process, that of conveying to another mind a new simple idea, that is, a simple idea with which hitherto it had not been familiar, can only be done by a certain system of abstraction, that is, by presenting to it some more complex idea, in which the first simple idea is involved with others with which the mind is already familiar, by the successive abstraction of which it is left in possession of the simple idea. Let us take, as an illustration, the process by which the idea of a mathematical point would be conveyed to the mind of a person for the first time. We should first present to him the complex idea of a physical point, such as the point of a pencil or of a needle, with which he would already be familiar; we should then explain to him that the physical point involved two ideas, one of position and one of magnitude; and further, that the less the magnitude was supposed to become (or, in ordinary terms, the finer we supposed the point to be), the more precise and definite would become the position which the point occupies, and serves to mark or identify; and we should thus lead him, by the gradual abstraction of the idea of the magnitude of the point, to look upon it as infinitely small, and only to associate with it the idea of its position; and thus he would realize the idea of a mathematical point. In like manner, with a mathematical line, we should first present to him a line such as a pencil would trace on a sheet of paper, and direct his attention to the fact that the line so drawn was in reality a solid or magnitude having three dimensions, namely length, breadth, and thickness, the breadth and thickness of the black lead left by the pencil on the surface of the paper, and which constitutes the physical line presented to the eye. We should then point out the extreme minuteness of the two last dimensions of the line as compared with its length, and ask him to conceive these dimensions as becoming less and less until the idea of the line presented itself to his mind as only possessing length, but devoid of breadth and thickness. Such, then, being the process by which the mind arrives at the true idea which such words as point, line, and surface represent, it would be impossible, by any formal definition of a new term, to convey for the first time, accurately, the idea which it represented. Thus, to a mind that had never heard of a mathematical point, the definition which we have given above of such a point would convey no idea. It must not, however, be therefore supposed that the foregoing definitions are useless; they serve to show the precise and limited sense in which the words are used in the following work, and so avoid the ambiguity which would arise if employed with the same latitude as in ordinary conversation. They show, for example, that the only quality of a point recognised in geometry is its position; of a line, its direction and length; and of a surface, its position and extent. 7. PARALLEL STRAIGHT LINES are straight lines in the same plane, which, being produced to any extent in both directions, would never meet. 8. A RECTILINEAL ANGLE is a magnitude formed by the inclination (ie., opening or divergence) of two straight lines to one another, which meet in a point. SCHOLIUM. We have here defined an angle to be a magnitude, and we are anxious to impress the idea of its being so on the mind of the mathematical student, as it will materially assist his progress when pursuing his studies in the higher branches of analysis. In conceiving, however, an angle as a magnitude, we must be careful not to confound with it the notion of the surface situated between the two lines which form the angle, or to look upon its magnitude as in any way affected by the length of those lines. By way of explanation, let us borrow an illustration from the hands of a clock, and regard the angle formed by their center lines. Now at twelve B o'clock, as the hands coincide, no angle is formed by them; but from that moment they cease to coincide, and the magnitude of the angle becomes every instant greater as the minute hand moves away from the other. Now at any definite portion of time, such, for instance, as ten minutes, the hands of the clock form an angle of a certain definite magnitude, which is precisely the same whatever may be the length of those hands, whether they belong to a pocket watch or to a turret clock; in either case the interval of time is indicated by the divergence of the hands, or, in other words, by the magnitude of the angle which they form. The student has further been probably accustomed to regard the angle formed by two lines as being necessarily less than two right angles; so that if two lines were situated as in the margin, he would find it difficult to regard the angle which they formed as being that which is shaded, and would probably only be able to conceive it as the white opening BAČ. Or, to revert to the hands of the clock, at three o'clock he would regard the hands as forming an angle of 90°, or the fourth part of a circle; but as the angle which they form is the measure of the angular distance that the minute hand has moved from the hour hand, we must measure that distance in the direction of its motion, and we thus find that angular distance to be 270°, or three-fourths of a circle. And by thus regarding the magnitude of an angle as the measure of the angular distance that a line revolving about one of its extremities has moved from its normal or first position, we are enabled to realize the idea of an angle greater even than four right angles or an entire revolution. We have only to conceive the line as moving at a uniform rate, so that the magnitude of the angle may be estimated by the length of time that it has been in motion, to see that, if the time exceeded that in which an entire revolution was performed, the angle whose magnitude it indicated had become greater than four right angles; and in like manner, after the interval required to complete two revolutions, that the angle had become greater than eight right angles, and so forth. B The point in which the two lines forming an angle meet is termed the vertex, and the two lines are termed the sides. The angle is referred to by a letter placed at the vertex, as the angle A; but if more than one angle is formed at the same point, it is then designated by three letters, one on each side and one at the vertex, the latter always being placed between the others, as the angle BCD. The angles formed by the sides of rectilineal figures derive a variety of designations according to their relative positions and magnitudes, which will be defined in subsequent scholia. 