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mon editions of Euclid. It will be seen that the lines which are most strongly marked form the immediate object of the proposition; lines of secondary importance to these are more faintly traced; whilst the dotted lines are those which are merely introduced for the sake of the proof, &c. Given points, lines, and angles are also distinguished by larger letters. The learner will find it serviceable to have the improved figure before him while learning a proposition, for which reason the same letters are used as in the usual editions of the Elements.
I will here add a few general observations on the best method of reading Euclid. The Golden Rule for the learner of Geometry is never to pass on to a new proposition, till he thoroughly understands the preceding one. For in this science the higher steps so completely depend on the lower ones, that it is next to impossible, in Euclid, (whatever may be the case with Algebra) to “get up," a set of the more important propositions without comprehending those on which they depend. When a proposition has been carefully studied, the pupil should exercise himself by writing it out, with a correct figure, and it will be more improving for him to use different letters from those employed in his book. In drawing a figure, the learner should be particular always to draw every line exactly in the order in which it is directed to be drawn in Euclid. Take, for instance, the second problem.— A learner would perhaps begin by drawing the larger circle, then the smaller one, and so on; and consider it immaterial with what part he began, so that his figure, when completed, was similar to that in the
book; but he will find that by making first the given line, then the given point, &c., according to the present direction, he will fix the greater part of the demonstration itself firmly in his mind.
What Edition of Euclid is used is also a point of importance to the learner. There can be no doubt that the Symbolical Euclid is by far the best.-It is the most perspicuous. It is the shortest. The propositions in it are in the precise form in which the student will be expected to write them in his examination. I must, however, here caution the beginner to be very particular in making himself thoroughly acquainted with the meaning of the symbols *, before he proceeds to read a single proposition, otherwise he will be perplexed and perhaps disheartened by his first attempt. One who is quite a novice in the use of symbols will do well to get an old edition of Euclid and compare it with the symbolical, as he will thereby best perceive the signification of the signs, and the advantages attending their use.
The most agreeable and the surest way of learning the definitions is to learn them as they are wanted in the course of reading, and not, as is generally done, to commit the whole of them to memory before reading a single proposition.
One of the first things of which it is important to have a correct view, is the meaning of an angle, and
is a better sign for a parallelogram than simply In the Second Book the reader will find an explanation of the symbol AB X CD, as used for the rectangle contained by AB and CD.
how one angle is said to be greater or less than another; and not a few have been retarded through forming an incorrect or an imperfect notion on this point, simple as it may appear. Let it be observed then, that an angle is described to be the inclination, that is, the degree of increasing width or separation, between the lines which form it, and does not at all depend on the length of those lines. This will be readily understood by an example. Let a be a pair
of compasses opened to a certain width; в a foot-rule bent in such a manner, that when applied to the compasses, as in fig. 2., the legs of one instrument may lie exactly on those of the other; then the angle formed by the compasses is exactly equal to that formed by the rule. According, also, as the degree of opening width is greater or less, the angle is greater
or less, thus, the angle A is greater than the angle B, and the angle в is greater than the angle c.
The last general direction is, never attempt to learn propositions without understanding them; many have attempted to do this, but I never knew one to succeed.
Note. To such Demonstrations as are very easy and very short, the steps of the Argument are not given; and the figure alone is drawn, as for such no further assistance can be required.