The pupil now comes to the study of the propositions prepared with a knowledge of the facts of which he is about to establish the connexion, and some idea of the technical part of the reasoning with which he is to be made conversant. We would not yet recommend that he should be thrown at once into the chain of proposi tions. We think he will more clearly see the nature of the step which he is about to make if two known results, of which one is easily deduced from the other, are first connected as a specimen. For example, suppose it known that the angles at the base of an isosceles triangle are equal, and also that the exterior angle of a triangle is equal to the sum of the interior and opposite angles. Suppose it also indisputable, that if A and B be respectively double of C and D, the sum or difference of A and B is double of the sum or difference of C and D. It can thence be immediately proved that the angle at the centre of a circle is double of the angle at the circumference, and that all the angles in the same segment are equal. These we suppose to be also known facts, our object being to impress on the mind of the pupil. that one known fact may be connected with another, so that the second, if not known, might have been found out by means of the first. The reasoning will then stand thus:-If it be true that the angles at the base of an isosceles triangle are equal, and also that the exterior angle of a triangle is equal to the sum of the interior and opposite angles, then it follows that the angle at the centre of a circle is double of that at the circumference, &c.; so that the last proposition is proved so soon as the two first shall be proved. A few instances selected for their simplicity, the facts contained in them not being of the most self-evident character, would open the subject in a more striking way, than the (to beginners) most incom

prehensible superfluity of demonstrating that from the greater of two lines a part can be cut off equal to the less.

This brings us to the postulates, or problems which are treated as axiomatic.* The student should be told that geometrical demonstration consists, not only in employing correct reasoning upon things which are in their own nature self-evident, but in employing as few of the latter as possible. The ruler and compasses should now be given to him, the former not being divided into a scale. The circle having been defined, he is told that the only use allowed to be made of the compasses, is to draw a circle with a given point as a centre, and the distance between that given point and another given point as the radius. As an illustration, the instructor draws two lines, and gives the compasses to the pupil, requiring him to cut off from the greater a part equal to the less. The pupil accordingly measures the smaller line, and is about to transfer the compasses to the greater, when the instructor interrupts him, shuts the compasses, and tells him that he is only to draw circles with them, but must not transfer any length, that is, must close them the moment he takes them off the paper. This appears to the student a perfect contradiction: on which the instructor goes through the process of the second and third propositions of the first book-closing the compasses every time they are removed from the paper, and calling the attention of the student to this circumstance each time it occurs. To the question, whether it would not be much more simple to transfer the length in the common way, the answer is, that it would be so in prac

*This is the definition now usually given of a postulate; but we believe that the arrangement of Euclid has been altered.See Penny Cyclopædia, Article AXIOM.

tice, but not in reasoning: that it is the object of the latter to be content with as little as possible in the form of assumption; and since it can be proved that with the compasses as a means of drawing circles only, and an undivided ruler, a line equal to the less can be cut off from the greater, it is considered right not to take the latter for granted, on the principle of dispensing with every assumption which is not absolutely indispensable.

The treatment of the first three propositions in this manner is a most advantageous opening of the subject, because it must give the pupil a clear idea of the very close nature of the reasoning which will be expected from him. What license can be allowed in a science where it is not taken for granted that from the greater of two lines a part can be cut equal to the less? In the common method of saying the propositions the very contrary effect is produced: the pupil is not made aware of the limited nature of the assumptions, which render the third proposition a real consequence: he imagines that a cumbrous machinery is put in action to prove what has been virtually assumed, and his inference, and a very just inference, is, that anything which comes after might as well have been assumed as proved. Not that he definitely maintains this; but he acts upon it in the degree of attention which he gives to succeeding propositions. The book most in use in this country, by which the elements of geometry are taught, is Simson's Euclid. The only fault that can be found with this translation is its occasional inexpressive and even confined method of rendering the Greek, by too close an adherence to a literal version, and its preserving the unmeaning repetitions of the original, which, however excusable in Euclid, who wrote against objectors, of whom there were many in his day, is useless and

66

tedious for the purposes of elementary instruction. We may add that it abounds in Hellenicisms, where common English would better have served the purpose; but this is also a fault of the preceding Latin versions. Will it be believed, that there is at least one Greek word in the Latin versions and in Simson which is not in the original? That which we call an "axiom," which will make our readers give Euclid credit for the particular use of the word "atiwua," is by him called “xolvy Evvola," two common words, implying "that which is in the understanding of every man." We should recommend all instructors to get rid of as much Greek as possible, and also of many superfluous phrases. For instance, "if there be two triangles which have two sides of the one equal to two sides of the other, each to each, &c." The phrase in italics is not an English idiom, but the literal translation of the Greek ixarέpa enaresa: it conveys no meaning to a person unaccustomed to it, and requires a definition itself. Nor indeed is it easy to express the idea of this equality in a condensed form: the word "respectively" is sometimes employed, but this is not a good term; we have known a pupil assert that AB and CD were 66 respectfully" equal to EF and GH. Those who do think, frequently imagine we mean that the sum of AB and CD, is equal to that of EF and GH. We should recommend it to be said, "If two triangles have one side of the first equal to one side of the second, and a second side of the first equal to a second side of the second, &c." Again, it is not necessary to repeat the verbal enunciation of a proposition every time it occurs in those which succeed. For instance, if it should happen that two sides and the included angle are respectively equal in two triangles, if the pupil cannot be then trusted to

point out which other parts are equal "each to each," he does not become more fit to do so after muttering "therefore, the bases or third sides are equal, and the other angles are equal, each to each, namely, those to which the equal sides are opposite." We make no apology for insisting upon such matters; they create confusion.

With the knowledge which we have supposed the. pupil previously to acquire, he will have no more difficulty in putting a proposition in a syllogistic form, than in understanding the reasoning. We have proposed a method in the article already cited, vol. II. of the Journal of Education, by which, when the pupil has had a little previous training in the syllogistic forms, the strictness of the method may be relaxed*. If to this be added, that no proposition should be passed until the learner can give the demonstration with dif ferent letters from those used in the book, or, which will be still better, with numerals instead of letters, and also with the figure drawn in any manner upon the paper, we should think a very sufficient check would be provided against the study of geometry being nothing but an exercise of memory.

The most serious embarrassment in the purely reasoning part is the reductio ad absurdum, or indirect demonstration. This form of argument is generally: the last to be clearly understood, though it occurs almost on the threshold of the elements. We may find the key to the difficulty in the confined ideas which prevail on the modes of speech there employed. Let the student return to his primitive forms of assertion, and first ascertain which are contradictory, that is, taking

See also the treatise on the "Study of Mathematics," Cap. XIV., for an example of both methods.