a certain condition amongst themselves. And this condition is determined in the twenty-second Proposition of the third Book. Again, since a straight line can be made to pass through any two given points and no more, and a circle through any three, it is plain that a straight line and circle may have two points in common, but cannot have more. Such a straight line is called a "Secant," (1) and the part of it (PQ) R 1. within the circle is called a Chord of the circle; and the portion of the circular plate contained by the arc P Q and the chord PQ is called a Segment of the circular plate. Or in the language of Euclid, "A Segment of a circle is the figure contained by a straight line or chord, and the arc or part of the circumference which it cuts off." If however it happen that the two points which the straight line and circle have in common, be consecutive, the straight line it is called a Tangent" to the circle, (2, 3,) so that the defini 66 tion of a Tangent is, " A straight 2. 3. line which has two consecutive point in common with the circle." And since no straight line can have more than one point in common with another, there can be only one tangent to a circle at any given point in it. This theorem is found in the corollary to Euclid III. prop. xvii. and proves that our definitions are not opposed to his in reality though they may be in appearance. The above definition therefore which obtains in all the higher Mathematics may perhaps be allowed to explain Euclid also. Thus, as we shall see more clearly in the next chapter, a circle may be conceived to be made up of straight lines, each of two points only, which lines when produced are tangents to the circle. Now when a circular substance comes in contact with one that is straight, the only part of the circle which will be in actual contact is that which forms the tangent of the point. For example, a wheel on a level road is only in contact with the road at the bottom of the wheel, and the road is a tangent to the wheel. Similarly when a ray of light, or a column of sound strikes a circle, it comes in direct contact only with the tangent to the curve: and the angle at which it strikes the curve, is the angle contained between its line of motion, and the tangent to the curve at the point of contact. This also will be clearer presently. In our inquiry into the straight line we saw that much depended upon the angle. To this subject we now return;-a subject perhaps of somewhat singular interest to us, to whose forefathers “that little corner" of Denmark gave its name of Angli; a name still living in "England," "Angleterre," and "Engelland." Nor let the disciple of Izaak Walton forget that his sport depends upon the varying angle between the rod and line, and thence receives its well loved name of Angling. We have seen that an angle exists between every three consecutive points in a line, and depends upon their order of succession. We found also that as the point which described the circle revolved, the superior angle between the radius and the axis gradually increased while the inferior decreased equally. Hence since the angle is increased by the motion of the same line the end of which generates the circle, and inasmuch as the circle increases exactly as the angle between the radius and the axis increases-therefore the size of the angle may be measured by the size 1. B of the arc of the circle which the angle subtends, i.e. the arc to which the angle is opposite (1.) Thus, the circumference of a compass is divided into 32 "points;" and sailors are accustomed to measure by them the various angles they may require. Thus a ship is said to be "eight points from the wind" when she is at right angles to it, or to have gone off a certain number of points, when she has changed her course of sailing. And since in a straight line the superior is equal to the inferior angle, therefore the portions of the circle above and below the axis are equal, and consequently the angle in a straight line subtends a semicircle. Now for convenience' sake a circle is said to be divided into 360 equal parts, which are called degrees. The angle in a straight line, therefore, will subtend, that is be opposite to 180 of them, and is called "an angle of 180 degrees," which is commonly written "180°.” 2. Suppose now that there is a straight line AB and a point in it C, and at this point a line CD revolving from Aright to left (2.) Then there will be an angle DCB increasing and an angle DCA decreasing as CD revolves from right to left, and CD will pass through a position, similar to the last case in which the angle DCB is equal to the angle DCA (3.) When the straight line DC is in this D' 3. D B A B C position with regard to another straight line AB, each of these angles is called a right angle, and we have this definition: "When one straight line standing upon another straight line makes the adjacent angles equal to each other, each of these angles is called a right angle, and the straight lines are said to be perpendicular to each other." And as the angle in a straight line was shown to subtend a semicircle, so a right angle may be shown to subtend a "quadrant" or quarter of a circle, and therefore to be an "angle of 90 deg." And this has been thought by some to be the true definition of a right angle, that given above being more properly the definition of 5. P 4. two right angles, or perhaps of perpendicularity. 6. Р. An angle which subtends the Quadrant of a Circle is called a Right Angle (4.) An angle which subtends less than the Quadrant of a Circle is called an Acute Angle (5.) An angle which subtends more than the Quadrant of a Circle is called an Obtuse Angle (6.) It is to be observed also (7,) from the manner in which we obtained the angle BAP,, that it is composed of the angles BAP and PAP,, inasmuch as the angle P'AB was first formed by the revolution of Hence, 7. Consequently if at any point in a straight line there be drawn another straight line, the two angles which the second line makes with the first are together equal to the angle in a straight line, that is, to 180°. When in practice it is required actually to measure the number of degrees in any angle, an instrument is used called a Theodolite the action of which it is not necessary to explain here. SECTION III. CHAPTER XVII. CONTINUITY.-FIRST CONDITION OBTAINED. If we now compare the two lines we have obtained we shall find at once this striking difference. In circular motion after a certain time we arrive again at the place from which we started: in moving in a straight line, this, to all appearance, at least, will never be the case. This Enclosure of space is an idea which we have not yet discussed, and it is both important and difficult. The question may be put in this form: According to what law must a point move in order that all its positions may be connected, or in other words, in order that it may return to the point from which it set out? Now first since the space must consist of both length and breadth, and since any two points can be in a straight line which of course consists of length alone; therefore in order that space may be enclosed, there must be at least three points which are not in the same straight line, or which is equivalent, the point by the motion of which the space is to be enclosed must make at some point an angle of less than 180° with the straight line which it describes at first. Suppose then a point P moving in a straight line from A, and that we wish to alter its course so that space may be enclosed. Then, at A 1. C |