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And because DE is parallel to BG,

the alternate angles EDT, BGT are equal; (xI. 29.) and the angle DTY is equal to the angle GTS: (1.15.) therefore in the triangles DTY, GTS there are two angles in the one equal to two angles in the other, and one side equal to one side, opposite to two of the equal angles, viz. DY to GS;

for they are the halves of DE, BG:

therefore the remaining sides are equal, each to each :_(1. 26.) wherefore DT is equal to TG, and YT equal to TS. Wherefore, if in a solid, &c.

Q. E. D.


If there be two triangular prisms of the same altitude, the base of one of which is a parallelogram, and the base of the other a triangle; if the parallelogram be double of the triangle, the prisms shall be equal to one another.

Let the prisms ABCDEF, GHKLMN be of the same altitude, the first whereof is contained by the two triangles ABE, CDF, and the three parallelograms AD, DĚ, EC; and the other by the two triangles GHK, LMN, and the three parallelograms LH, HN, NG; and let one of them have a parallelogram AF, and the other a triangle GHK, for its base.

If the parallelogram AF be double of the triangle GHK, the prism ABCDEF shall be equal to the prism GHKLMN.

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Complete the solids AX, GO:

and because the parallelogram AF is double of the triangle GHK; and the parallelogram HK double of the same triangle;

(1. 34.) therefore the parallelogram AF is equal to HK: but solid parallelopipeds upon equal bases, and of the same altitude, are equal to one another; (xI. 31.)

therefore the solid AX is equal to the solid GO:

and the prism ABCDEF is half of the solid AX; (x1. 28.) and the prism GHKLMN half of the solid GO: (x1. 28.) therefore the prism ABCDEF is equal to the prism GHKLMN. Wherefore, if there be two, &c.

Q. E. D.


THE Eleventh Book of the Elements commences with the definitions of the Geometry of Planes and Solids, and then proceeds to demonstrate the most elementary properties of straight lines and planes, solid angles and parallelopipeds.

The solids considered in the eleventh and twelfth books are Geometrical solids, portions of space bounded by surfaces which are supposed capable of penetrating and intersecting one another.

In the first six books, all the diagrams employed in the demonstrations are supposed to be in the same plane, which may lie in any position whatever, and be extended in every direction, and there is no difficulty in representing them roughly on any plane surface; this, however, is not the case with the diagrams employed in the demonstrations in the eleventh and twelfth books, which cannot be so intelligibly represented on a plane surface on account of the perspective. A more exact conception may be attained, by adjusting pieces of paper to represent the different planes, and drawing lines upon them as the constructions may require, and by fixing pins to represent the lines which are perpendicular to, or inclined to any planes.

Any plane may be conceived to move round any fixed point in that plane, either in its own plane, or in any direction whatever; and if there be two fixed points in the plane, the plane cannot move in its own plane, but may move round the straight line which passes through the two fixed points in the plane, and may assume every possible position of the planes which pass through that line, and every different position of the plane will represent a different plane; thus, an indefinite number of planes may be conceived to pass through a straight line which will be the common intersection of all the planes. Hence, it is manifest, that though two points fix the position of a straight line in a plane, neither do two points nor a straight line fix the position of a plane in space. If however, three points, not in the same straight line, be conceived to be fixed in the plane, it will be manifest, that the plane cannot be moved round, either in its own plane or in any other direction, and therefore is fixed.

Also any conditions which involve the consideration of three fixed points not in the same straight line, will fix the position of a plane in space; as two straight lines which meet or intersect one another or two parallel straight lines in the plane.

Def. v. When a straight line meets a plane, it is inclined at different angles to the different lines in that plane which may meet it; and it is manifest that the inclination of the line to the plane is not determined by its meeting any line in that plane. The inclination of the line to the plane can only be determined by its inclination to some fixed line in the plane. If a point be taken in the line different from that point where the line meets the plane, and a perpendicular be drawn to meet the plane in another point; then these two points in the plane will fix the position of the line which passes through them in that plane, and the angle contained by this line and the given line, will measure the inclination of the line to the plane; and it will be found to be the least angle which can be formed with the given line and any other straight line in the plane.

If two perpendiculars be drawn upon a plane from the extremities of a straight line which is inclined to that plane, the straight line in the plane intercepted between the perpendiculars is called the projection of the line on that plane; and it is obvious that the inclination of a straight line to a plane is equal to the inclination of the straight line to its projection on the plane. If however, the line be parallel to the

plane, the projection of the line is of the same length as the line itself; in all other cases the projection of the line is less than the line, being the base of a right-angled triangle, the hypothenuse of which is the line itself.

The inclination of two lines to each other, which do not meet, is measured by the angle contained by two lines drawn through the same point and parallel to the two given lines.

Def. VI. Planes are distinguished from one another by their inclinations, and the inclinations of two planes to one another will be found to be measured by the acute angle formed by two straight lines drawn in the planes, and perpendicular to the straight line which is the common intersection of the two planes.

