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As the theoretical deductions of Geometry depend almost entirely upon the results of close and accurate reasoning, its study is eminently calculated to fix the attention, to exercise the memory, and to enlarge the powers of the mind. In its practical application, on the other hand, it enters into all the various occupations of mankind, from the work of the humblest artificer, to the sublime investigations of the astronomer engaged in the solution of the most complex problem connected with the system of the universe. Euclid's “ Elements” have ever been regarded as the groundwork of the science, and it is scarcely probable that they will be superseded by a system of superior excellence, so far as the subject of Geometry is concerned. Of the fifteen books, the first six treat of plane, and the eleventh and twelfth of solid, Geometry; the rest being chiefly concerned with the principles of numbers, which have been more fully and fitly investigated by modern arithmeticians. The first book is mainly occupied with the relations subsisting between the sides and angles of triangles, and the conditions of equality which obtain between two or inore of these, the simplest of all geometrical figures. Some properties of parallelograms are also investigated in connection with those of triangles ; together with those particulars respecting perpendicular and parallel straight lines, which are essential to the proof of each succeeding proposition.
1. A Point is that which has no parts, and which has no magnitude.
Definitions are concise explanations of certain elementary principles upon which a science depends; and those which are prefixed to this book are, for the most part, peculiarly simple, exact, and perspicuous. To the first, however, it is objected that it is entirely negative, and that it is not convertible ; for though a point is without extension, every thing unextended-as, for example, an instant of time-is not a point. Other definitions have been proposed, of which that given by Professor Playfair is as good as any; viz., that a point is that which has position, but not magnitude. The fact is, that abstract ideas, from which the first principles of Geometry take their rise, scarcely admit of strict definition, marking some specific distinction; but Euclid has arranged the few elementary notions, so to speak, upon which the science rests, with such admirable nicety, that their import is clearly understood, though it may not have been perfectly developed.
2. A line is length without breadth. 3. The extremities of a line are points. This is not so much a definition, as an inference from the two which precede it. As a line has no breadth, its extremity can have none; and if it had length, it would be a continuation of the line, not a termination. Having, therefore, position without magnitude, it is a point. Lines which have no termination-as circles and other curvilinear figures—are not of course taken into the account.
4. A straight line is that which lies evenly between its extreme points.
In other words, it is the shortest of all lines which have the same extreme points, according to the definition of Archimedes. Thus AB is evidently shorter than ACB, ADB, AEB, or any other line which does not lie evenly between Al A and B.
5. A superficies is that which has only length and breadth.
6. The extremities of a superficies are lines. The remarks which were made under the third definition will, mutatis mutandis, equally apply here; spherical or curved superficies or surfaces (super and facies), which have no boundaries, not being taken into consideration. It will also materially conduce to a right understanding
of the definitions of a point, a line, and a superficies, if we consider the nature of a solid, in which they all originate, and in which exist the several dimensions of length, breadth, and thickness. For if the solid ABCDEFGH is considered to be composed of the two laminæ or smaller solids AM, BM, the superficies KLMN is the common boundary of these solids; and it cannot be a part of the thickness of the solid Am, because if this A solid be removed, it still remains as the boundary of the solid BM.
For a like reason, it cannot be part of the thickness of the solid bm. Therefore the superficies KLMN has no thickness, but only length and breadth. In the same manner the line kl is the common boundary of the two superficies al, KC; and, as it partakes of the breadth of neither, it has length only without breadth. So again the point k is the common extremity of two lines AK, BK; and, partaking of the length of neither, has neither length, breadth, nor thickness, but simply position without magnitude. Hence the common intersection of one line with another are points ; of one superficies with another, lines ; and of one solid with another, surfaces. It is clear also that straight lines can intersect each other in one point only. In practice the mathematical ideas of a point, line, and surface no longer exist, since even a point must have some magnitude, nor is a line or surface absolutely without breadth or thickness. Still it is only necessary to regard these several dimensions as continually diminishing to the verge of evanescence, and the theory and practice are easily reconciled.
7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.
A plane superficies is more frequently called simply a plane; and the derivation of the word is from the Latin planus, signifying flat, or level. In the superficies ABCD,
coinciding with the plane of the paper, the line joining any two points, E and F, lies clearly within the superficies; but in abcd, which stands out of the plane of the paper, the line joining E and F does not so.
8. “A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.”
This definition comprehends the angles formed by the meeting of curved, as well as straight lines ; but it is with the latter alone, of which the next definition treats in particular, that the Elements of Euclid are concerned. The word angle is manifestly derived from the Latin angulus, a corner : but an angle is a very distinct thing from a corner, and the student should be careful to obtain an accurate notion of their difference. The one is an external edge, the other an internal inclination.
9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
N. B.-When several angles meet at a point, each of them is indicated by three letters, of which the one at the vertex of the angle is read between the other two. Thus the angles ABC, ABD, DBC are formed at B by the lines AB, BC; AB, BD; and BD, BC; respectively. A single letter is sufficient, when there is only one angle at a point; as the angle E.
It is clear that angles are altogether unaffected by the length of the lines, by E the meeting of which they are formed. See the Observation after Definition 18.
10. When a straight line, standing on another straight line, makes the adjacent an
gles 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.
11. An obtuse angle is that which is greater than a right angle.
12. An acute angle is that which is less than a right angle.
Thus the angles ABC, ABD are each of them right angles ; EBC is an obtuse, and EBD an acute angle; and the line AB is a perpendicular
13. “A term or boundary is the extremity of any thing."
14. A figure is that which is enclosed by one or more boundaries.
The quantity of space contained in any figure is called its area.
15. A circle is a plane figure contained by one line, which is called the circumference; and is such that all straight lines, drawn from a certain point within the figure to the circumference, are equal to one another :
16. And this point is called the centre of the circle.
17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
18. A semicircle is the figure contained by a diameter, and the part of the circumference cut off by the diameter.