equal parts, or what comes to the same thing, of drawing from the centre of the circle n radii such that the angles between consecutive radii are equal, that is, to divide the space about the centre into n equal angles. Thus, if it is required to inscribe a square in a circle, we have to draw four lines from the centre, making the four angles equal. ”

“ If the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle, and if the projection is, as in our case, orthographic, then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere. We get such a circle in the plan and another in the elevation, their centres being plan and elevation of the centre of the sphere. ”

“ Their self-similarity. Every inch of a straight line coincides with every other inch, and off a circle with every other off the same circle. Where, then, did Euclid fail? In not introducing the third curve, which has the same property—the screw. The right line, the circle, the screw—the representations of translation, rotation, and the two combined—ought to have been the instruments of geometry. With a screw we should never have heard of the impossibility of trisecting an angle, squaring the circle, etc.—De Morgan, A.
Quoted in Graves’ Life of Sir W. R. Hamilton, Vol. ”

“ A circle of unequall isoperimetrall plaines is the greatest.
The reason is because it is the most ordinate, and [257*] comprehended of most bounds; see the 7, and 15 e iiij. As the Circle a, of 24 perimeter, is greater then any rectilineall figure, of equall perimeter to it, as the Quadrate e, or the Triangle i.
Of Geometry the twentieth Booke, Of a Bossed surface.
1. A bossed surface is a surface which lyeth unequally betweene his bounds.
It is contrary unto a Plaine surface, as wee heard at the 4 e v. [258*]
2. ”

“ A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis—considered as a circle of infinite radius. ”

“ The circle whose radius is the maximum line (PA) has contact of the second kind. 3. A circle having any of the remaining lines (PF) as radius intersects HDE in two points (F, I) .
PROP. IX.—Theorem.
A point (P) within a circle (ABC) , from which more than two equal lines (PA, PB, PC, &c.) can be drawn to the circumference, is the centre.
Dem.—If P be not the centre, let O be the centre. Join OP, and produce it to meet the circle in D and E; then DE is the diameter, and P is a point in it which is not the centre: therefore [vii.] only two equal lines can be drawn from P to the circumference ”

“ In any triangle, the radius of the circumscribed circle is to the radius of the circle which is the locus of the vertex, when the base and the ratio of the sides are given, as the difference of the squares of the sides is to four times the area. ”

“ PROP. XVI.—Theorem.
1. The perpendicular (BI) to the diameter (AB) of a circle at its extremity (B) touches the circle at that point. 2. Any other line (BH) through the same point cuts the circle.
Dem.—1. Take any point I, and join it to the centre C. Then because the angle CBI is a right angle, CI2 is equal to CB2 + BI2 [I. xlvii.] ; therefore CI2 is greater than CB2. Hence CI is greater than CB, and the point I [note on I., Def. xxxii.] is without the circle. In like manner, every other point in BI, except B, is without the circle. ”

“ A circle may be a great circle of either the celestial sphere, the earth, or even of a baseball. The poles of a great circle are the points on the surface of its sphere, penetrated by the diameter perpendicular to the plane of the great circle. As for example, the poles of the earth are connected by the diameter that is perpendicular to the plane of the equator. An angle at any pole is measured on the great circle which subtends it. For instance, angles at the poles of the earth are measured on the equator; angles at the zenith on the horizon. ”

“ He affirms therefore, that though it be true in the Theory, that a Circle, whose Plain is inclined to the Axis of the Sphere by an Angle, whereof half the Diameter is the Sine, and which touches the Sphere in its Pole, will touch in all its parts a spherical Surface, that shall turn upon that Axe. But that it is true also, that that must be but a Mathematical Circle, and without Breadth, and which precisely touches the Body in its middle: Whereas in the practice, a Circle capable to keep Sand and Putty, must be of some breadth ”

“ As such curves can be described with quite as much accuracy as the circle—for, be it remembered, an absolutely perfect circle has never yet been drawn—we see that it is only the limitations which geometers have themselves invented that give this problem its difficulty. ”

“ On the other hand, the fact that a straight line does or does not cut a circle is independent of measurement, it being dependent only upon the mutual “position” of the line and the circle. This difference becomes clearer if we project any figure from one plane to another (see Projection) . By this the length of lines, the magnitude of angles and areas, is altered, so that the projection, or shadow, of a square on a plane will not be a square; it will, however, be some quadrilateral. Again, the projection of a circle will not be a circle, but some other curve more or less resembling a circle. ”

“ The chief difficulty of this subject, however, consists in the fact that mathematical or other certain laws may entirely forbid the existence of some combinations. The circle may be defined as a plane curve of equal curvature, and it is a property of the circle that it contains the greatest area within the least possible perimeter. May we then contemplate mentally a circle not a figure of greatest possible area? ”

“ In the case of circles, we know exactly what it means to say that there are no circles in nature: no natural systems exist which are precisely isomorphic to the equation that describes a geometric circle: things (e.g. basketball hoops) are circular, but on close enough examination they turn out to be rough and bumpy in a way that a mathematical circle is not. ”

“ The circle being drawn, a line A, 179, is drawn through its centre, and from its intersection with the circle as at b, here a step on each side is marked as c, d, then lines c to d, and c and d to e, where A meets, the circle will describe a trigon. If the figure is to stand vertical, all that is necessary is to draw the line a vertical, as in Figure 180. A ready method of getting the dimension across corners, across the flats, or the length of a side of a given polygon, is by means of diagrams, such as shown in the following figures, which form excellent examples for practice. ”

“ He hath left us also his attempts upon the circle: he proved that a circle is equal to a right angled triangle, whose base is equal to the circumference, and its altitude equal to the radius; and consequently that its area is found by drawing the radius into half the circumference; and so reduced the quadrature of the circle to the determination of the ratio of the diameter to the circumference ”