NOTES. I. Upon certain Names and Definitions. SOME new expressions and definitions have been introduced into this work which tend to give to the language of geometry more exactness and precision. We proceed to give an account of these changes, and to propose certain others, which might fulfil more completely the same purposes. In the ordinary definition of a rectangular parallelogram and of a square, it is said that the angles of these figures are right angles; it would be more exact to say, that their angles are equal. For, to suppose that the four angles of a quadrilateral may be right angles, and also that these right angles are equal to each other, is to suppose propositions which require to be demonstrated. This inconvenience and several others of the same kind might be avoided, if, instead of putting the definitions, as is usual, at the head of a section, we distributed them through the section each in the place where the proposition implied is demonstrated. The word parallelogram according to its etymology signifies parallel lines; it answers not better to a figure of four sides than to one of six, eight, &c., the opposite sides of which are parallel. Likewise the word parallelopiped signifies parallel planes; it does not designate a solid of six faces any more than one of eight, ten, &c., of which the opposite ones are parallel. It seems then that the denominations of parallelogram and parallelopiped, which have besides the inconvenience of being very long, ought to be banished from geometry. We might substitute in their place those of rhomb and rhomboid, which are much more convenient, and preserve the name of lozenge to denote a quadrilateral, the sides of which are equal. The word inclination ought to be understood in the same sense as that of angle; each indicates the manner of being of two lines, or of two planes, which meet, or which produced would meet. The inclination of two lines is nothing, when the angle is nothing, that is, when the lines are parallel or coincident. The inclination is greatest, when the angle is greatest, or when the two lines make with each other a very obtuse angle. The quality of leaning is taken in a different sense; a line leans so much the more with respect to another, as it departs more from a perpendicular to this last. The denomination of equal angles is given by Euclid and others to those triangles, which are only equal in surface; and that of equal solids to those which are only equal in solidity. It appears to us more proper to call the triangles as well as the solids in this case cquivalent, and to restrict the denomination of equal triangles and equal solids to those which would coincide upon being applied. It is, moreover, necessary to distinguish among solids and curved surfaces two different kinds of equality. Indeed two solids, two solid angles, two spherical triangles, or two spherical polygons, may be equal in all their constituent parts without coinciding when applied. It does not appear that this observation has been made in elementary books; and, for want of having regard to it, certain demonstrations, founded upon the coincidence of figures, are not exact. Such are the demonstrations by which several authors pretend to prove the equality of spherical triangles in the same cases and in the same manner as they do that of plane triangles. We are furnished with a striking example of this by Robert Simson, who, in attacking the demonstration of the 28th proposition of the eleventh book of Euclid, fell himself into the error of founding his demonstration upon a coincidence which does not exist. We have thought it proper, therefore, to give a particular name to this kind of equality, which does not admit of coincidence; we have called it equality by symmetry; and the figures which are thus related we call symmetrical figures. Thus the denominations of equal figures, symmetrical figures, equivalent figures, refer to different things and ought not to be confounded. In the propositions, which relate to polygons, solid angles, and polyedrons, we have expressly excluded those which have re-entering angles. For, in addition to the advantage of considering in the elements only the most simple figures, if we had not thus restricted ourselves, certain propositions would either not have been true, or would have required to be modified. We have, therefore, confined ourselves to the consideration of lines and surfaces, which we call convex, and which are such that they cannot be cut by a straight line in more than two points. We have often used the expression product of two or of a greater number of lines, by which we mean the product of the numbers which represent these lines, they being estimated according to a linear unit taken at pleasure. The sense of this word being thus fixed, there is no difficulty in making use of it. The same is to be understood of the product of a surface by a line, of a surface by a solid, &c. It is sufficient to have established once for all that these products are or ought to be considered as the products of numbers, each of a kind that is adapted to it. Thus the product of a surface by a solid is nothing else than the product of a number of superficial units by a number of solid units. We often use the word angle in common discourse to designate the point situated at its vertex; this expression is faulty. It would be more clear and more exact to denote by a particular name, as that of vertices, the points situated at the vertices of the angles of a polygon, or of a polyedron. In this sense is to be understood the expression vertices of a polyedron, which we have used. We have followed the common definition of similar rectilineal figures; but we would observe, that it contains three superfluous conditions. For, in order to construct a polygon of which the number of sides is n, it is necessary in the first place to know a side, and then to have the position of the vertices of the angles situated without this side. Now the number of these angles is n 2, and the position of each vertex requires two data; whence it follows that the whole number of data necessary to construct a polygon of n sides is 1+ 2n 4, 3. But in the similar polygon there is one side to be taken at pleasure; thus the number of conditions, by which one polygon becomes similar to a given polygon, is 2 n 4. But the common definition requires, 1. that the angles should be equal, each to each, which makes n conditions; 2. that the homologous sides should be proportional, which makes n 1 conditions. There are then in all 2n-1 conditions, or three too many. In order to obviate this inconvenience we can resolve the definition into two others, in this or 2 n manner. 