TABLE OF CIRCUMFERENCES AND AREAS OF CIRCLES CORRE- A TREATISE ON GEOMETRY, AND ITS APPLICATION IN THE ARTS. CHAPTER I. OF STRAIGHT LINES AND PLANE SURFACES. (1.) THE Science which in the present advanced state of knowledge under the title of Geometry comprehends so vast and important a field of human inquiry, was at its origin confined in its application to the art of measuring small portions of the earth's surface, and probably had no higher object than to determine the magnitude and fix the limits of property. The annual overflowings of the river Nile obliterated the ordinary boundaries by which the land was subdivided and appropriated, covering the surface with mud. It was therefore necessary to possess some means by which these artificial limits could be from time to time renewed, so that a map of the land being preserved, the property of each person could be re-established. exigency is said to have directed the attention of the Egyptians to the general properties of geometrical figures; and that as their beautiful relations were gradually developed, the art rose to more noble objects, and was regarded as a subject of higher speculation. This When, however, we consider the multitude of instances of the inevitable application of geometrical principles in the arts of life, even in the first stages of civilisation, it is impossible to conceive that the B discovery or observation of the most simple and obvious properties of geometrical figures could be confined to one country, or could be postponed beyond a very early date in the history of the human race. The natural forms presented by the animal, vegetable, and mineral worlds, the diversified appearance of the surface of the earth, as varied by hill and valley and intersected by seas and rivers, not to mention the equally obvious appearances of the firmament, could not fail to have suggested to the mind the relations of lines and angles, of surfaces flat and curved, and, in short, to have furnished a family of ideas which could not have been long contemplated, without producing some conceptions of general geometrical relations. It may, however, be admitted that such notions may have existed for a period of time, more or less considerable, in a separate and unconnected form, and that the peculiar physical circumstances, incidental to the country of the Nile, united with the early epoch of its civilisation, afford probable grounds for conjecture that these scattered principles, which the constant experience of life must have forced upon every mind, there first received a high degree of generality, and coalesced into a body under the badge of a distinct science. It was, however, after its importation into Greece, that geometry was brought to that state of perfection in which it has been handed down to modern times, having, fortunately, in the works of Euclid and others survived the dark ages. This science, considered as a part of public instruction, has two distinct objects. First, it may be regarded as an exercise by which the faculty of thinking and reasoning may be strengthened and sharpened. It is peculiarly fitted for this purpose by the accuracy and clearness of which its investigations are susceptible, and the very high certitude which attends its conclusions. Secondly, it is the immediate and only instrument by which almost the whole range of physical investigation can be conducted; without it we could not advance a step beyond the surface of the earth in our knowledge of the universe; without it we could obtain no knowledge of the figure or dimensions of the earth itself, nor of the mutual mechanical operation, or influence of bodies upon it. In fact there is scarcely a part of natural science in which geometry is not an indispensable instrument of inquiry. According as one or other of these objects have been kept in view, writers on geometry have imparted more or less rigour to their reasonings, and limited their inquiries to topics having more or less immediate application to the arts of life. In the course of instruction followed by the great mass of students in our universities, geometry has been regarded almost exclusively as a system of intellectual gymnastics; while, on the other hand, owing to the very stinted portion of instruction attainable by those who are engaged in the useful arts, the science is with them almost degraded to a mass of rules, without reasons, and dicta, the truth of which is expected to be received on the authority of the writer, and of which the reader is not put in a con-dition to judge. Such are the extremes of exclusively practical and exclusively theoretical works. Treatises on this subject, holding an intermediate position, and combining to a considerable extent that rigour of reasoning which has conferred so much beauty and celebrity on the science, with a portion of its useful applications, are less common in this country than in other parts of Europe, where the business of education is conducted with less confined objects. It is our present purpose to endeavour to supply such views of this science as will be found useful to those classes, who while they do not pursue geometry as a mere intellectual exercise, are capable, nevertheless, of appreciating its clearness and certainty, and are unwilling to receive a proposition as true without a proof of it, where a proof may be obtained; and who, on the other hand, also delight to contemplate some of the most important useful purposes to which the abstract principles of the science have been applied. There is no part of geometry which has given rise to so much and so unprofitable discussion as the formal explanation of those terms which express the primary notions involved in geometrical investigations. According to the rigorous method of treating of the science, it has been thought indispensable to lay down in the first instance certain formal definitions of the objects or notions which constitute the subjects of investigation, and from those and certain propositions called axioms to deduce all the conclusions of the science. The meaning of a term may be made known in either of three ways::- First, by another term synonymous with it, the import of which may happen to be better understood; Secondly, by shewing the object or thing signified by the term to be explained; Thirdly, by a sentence composed of several terms not synonymous with each other, but signifying collectively the meaning of the term to be explained. It is the last alone which can be properly called a definition. A synonymous term may not be better understood than the term to be explained, and will itself stand equally in need of definition. To show the object will be effectual, when an object can be found which is a strict representative of the term in question. This, however, is not always the case. The explanation of a term by several other terms not synonymous with each other, is applicable only to terms expressing compounded notions, and cannot have any application to terms of simple and uncompounded meaning, because the several terms of which such a definition is composed, signifying many different conceptions of the mind, cannot represent a term which signifies one uncompounded conception.* It is obvious that definition must stop somewhere. Since one term can only be defined by other terms, these others themselves must be defined; and it is clear that we must ultimately come to a term, the meaning of which must be obtained by some means independent of mere language. Now it so happens, that all these * See Locke on the Human Understanding, Book III. |