the hour's labour which it requires to learn dexterity and accuracy in the practice of them; but we can only recommend that, and pass to another section and branch of the science.

SECTION IX.

PRELIMINARY NOTIONS OF GEOMETRY.

As there are some of those considerations in the general science of quantity, which come next to be examined in that order which we conceive to be most conducive to clear and simple views of the elements of the whole, which have as much reference to geometry as either to arithmetic or to quantity generally, it will be proper to take some notice of the simpler portion of that branch, or rather application, of the general science.

It has been already mentioned, that geometry is the science of magnitude, or quantity considered as extended in some way or other, so as to occupy space; but that which occupies space is not the only subject of geometrical investigation, for, before we can arrive at a distinct notion of even the simplest body which can be supposed to occupy space, there are many elements to be considered, and many relations to be understood; for, though we do not, in our geometrical inquiries, trouble ourselves about those physical qualities of real bodies which form the distinctions between one kind of body and another, and which give to each of them its peculiar practical value, yet that which we call a body, or solid, is not an original perception of our senses, but a compound result of various relations of elements, all of which we must thoroughly understand before

we can have a proper geometrical notion of the body or solid itself; and if any of those relations happen to be indeterminate, our knowledge of the solid is vague and imperfect. Thus, for instance, though we have a general notion that the earth which we inhabit approaches pretty nearly in form to a round ball, or globe, as we call it; that the diameter, or measure straight through the centre of it, is nearly 8,000 miles; that the diameter measured at the equator, or that region where day and night are about equal at all times of the year, is about 30 miles longer than the cross diameter extending from pole to pole, or from the one or the other of those places where we know, by wellfounded inference, though not by positive observation, that the year consists of one day and one night, of nearly, but not exactly, equal length;—though we know all this, and though some of the most able men of all ages, having the best claim to be considered scientific, have devoted their best attention to the determining of the earth's figure, yet there are many particulars which prevent the results of their labours from being perfectly accurate, and even deprive them of the means of ascertaining the degree of accuracy which they actually attain. Nor is this the case with the globe itself merely, but with every portion of it, and with every portion of matter to which we practically apply, or can apply our geometry. If a field of ground, or even the length of a road, or the breadth of a river, or the distance of any one point of the earth's surface from any other point, is measured with the greatest care by two different surveyors, or even twice over by the same surveyor, the chance, nay, almost the certainty is, that the results will not correspond, and one would be half inclined to suppose that things alter their shapes and sizes for the very purpose of perplexing us in our measurements. Even the standard of our mensuration is liable to vary: if it is a piece of tape, or a twisted cord,

it is shorter when the weather is damp than when it is dry; and if it is a metal chain or rod, it is longer when the weather is warm than when it is cold.

There are many other known causes of incorrectness than these, and of what are not known, but may exist, we are of course ignorant. Those circumstances are mentioned, not with a view of decrying the merits of geometry, but merely to show that, however any of our sciences may profess to be perfect in theory, we must not rest satisfied with a single branch, when we come to apply them to real practice, but must know the properties of things themselves, and the variations to which they are liable, as well as those abstract sciences of which we make them the subjects.

It will be readily admitted, that, as we have so many causes of error to contend with in the application of our geometry, it behoves us to be very accurate in that geometry itself, and not to allow ourselves to add the disadvantage of an imperfect and badly understood tool, to that of ungainly or unmanageable materials.

Many of the words which we use in a strict sense in geometry, are used much more loosely in common language, and therefore we require to make ourselves well acquainted with the difference between the scientific and the popular meaning. Indeed it is the want of attention to these differences, and the consequent looseness of our fundamental definitions in science, that the greater number of the hardships which we feel in the study of it, and the blunders which we make in its application, are mainly to be attributed.

Even the word solid has popular meanings different from its geometrical one; and the geometrical solid is not necessarily a solid body, or a body which has real existence at all; it is a certain portion of space, the boundaries of which, in all their

dimensions and relations to each other, are known and determined; and if those dimensions and their relations to each other are the same, it is of no consequence whether the space which they bound is filled with any one kind of matter, solid, or liquid, or air, or whether it is empty space, or even has any existence. Its existence, full or empty, must, however, be possible upon geometrical principles; that is, there must be nothing absurd in any of the relations which the dimensions or other characters of the geometrical solid bear to each other; as, for instance, the solid must have length, and breadth, and thickness, that is, three dimensions situated with regard to each other, in directions afterwards to be explained; but whether any of these are the same with each other, or whether any one of them is the same at two points of the solid, must depend upon circumstances. So also the solid, in order to be geometrical, must have boundaries which inclose it everywhere, but do nothing more, and which, from their known figures, dimensions, and situations with regard to each other, determine the form of the solid.

We can determine nothing geometrically without measuring, and measuring by the application of a standard, which, as must be the case with all standards, must be of the same kind with that which we can measure; and not only must we have a standard and apply it, but we must begin at some beginning, and this beginning is the primary and simplest of all geometrical considerations. In order that we may have a definite or known measurement, we must have an end as well as a beginning; and as any distance or extent in space is the same, whether we measure it in one direction or in the opposite, the end and the beginning stand precisely in the same relation to the measure of which they are the terminations. Thus, for example, it would take exactly the same number of revolutions

of the same carriage-wheel to pass over the road between London and York, whether the carriage began the journey at London and ended at York, or began it at York and ended it at London.

This simplest of geometrical elements is called a POINT, and, simple as it is, it is necessary to have very clear notions of it, because even it has given rise to some idle use of words. A point means position merely, and therefore it is not considered as occupying any portion of that space in which it is situated; and when we consider nothing but a mere point and space, we may regard any point, wherever we imagine it to be situated as the centre of space, that is, as being in the very middle of space, or having equal measures of extension in all directions around it. Thus, in whatever part of its orbit the earth may be, oron whatever part of its surface a spectator may be situated, the centre of his eye, while he surveys the heavens around him, is to him the centre of space. But when a point has reference to any particular dimension or measure, it may be either of the extremities of that measure, or anywhere between them, to which allusion may have occasion to be made; yet, having no dimensions itself, it can form no part of any dimension; it bears, in fact, to extension nearly the same relation that 0 bears to number in arithmetic, for as no multiplication of O can produce even the smallest possible number, so no repetition of a point can produce even the smallest dimension; and so also, as no division by 0 can in the slightest degree diminish the smallest number, no division by points can diminish the smallest possible dimension.

A LINE is the geometrical element next in simplicity. There are various kinds of lines, straight lines and crooked or curved lines, and the latter may have single or double curvatures: thus, a road which makes a sweep upon perfectly level ground,