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PROPOSITION D. THEOREM.
Solid parallelopipeds contained by parallelograms equiangular to one another, each to each, that is, of which the solid angles are equal, each to each, have to one another the ratio which is the same with the ratio compounded of the ratios of their sides.
The bases and altitudes of equal solid parallelopipeds, are reciprocally proportional: and conversely, if the bases and altitudes be reciprocally proportional, the solid parallelopipeds are equal.
PROPOSITION XXXV. THEOREM.
If, from the vertices of two equal plane angles, there be drawn two straight lines elevated above the planes in which the angles are, and containing equal angles with the sides of those angles, each to each; and if in the lines above the planes there be taken any points, and from them perpendiculars be drawn to the planes in which the first named angles are; and from the points in which they meet the planes, straight lines be drawn to the vertices of the angles first named; these straight lines shall contain equal angles with the straight lines which are above the planes of the angles.
COR. From this it is manifest, that if from the vertices of two equal plane angles, there be elevated two equal straight lines containing equal angles with the sides of the angles, each to each; the perpendiculars drawn from the extremities of the equal straight lines to the planes of the first angles, are equal to one another.
PROPOSITION XXXVI. THEOREM.
If three straight lines be proportionals, the solid parallelopiped described from all three, as its sides, is equal to the equilateral parallelopiped described from the mean proportional, one of the solid angles of which is contained by three plane angles equal, each to each, to the three plane angles containing one of the solid angles of the other figure.
PROPOSITION XXXVII. THEOREM.
If four straight lines be proportionals, the similar solid parallelopipeds similarly described from them shall also be proportionals: and, conversely, if the similar parallelopipeds similarly described from four straight lines be proportionals, the straight lines shall be proportionals.
PROPOSITION XXXVIII. THEOREM.
"If a plane be perpendicular to another plane, and a straight line be drawn from a point in one of the planes perpendicular to the other plane, this straight line shall fall on the common section of the planes.”
PROPOSITION XXXIX. THEOREM.
In a solid parallelopiped, if the sides of two of the opposite planes be divided, each into two equal parts, the common section of the planes passing through the points of division, and the diameter of the solid parallelopiped, cut each other into two equal parts.
PROPOSITION XL. THEOREM.
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.
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.
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. Στερεὰ γωνία ἐστὶν ἡ ὑπὸ πλειόνων ή δύο γωνιῶν ἐπιπέδων περιεχο μένη, μὴ οὐσῶν ἐν τῷ αὐτῷ ἐπιπέδῳ πρὸς ἑνὶ σημείῳ, συνισταμένων. The rendering by Simson of this definition may be slightly amended. The word wepiexoμévn is rather comprehended or contained than made: and ovviotaμévwv means joined and fitted together, not meeting. "A Solid angle is that which is contained by more than two plane angles joined together at one point, (but) which are not in the same plane."
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 the Greek text of Euclid. 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. XIV. The sphere, as well as the cone and the cylinder are defined by a mode in which the figures may be conceived to be generated. Here motion is for the first time introduced in defining of Geometrical figures. In these motions, the successive change of position only is considered which a figure undergoes, and the figure traced out in consequence of the motion. The velocity with which the new figure is traced out, as well as the time and force requisite for effecting it, are considerations which do not enter into the subject of Geometry.
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 rectangles in the notes to Book 11, p. 81; and every rightangled 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 4B, 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.
And since the rectangle ABEC contains 5 x 4 square units, (note, p. 81.) and that for every linear unit in AD there is a layer of 5 × 4 cubic units corresponding to it;
consequently, there are 5 x 4 x 3 cubic units in the whole parallelopiped 4G. 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 a a a, or a3.
It may easily be shewn Algebraically that the volumes of similar rectangular parallelopipeds are proportional to the cubes of their homologous edges.
Let the adjacent edges of the parallelopiped AB contain a, b, c units, and those of another similar parallelopiped A'B' contain a', b', c' units respectively. Also, let V, V' denote their volumes.
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. 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 BÉD.
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.
The Demonstrations of the last Nineteen Propositions of the Eleventh Book have been omitted, as they are not included in the course of reading prescribed for Mathematical Honours at Cambridge.
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.
and from the remainder more than its half, and so on;
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 ВÍ taken from AB is greater that 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.