1. Two triangles are similar, when they have two angles equal, each to each. 2. Two polygons are similar, when there can be formed in the one and the other the same number of triangles similar, each to each, and similarly disposed. But, in order that this last definition should not itself contain superflous conditions, it is necessary that the number of triangles should be equal to the number of sides of the polygon minus two, which may take place in two ways. We can draw from two homologous angles diagonals to the opposite angles; then all the triangles formed in 1 each polygon will have a common vertex, and their sum will be equal to the polygon; or rather we can suppose that all the triangles formed in a polygon have for a common base a side of the polygon, and for vertices those of the different angles opposite to this base. In each case the number of triangles formed being n 2, the conditions of their similitude will be equal to the number 2 n 4; and the definition will contain nothing superfluous. This new definition being adopted, the ancient one will become a theorem, which may be demonstrated immediately. If the definition of similar rectilineal figures is imperfect in books of elements, that of similar solid polyedrons is still more so. In Euclid this definition depends upon a theorem not demonstrated; in other authors it has the inconvenience of being very redundant; we have, therefore, rejected these definitions of similar solids.† The definition of a perpendicular to a plane may be regarded as a theorem; that of the inclination of two planes also requires to be supported by reasoning; the same may be said of several others. It is on this account that, while we have placed the definitions according to ancient usage, we have taken care to refer to propositions where they are demonstrated; sometimes we have merely added a brief explanation which appeared sufficient. The angle formed by the meeting of two planes, and the solid angle formed by the meeting of several planes in the same point, are distinct kinds of magnitudes, to which it would be well perhaps to give particular names. Without this it is difficult to avoid obscurity and circumlocutions in speaking of the arrangement of planes which compose the surface of a polyedron; and as the theory of solids has been little cultivated hitherto, there is less inconvenience in introducing new expressions, where they are required by the nature of the subject. I should propose to give the name of wedge to the angle formed by two planes; the edge or height of the wedge would be the common intersection of the two planes. The wedge would be designated by four letters, of which the two middle ones would answer to the edge. A right wedge then would be the angle formed by two planes perpendicular to each other. Four right wedges would fill all the solid angular space about a given line. This new denomination would not prevent the wedge always having for its measure the angle formed by two lines drawn from the same point, the one in one of The author here refers to a distinct note on the equality and similitude of polyedrons, not given in this translation. the planes and the other in the other, perpendicularly to the edge or common intersection. Cambridge Mail matics.. II. By the Translator. THE improvements referred to in the preceding note, so far as they have been adopted by the author, have been carefully preserved in the translation. Indeed it has been found necessary in a few instances to use English words in a sense somewhat different from their ordinary acceptation. The word polygon is generally restricted to figures of more than four sides. It is used in this work with the latitude of the original word polygone to stand for rectilineal figures generally; and polyedron is adopted in a similar manner for solids. Quadrilateral is employed as a general name for four-sided figures. The word losenge is rendered by rhombus, and trapéze by trapezoid, the English words, as they are commonly used, corresponding to the French. The perpendicular let fall from the centre of a regular polygon upon one of its sides is called in the original apothéme. It occurs but a few times, and as there is no English word answering to it, it is rendered by a periphrasis, or simply by the word perpendicular. The portion of the surface of a sphere comprehended between the semicircumferences of two great circles is denoted in the original by fuseau; Dr. Hutton uses the word lune in the same sense; others have employed lunary surface; as lune properly stands for the surface comprehended between two unequal circular curves, the latter denomination was thought the least exceptionable, and is adopted in the translation. III. On the Demonstration of the Proposition of Article 58. THE proposition of art. 58 is only a particular case of the celebrated postulate upon which Euclid has established the theory of parallel lines, as well as the theorem upon the sum of the three angles of a triangle. This postulate has not yet been demonstrated in a manner entirely geometrical, and independent of the consideration of infinity, which is undoubtedly to be attributed to the imperfection of the definition of a straight line, which serves as the basis of the elements. But, if we consider this subject in a point of view more abstract, analysis offers a very simple method of demonstrating the proposition rigorously. |