It is also obvious that the inclination of one plane to another will be measured by the angle contained between two straight lines drawn from the same point, and perpendicular, one on each of the two planes.

The intersection of two planes suggests a new conception of the straight line.

Def. IX. When a solid angle is contained by three plane angles, each plane which contains one plane angle, is fixed by the position of the other two, and consequently, only one solid angle can be formed by three plane angles. But when a solid angle is formed by more than three plane angles, if one of the planes be considered fixed in position, there are no conditions which fix the position of the rest of the planes which contain the solid angle, and hence, an indefinite number of solid angles, unequal to one another, may be formed by the same plane angles, when the number of plane angles is more than three.

Def. x is restored, as it is found in the editions of Euclid, by Dr Barrow and others. It appears to be universally true, supposing the planes to be similarly situated, in which are contained the corresponding equal plane angles of each figure.

Def. A. Parallelopipeds are solid figures in some respects analogous to parallelograms, and remarks might be made on parallelopipeds similar to those which were made on parallelograms in the notes to Book 11, p. 67; and every right-angled parallelopiped may be said to be contained by any three of the straight lines which contain the three right angles by which any one of the solid angles of the figure is formed; or more briefly, by the three adjacent edges of the parallelopiped.

As all lines are measured by lines, and all surfaces by surfaces, so all solids are measured by solids. The cube is the figure assumed as the measure of solids or volumes, and the unit of volume is that cube, the edge of which is one unit in length.

If the edges of a rectangular parallelopiped can be divided into units of the same length, a numerical expression for the number of cubic units in the parallelopiped may be found, by a process similar to that by which a numerical expression for the area of a rectangle was found.

Let AB, AC, AD be the adjacent edges of a rectangular parallelopiped AG, and let AB contain 5 units, AC, 4 units, and AD, 3 units in length.

Then if through the points of division of AB, AC, AD, planes be drawn parallel to the faces BG, BD, AE respectively, the parallelopiped will be divided into cubic units, all equal to one another.

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And since the rectangle ABEC contains 5 × 4 square untis, (note, p. 68.) and that for every linear unit in AD there is a layer of 5 × 4 cubic units corresponding to it;

consequently, there are 5 × 4 × 3 cubic units in the whole parallelopiped AG. That is, the product of the three numbers which express the number of linear units in the three edges, will give the number of cubic units in the parallelopiped, and therefore will be the arithmetical representation of its volume.

And generally, if AB, AC, AD; instead of 5, 4 and 3, consisted of a, b and c linear units, it may be shewn, in a similar manner, that the volume of the parallelopiped would contain abc cubic units, and the product abc would be a proper representation of the volume of the parallelopiped.

If the three sides of the figure were equal to one another, or b and c each equal to a, the figure would become a cube, and its volume would be represented by aaa, or a3. Prop. VI. From the diagram, the following important construction may be made. If from B a perpendicular BF be drawn to the opposite side DE of the triangle DBE, and AF be joined; then AF shall be perpendicular to DE, and the angle AFB measures the inclination of the planes AED and BED.

Prop. XIX. It is also obvious, that if three planes intersect one another; and if the first be perpendicular to the second, and the second be perpendicular to the third; the first shall be perpendicular to the third; also the intersections of every two shall be perpendicular to one another.

Prop. XXXIII. Algebraically.

Let the adjacent edges of the solid AB contain a, b, c units,
and those of the solid CD contain a', b', c' units respectively.
Also, let V, V' denote their volumes.

Then Vabc, and V' a'b'c'.


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In a similar manner, it may be shewn that the volumes of all similar solid figures

bounded by planes, are proportional to the cubes of their homologous edges.



If from the greater of two unequal magnitudes, there be taken more than its half, and from the remainder more than its half; and so on : there shall at length remain a magnitude less than the least of the proposed magnitudes. (Book x. Prop. 1.)

Let AB and C be two unequal magnitudes, of which AB is the greater.
If from AB there be taken more than its half,

and from the remainder more than its half, and so on;
there shall at length remain a magnitude less than C.

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For C may be multiplied so as at length to become greater than AB. Let it be so multiplied, and let DE its multiple be greater than AB, and let DE be divided into DF, FG, GE, each equal to C. From AB take BH greater than its half,

and from the remainder AH take HK greater than its half, and so on, until there be as many divisions in AB as there are in DE: and let the divisions in AB be AK, KH, HB;

and the divisions in DE be DF, FG, GE.

And because DE is greater than AB,

and that EG taken from DE is not greater than its half, but BH taken from AB is greater than its half;

therefore the remainder GD is greater than the remainder HA. Again, because GD is greater than HA, and that GF is not greater than the half of GD, but HK is greater than the half of HA; therefore the remainder FD is greater than the remainder AK: and FD is equal to C,

therefore C is greater than AK;

that is, AK is less than C. Q. E. D.

And if only the halves be taken away, the same thing may in the same way be demonstrated.

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Similar polygons inscribed in circles, are to one another as the squares of their diameters.

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