9. A RIGHT ANGLE is half the angle, formed by a straight line with its continuation. 1 SCHOLIUM. The line which divides the angle formed by a straight line with its continuation, into two right angles, is said to be perpendicular to that straight line. Thus the line CD is perpendicular to the line AB, and the angles ACD and BCD are both right angles. 10. An OBTUSE ANGLE is an angle which is greater than a right angle. 11. AN ACUTE ANGLE is an angle which is less than a right angle. 12. A PLANE FIGURE is a plane surface which is bounded on all sides by one or more lines. SCHOLIUM. The bounding line of a plane figure is termed its perimeter, and the space which is contained within the same is termed the area of the figure. 13. A CIRCLE is a plane figure bounded by one curved line, which is such that all straight lines drawn from it to a certain point within the figure are equal. 14. The CIRCUMFERENCE of a circle is the curved line by which it is bounded. 15. The CENTER of a circle is a point within the figure equally distant from its circumference. 16. A RADIUS of a circle is a straight line drawn from the center to the circumference. 17. A DIAMETER of a circle is a straight line drawn through the center and terminated both ways by the circumference. SCHOLIUM. Thus the curved line ABCDF is the circumference of a circle, of which E is the center, BD a diameter, and AE a radius. Any portion of the circumference, as AFD, is termed an arc of the circle; the straight line AD joining its extremities is termed its chord; and the figure ADF contained by the arc and its chord is termed a segment. The space contained by two arcs of circles of different radii is termed a lune, as GHI. 18. A RECTILINEAL FIGURE is a plane surface, bounded on all sides by straight lines. SCHOLIUM. The straight lines by which a rectilineal figure is bounded are termed its sides, which are said to contain the figure. 19. A TRIANGLE is a rectilineal figure which is bounded by three straight lines. SCHOLIUM. For convenience one of the lines by which a triangle is contained is termed the base of the triangle, the other two lines being termed its sides, and the point in which the two sides meet is termed the vertex. 20. An EQUILATERAL TRIANGLE is a triangle which has three sides equal. 21. An ISOSCELES TRIANGLE is a triangle which has only two sides equal. SCHOLIUM. When all the three sides of a triangle are unequal, it is sometimes termed a scalene triangle. 22. A RIGHT-ANGLED TRIANGLE is of which form a right angle. triangle two sides SCHOLIUM. In a right-angled triangle the third side opposite to the right angle is termed the hypotenuse; and in any triangle any side is said to subtend the angle opposite to it; thus the hypotenuse subtends the right angle. 23. An OBTUSE-ANGLED TRIANGLE is a triangle two sides of which form an obtuse angle. 24. An ACUTE-ANGLED TRIANGLE is a triangle the sides of which form three acute angles. 25. A QUADRILATERAL FIGURE is a rectilineal figure which is bounded by four straight lines. SCHOLIUM. A straight line drawn from any two opposite angles of a quadrilateral figure is termed a diagonal. 26. A PARALLELOGRAM is a quadrilateral figure whose opposite sides are parallel. SCHOLIUM. If a diagonal AC be drawn to any parallelogram ABCD, and lines GH and EF be drawn respectively parallel to two contiguous sides of the same, so as to intersect in some point K of the diagonal, the parallelogram will be divided into four parallelograms, two of which, AEKH and KGCF, are said to be about the diagonal, and the other two of which, EBGK and HKFD, are termed the complements of the former. E H For brevity parallelograms are frequently designated by only two letters placed at the opposite corners, as the parallelogram EG, instead of EBGK. 27. A RECTANGLE is a parallelogram two of whose sides form a right angle. SCHOLIUM. As a rectangle is contained under four lines, two of which, AB and BC, are equal the other two, CD and AD, it is designated as the rectangle under those two lines; thus the rectangle ABCD would be termed the rectangle under AB and BC. 28. A SQUARE is a quadrilateral figure which has all its sides equal, and two sides of which form a right angle. A B SCHOLIUM. As a square is contained by four equal lines, upon either of which it may be conceived as constructed, it is designated as the square on one of those lines, as the B square on the line AB. 29. A POLYGON is a rectilineal figure which is bounded by more than four sides. E SCHOLIUM. In any rectilineal figure, ABCDEF, the angles formed by its several sides on the inner side and distinguished by being shaded, are termed the internal or interior angles; when the internal angle is less than two right angles, it is termed a salient angle, as the angle E; but when greater, it is termed a reëntrant angle, as the internal angle at F. If any side is produced, the angle which its production makes with the contiguous side is called the exterior or external angle. Thus, if the side AB A is produced to G, the angle CBG is termed the external angle at B. When two straight lines intersect, as AB and CD, the two opposite equal angles, as AEC and DEB, are termed vertical angles; while those contiguous, as AEC and CEB, are termed adjacent angles. When a straight line, as AB, intersects two other straight lines, as CD and EF, the angles CGH and GHF are said to be alternate angles, as are also DGH and GHE. POSTULATES. Let it be granted:— 1. That a straight line may be drawn from any point to any other point. 2. That any finite straight line may be extended, or produced, to any length. 3. That a circle may be described from any center with any radius. SCHOLIUM. A Postulate is a problem the solution of which is self-evident, and therefore requiring no demonstration; it will be observed that they are only subsequently employed in the construction of theorems or the solution of problems, but never in the demonstration. The third postulate points out the restricted use allowed to the compasses, namely, only to describe circles, but never to measure or carry distances. The compasses must be conceived as closing whenever removed from actual contact with the surface of the paper. |