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and the pyramid F-ABC have equal bases ABC, DEF; they have also the same altitude, namely, the distance of the parallel planes ABC, DEF; hence the two pyramids are equivalent. Now the pyramid F-CDE has already been proved equivalent to F-ACE ; consequently the three pyramids F-ABC, F-CDE, F-ACE, which compose the prism ABD are all equivalent. Therefore the pyramid F-ABC is the third part of the prism ABD, which has the same base and the same altitude.

570. Corollary. The solidity of a triangular pyramid is equal to a third part of the product of its base by its altitude.

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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 desiga nate 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 Geom.

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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 equivalent, 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, equivulent 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 conver, 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

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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, or 2 n - 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.

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manner.

1. Two triangles are similar, when they have two angles equal, each to each.

2. To 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

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