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DOCTRINE OF EQUALITY.
supposing there is equality or inequality inferrible, in cases where no inference can be drawn. The inefrence of the area is always from the product of the length and breadth, and if these are known, the product or area is known; but if the product or area only is known, the factors of that product-the length and breadth—are quite indeterminate, only we may gather from the case of equal factors of 16, that the factors of any product cannot together be less than the two equal factors which can form that product, and that these two equal factors must be the same for the same product.
The doctrines of equality and its opposite, which we have endeavoured to explain in this section, in a manner the most general, and the most simple as well as comprehensive in its application, not only to mathematical subjects but to all subjects where a question of equality can occur, is one which requires to be studied with the greatest care, because it is the foundation of very much of our accurate judgment, upon almost every question that can be named as determinate in its evidence, and also our best security against error in cases which involve uncertainty. The reader who wishes to profit by this book will therefore find his advantage in giving this particular section a second perusal.
INTERSECTION OF LINES, ANGLES, AND SIDES AND ANGLES OF
AFTER having obtained some general notion of the subjects of Geometry, as mentioned in section X., and the leading prin
ORDER OF PROCEEDING.
ciples of Geometrical investigation, as in section XI., we have to consider the order of the subjects, and to take them in that which appears to be at once the most simple and the most natural. For popular purposes the following is perhaps as convenient as any.
First, Lines, including straight lines and also the circle, with the intersections of straight lines, the angles which these form, and the connection between plane rectilineal angles and the circle. This is the proper foundation of the science, and contains the elements of the boundaries of the more simple elementary figures.
Secondly, SURFACES, that is, plane surfaces, or areas, considered in their extent, and with reference to their boundaries. But as an area is determinable only by an arithmetical multiplication, in any particular case, and as, consequently, the general investigation is a matter of quantity and operation jointly, it will be necessary, before we proceed to this, to consider the doctrine of proportion, the powers of quantities, and the arithmetic of exponents, each of which will form the chief subject of a section ; but, as they are all intimately connected, much reference from the one to the other will be required.
Thirdly, the INTERSECTIONS OF PLANEs, by means of which the forms of plane solids are determined; and this will include the doctrine of SOLID ANGLES, or of more planes than two meeting in the same point.
Fourthly, the CONTENT OR CAPACITY OF SOLIDS, taken in conjunction with the planes which form their boundaries, and the lines and angles made by the intersections of those planes.
These will put us in possession of the principles of elementary geometry, as far as they are necessary in the ordinary business of life ; and then we can return to the general science of quantity, and if our limits permit, to the applications. We
shall devote the remainder of this section to the consideration of lines and angles, the simplest case of which is that of a point.
Now any point, as, for instance, the point at A,
may be regarded as the centre of a plane, extending equally and immeasurably on all sides, so as to bisect or divide into two parts exactly equal to each other the whole of space. It is perfectly indifferent where we consider this point to be situated, because, as we can no more conceive or imagine a boundary to space in one direction than in another, we may suppose any point whatever, be it situated where it may, as being the centre of space, this point being a mark of position only, and having no extent in any direction, may be considered as equally the centre of the plane which bisects the whole of space, in whatever direction that plane is situated. According to our common notions, in which we regard a straight line directed to the centre of the earth as being the perpendicular, the plane may be in the direction of this perpendicular; it may be in the cross direction to this, or in what we call the horizontal position or the level ; or it may be at any slope whatever, and may slope in any direction; but in all the endless variety of positions which we can with equal propriety suppose it to have, and in all the endless changes of position in the plane, we may still conceive the place of the point as remaining exactly the same in absolute space, and the plane extending indefinitely every way, but every way equally, and in all possible positions dividing the whole of space into two parts exactly equal to each other.
This notion of the perfect immovability of a point, and the possibility of turning a plane on this point in every imaginable position throughout absolute space, has not hitherto, we believe, been alluded to in books on elementary geometry; but it is, in
truth, the grand primary conception, by means of which the geometer is enabled at once to lay his grasp upon the whole universe; and, seizing element after element as they arise, in due order and according to proper laws, to map
the tablet of his mind all that creation which God has made, as far as the line and the angle, magnified by the utmost perfection of the instruments of observation, can carry him.
As the space marked out by a plane round any point extends equally in all directions, the best representation which we can have for it is a circle, as, for instance, the circle of which the centre is the point a.
It will be recollected that the very definition of a circle is, that the circumference, or line bounding it, is in all directions equally distant from the centre; and therefore, if we imagine the radius, or distance from the circumference to the centre to be indefinitely long, the circle becomes the best representation which we can have for a plane extending through all space ; and because the circle which we have described round the point A is any circle, we may regard its circumference as representing all space round the point A; and farther, as the point a is the only thing which is supposed to have position, that is, to be fixed or determined in space, we may regard this circle as the representative of all space in every possible direction, or that within it, in one or other of its possible positions, it can contain every line, every figure, and every solid which can by possibility exist in nature, or which the most fertile imagination can
TWO POINTS IN SPACE.
picture to itself. When, however, we refer to the circle in one position as a plane we can consider it as including plane surfaces only, and as including those only which are situated in the same plane with the circle.
Let us next suppose that there are two points, that they are both fixed in position, and that a line is drawn through them both, as, for instance, the line cd, which passes through the two points A and B, and is continued, or might be continued, to an indeterminate length, in the left hand direction toward C,
and in the right hand direction toward D.
It is evident that, because the two points A and B are supposed to have position, that is, to have fixed places in direction and in distance from each other, the whole line cd as it appears, or as it could exist, though drawn countless millions of miles both toward c and toward D, is also fixed or definite in the direction of A and B, or of any other points that can be imagined to be taken in it. If now, then, we suppose a circle to be described round any point in the line cd, as, for instance, round the point a,
it follows that the plane of this circle must be confined to the line cd in the direction of whatever points it may cut this line, as, for instance, the points B and g in the above example; but it would be the same in the case of any other two points in the line