lines. They meet together at the hinge. When the legs are drawn so far apart that they are exactly in opposite directions, they meet together, but are in the same straight line.1 The forming an angle by drawing the legs of a pair of compasses asunder teaches incidentally what should be early learned and always remembered-that the magnitude of an angle does not depend on the length of the straight lines containing it, but solely on the amount of opening between them. You may see as you look about you many examples of angles, the corner of the book before you is an angle, so is the corner of the table, or of the door, so again is the corner of a field where two hedge-rows meet. You may remember what Horace calls that cozy corner of his Tiburtine farm, where he hoped to sit in his old age, and drink his Falernian ANGULUS ille mihi ridet. The compasses used by smiths and carpenters are sometimes made in the following manner. From one АА A distance from each other. of the legs a curved rim pro- may be thus kept at any desired This form of compasses suggests a method of indicating an angle, very commonly used in books on trigonometry, namely, that of putting a dotted curved line between the straight lines which contain the angle. A letter (generally a Greek one, as a) is often inserted in the opening; and the angle between the straight lines is called the angle a. 1 Hence the following definition of an angle has been suggested: An angle is the difference of direction of two straight lines which meet. But the learner must not be misled by this figure, to imagine that the angle is the shut-up space, within which the letter a is placed. The dotted line is put there only to draw attention to the amount of opening between the straight lines. When only two straight lines meet at a point, Euclid indicates the angle between them by a letter (as A) placed at the point of meeting of the two straight lines, and the angle between the straight lines he calls the angle A. If more than two straight lines should meet at a point the single letter A would fail to tell whether it was intended to indicate the angle a or the angle ẞ, or indeed the larger angle y, made up of the angles a and ẞ together. A A ai B In this case Euclid uses three letters to indicate the angle. The angle a, that is the opening between the straight lines BA, A C, he indicates by the letters B A C or CAB. The angle ß, that is the opening between the straight lines CA, A D, he indicates by the letters CAD, or DAC; and the angle y, that is the opening between the straight lines BA, A D, he indicates by the letters BAD, or DAB, the letter at the angular point being always the middle letter of the three. This latter figure will help the learner to understand the adding angles together, or the subtracting one angle from another. The angle a, which is indicated by the letters B A C, may be added to the angle ß, indicated by the letters C A D, and taken together they make up the angley, indicated by the letters B A D. So the angle ẞ (that is C A D) may be taken from the angley (that is B A D), and the remainder will be the angle a (that is BA C). It may be well here to put the learner on his guard against an error, a very odd and unexpected one, some times made. If When a teacher says to a learner :- D it c is not uncommon for him to reply, 'If you take away the angle CAD, all that remains is the line A B.' Now this is an entire misapprehension. Angles and lines are different things altogether. If an angle is taken from an angle, it is not a line that remains, but an angle, just as when you take shillings from shillings it must be shillings, and not inches or ounces, that are left. In fact, the intermediate line CA belongs, so to speak, to both angles: when taken with A B, it makes the angle C A B (or a), and when taken with A D it makes the angle CAD (or 6). And when either of the two, as a, is taken from the whole angle 7, it is the other angle ẞ that remains. In other words; if from the angle BAD we take the angle BA C, the angle C A D remains. As it is of the highest importance to see clearly and quickly the angle which is contained between any two straight lines which meet, it would be well for the learner to draw some interlacing straight lines, and to examine himself, or to get some friend to examine him in pointing out the angle contained by any two named lines which meet; or, the angle being pointed out, in naming the straight lines which contain it. K B M For example. If the friend says, 'What is the angle contained by LB, B M?' let the learner place a little curved line between these straight lines, as in figure, to mark the angle; or if the friend puts the curved line between any two straight lines which meet, let the learner name to him the H two straight lines which contain F N the angle so marked. In naming the straight lines containing the angle pointed out, let the learner be advised to take the first line named towards the angular point, and the second from the angular point. For instance, if he wishes to name the straight lines containing the angle marked a, let him say that the angle is contained by the straight lines BA, AD, in preference to saying that it is contained by the straight lines A B, A D, or by BA, DA. This may be thought an immaterial remark, but it is found that learners who name the lines in the way here recommended, learn most quickly to distinguish the angle contained between two named straight lines. After one more remark in reference to angles, we will pass to the next definition. It will be seen (last fig.) that the angle contained by BA, AD, by BA, A C, or indeed by MA, A N, is one and the same angle-the one marked in the figure as the angle a. This teaches that it matters not at what points in the lines containing an angle we place the letters which indicate it. On Right Angles, Obtuse Angles, and Acute Angles. C Suppose A B to represent the edge of your desk or table, and CD your ruler. If you bring your ruler, CD, to the edge of your desk, AB, and hold it there, on end, in such a way that the angles CDA and CDB [these are called adjacent angles] are equal, each of the two is a right angle. D B If you hold the ruler so as to make one of the adjacent angles greater than the other, then the greater angle, C D A, is called an obtuse angle (from obtusus, blunt), and the lesser, CD B, is called an acute angle (from acutus, sharp). This illustration leads on to the Definitions 10, 11, and 12 of Euclid, which are as follows: Definition 10.-When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. Definition 11.-An obtuse angle is that which is greater than a right angle. Definition 12.-An acute angle is that which is less than a right angle. On the Circle. Euclid now passes on to a new kind of magnitude, which is introduced by the two following definitions : Definition 13.-A term or boundary is the extremity of anything. Definition 14.-A figure is that which is enclosed by one or more boundaries. The 'figure' so introduced is the Circle, which is thus defined: Definition 15.-A circle is a plane [or flat] figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within it to the circumference are equal to one He adds: another. Definition 16.-And this point is called the centre of the circle. A primitive mode then of describing a circle would be |