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level or uneven. Thus if a lake be perfectly calm, a stick floating upon it would be entirely upon the water; but if there be ever so small a ripple, the stick will be partly in the water and partly out. In the former case the surface is said in Geometry to be plane; and we have its definition as follows:
A Plane, or a Plane Surface, is that in which any two points being taken, the straight line between them lies wholly in that surface. We shall obtain a simpler definition in a future chapter.
In the present treatise plane surfaces only will be considered ; and all lines will be conceived to be “in the same plane.” We shall not therefore repeat the qualification, as it will be the same in all cases. These remarks apply of course to the deductions obtained in the last chapter.
MEANWHILE it must not be forgotten that in order to measure the angle in a straight line, we have been making a straight line revolve round one of its own extremities, and consequently by the force of what we said above (chap. viii.) have been describing a surface of some kind or other.
In examining the locus 1. thus formed as every point in the revolving straight line moves in a similar path (1) it will be sufficient to confine ourselves at first to the motion of a single point.
Let us take a straight line AB and examine
the motion of the point B only. Now as AB is of a fixed length the motion of B round A cannot alter its distance from A. The point B therefore must move in such a manner as to be always at the same distance from A. Hence the law of motion of B is known, and if we give to the new path thus formed the name of
Circle," we may define as follows :
A Circle is the locus of a point (B) which is always at the same distance from a given fixed point (A,) all the points being in the same plane.
A word or two of explanation is here needed. The locus of which we are now speaking is considered by Euclid not as a circle but as the circumference of a circle. The circle itself is the name he gives to the whole figure described by the revolving line, that is to the whole surface contained by the circumference; but we have ventured to adopt at once the language of higher mathematics, and to consider the circle as the locus of a point, and the surface as a circular plate, defining as follows :
A Circular Plate is the figure formed by the revolution of a straight line round one of its own ends which remains at rest.
But it may be asked, Why not employ Euclid's own terms to express his own ideas? We reply that not only do the ordinary diagrams cause us unconsciously to conceive a circle as a line (an idea which obtains in all the higher Geometry,) but that Euclid himself (or at least his translator) uses the word in a somewhat
“ Let it be granted,” he says, “ that a circle may be described from any centre, at any distance from that centre." How can the whole surface be at a distance from the centre when the centre itself is in the surface? At the same time the author cannot but feel that he is perhaps rendering himself liable to the charge of hypercriticism and even audacity in changing terms so clear and so venerable. His apology is that clear as many mathematical ideas are in themselves, he has found that a slight occasional confusion in their use prevented him both from a nice
perception of their meaning, and also from observing the exquisite exactness with which they grow out of and fall into each othera matter full of instruction and beauty. There is great pleasure in watching the prismatic colours of a diamond, but a grain of dust in the eye will spoil it all.
We shall now attempt first, to show how our definitions of the Circle and Circular Plate correspond with those which Euclid gives of the Circumference and Circle; and secondly, to examine some of the various parts of these figures.. “ A Circle,” says Euclid, “is a plane figure
1. contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. And this point is called the centre of the circle” (1.) (AB=AP.)
The definitions correspond exactly. For the figure is, as we have seen, bounded by the path described by the extreme point of the revolving line, i.e. it is contained by one line 2. which is called the circumference. Again, since the circumference is the locus of a point (2) which is always at the same distance from a fixed point, all straight lines drawn from the fixed point to any point in the locus will be equal to one anotherin other words, the fixed point is the centre, and all straight lines drawn from it to the locus of the revolving point, i.e., to the circumference, are equal to one another.
Next, to define a few of the parts both of the circle and the circular plate, using our own definitions.
The fixed extremity of the revolving line is called the Centre of the circle.
The revolving line itself is called the Radius.
In its first position, usually taken as horizontal, it is called the Prime Radius or Axis. And just as a point by its motion generates a line, that is, as we conceive it to leave itself behind as it were, while it moves on; so in the case of the circle the prime radius is conceived to be always at rest, while the revolving radius or rather its extreme point, traces out the line.
3. The path described by the moving point between any two given positions is called an Arc of the circle (3,) the latter name being
A given from its bow-like shape (BP.)
And the portion of the circular plate which is contained by any two radii and the arc between them is called a Sector of the circle (3.)
We may further notice that motion in a circle affords an easy method of fixing a point at any particular 4. distance from another point. Thus, if there be two points, A and B, at a certain distance from each other, and it be required to fix another point, C as far from A as B is, but A
5, in another direction (4,) take A as centre, and AB as radius, and make AB revolve till it lies in the required direction; then B is the required position (P.) (5.)
A Or again, suppose it be required to fix a point in such a position as to be at the same distance from two given points as these two points are from each other. For example, to plant a flag at such a distance from two flags already fixed, that the distance from any one of the three flags to another of them will be the same.
Then, first taking A as the fixed point, and making B perform a whole revolution;
And, secondly, taking B as the fixed point, and making A perform a whole revolution ;
6. we have two circles, in both of which C must lie (since it is to be equally distant from A and B,) and there are but two such points í namely, those which these circles have in common, one above and the other below AB. Either of these points therefore is in the required position (6.)
made to pass.
BEFORE concluding the subject let us as in the case of the straight line inquire through how many points a circle can be
It is formed, as we have seen, by the motion of a point which is always at a fixed distance from a given point. The size of the orbit therefore depends upon the distance of these two points from each other. Now we have seen that when a fixed distance is given, it is equivalent not to two but to three given points. And just as the straight line depended upon two points, and therefore given any two points a straight line could be drawn from one to the other; so given any three points, it is always possible to find some circle which will pass through them. How to determine the centre when the three points are given through which the circle is to pass, is the subject of the fifth proposition of Euclid's fourth Book.
If however there be four points through which it is required to make a circle pass, the relative positions of these points must fulfil