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retained, the learner should endeavour to dispense with diagrams and letters, and to express the proposition and its demonstration in general terms in the manner exemplified in Appendix (B), page 146. This indeed is at the commencement a task of considerable difficulty; but the exertion which it calls for at first, is amply repaid by its wonderful efficacy in strengthening the hold which the mind has of its mathematical acquisitions, and in giving increased facility in wielding them. Truths once clearly and accurately expressed in general language, are much more likely to be recalled to the mind and revived in meditation, than those the adequate expression of which depends in any degree on diagrams or other physical aids. The practice of demonstrating in general terms, conducts, it may be added, to habitual clearness and precision of language.
The learner, after having made himself master of the Propositions in Euclid's Elements, will do well to exercise his sagacity in the solution of new questions. A little perseverance will suffice to convince him that sagacity and the faculty of discovery, are, like our other faculties, capable of being immeasurably developed by judicious training. Among the Supplementary Propositions appended, he will find several theorems, which though not entering into the consecutive series essential to an elementary system of Geometry, are yet of much importance as giving an insight into the properties of figures, and add greatly to the resources of those who take pleasure in geometrical investigations.
In corroboration of the opinion expressed above, and from which many, perhaps, will be inclined to dissent, respecting the ease with which Geometry may be learned, if only care be taken to avoid hurry, we shall here close our remarks with a passage taken from the late Mr. WALKER'S edition of Euclid. That learned writer, who was an experienced teacher as well as able mathematician, expresses himself on that point in the following
"If in any part the Student finds himself unable to comprehend what he meets with, he may almost with
certainty conclude, that there is something in the previous part of which his comprehension has not been clear, or which is not perfectly retained. In fact, for one that fails in mathematical studies for want of intellectual abilities, there are hundreds that fail for want of mental patience. And, hazardous as the engagement might appear to some, my experience makes me not afraid to say, that I would undertake to make any one a mathematician, who should only combine patience of mind with the most moderate understanding, and would exert attention to the subject. The young student may also be assured, that in pursuing the study aright, however slow he may seem to proceed for a time, he will find difficulties rapidly lessen, and his progress become as smooth as it is sure.'
THE arguments employed to demonstrate any proposition in Geometry must be founded more or less on the antecedent Propositions, or on the Definitions, Postulates, Axioms, Hypothesis, or Construction. These are respectively referred to, in the following pages, under the abridged forms Def. Post. Ax. Hyp. and Const.
The Hypothesis is the condition assumed or taken for granted. Thus, when it is affirmed that in an isosceles triangle, the angles at the base are equal, the Hypothesis of the proposition is, that the triangle is isosceles, or that its legs are equal.
The Construction is the addition made to, or change made in the original figure, by dividing or drawing lines, &c., in order to adapt it to the argument of the demonstration or the solution of the problem. The conditions under which these changes are made, are as indisputable as those contained in the hypothesis. Thus, if we draw a line and make it equal to a given line, these two lines are said to be equal by Construction.
A+ B, or A plus B, A-B, or A minus B,
The signs and are to be read plus and minus. means A with B added to it, or the sum of A and B. means the difference of A and B, or A with B taken from it. The sign of multiplication is X; thus AX B signifies A multiplied by B.
In speaking of rectangles, the adjacent sides are written with a point between them; thus ABCD expresses the rectangle contained by the sides AB and CD. The square of AB, or AB AB, may be also written AB2.
Equality is expressed by the sign, which may be read equal to, or is equal to, or are equal to. Thus, A=B signifies that A is equal to B; A+B C, expresses that A and B together are equal to C.
The sign is the distinction of an angle. Thus / CAB means the angle CAB.
N.B. The portions between brackets occurring in the following pages, are editorial glosses, and not in the original text.
1. A POINT is that which has position, but not magnitude.
2. A LINE is length without breadth.
COROLLARY.-The extremities of a line are points; and the intersection of one line with another is also a point.
3. A right or straight line is that of which the successive points lie in the same direction.
COR.-Hence, two straight lines cannot enclose a space; for if they meet in two points, since they both lie in the same direction with those points, they must coincide between them, and they must form throughout one continued straight line.
4. SURFACE is that which has length and breadth, without thickness.
COR.-The extremities of a surface are lines.
5. A PLANE, or plane surface, is a surface in which any two points being taken, the straight line joining them, lies wholly in that surface.
6. A plane rectilinear ANGLE is the inclination of two straight lines to one another in the same plane; which
lines meet together, but do not lie in continuation of each other.
[The two straight lines which, meeting together, make an angle, are called the LEGS of that angle; and it must be observed, that the magnitude of an angle does not depend on the length of its legs, but solely on the degree of their inclination to each other. The point at which the legs meet is called the VERTEX of the angle.
An angle may be designated by a single letter when its legs are the only lines which meet together at its vertex thus if GC and EC alone met at C, the angle made by them might be called the angle C. But when more than two lines meet at the same point, it is necessary, in order to avoid confusion, to employ three letters to designate an angle about that point, the letter which marks the vertex of the angle being always placed in the middle: thus the lines GC and EC meeting together at C make the angle GCE or ECG: the lines GC and EC are the legs of the angle; the point C is its vertex. In like manner may be designated the angle GCA; or the angle ECA, which is the sum of GCE and GCA; and so of the other angles ECB, BCD, ECD, &c., round the same point.
When the legs of an angle are produced (or continued) beyond its vertex, the angles made by them on both sides of the vertex are said to be vertically opposite to each other thus, since GC is continued to D, and EC to F, the angles GCE and DCF are vertically opposite to each other. In like manner, GCF and DCE, FCA and ECB, ACG and BCD, ACE and BCF, GCB and DCA, are pairs of vertically opposite angles.]
7. When a straight line, standing on another straight line, makes with it the adjacent angles equal to each
other, each of these angles is called a RIGHT
at right angles with, or to be perpendicular, or a
8. An obtuse angle is an angle greater than a right angle.