CoR. 2. If a straight line in a plane be prolonged, its prolongation lies wholly in that same plane. For it was shown (in Prop. XIII above) that if any two points in a plane be joined, both the straight line between them and its prolongation either way, shall lie wholly in that plane. CoR. 3. A circle may be described about any centre, and with any radius. For a plane of unlimited extent (as by Cor. 1 above) may be made to pass through the proposed centre, and turned till it also pass through the other extremity of the proposed radius; by the revolution of the radius in which plane, a circle will be described. CoR. 4. All the radii of the same circle are equal. And circles that have equal radii, are equal. For in the same circle, the straight line by the revolution of which the circle was described can be made to coincide with all •INTERc. 1. of the radii respectively; wherefore the radii are all" equal. And if two circles have equal radii, the two extremities of any radius of the one may be applied to the two extremities of any radius of the other; and because these two straight lines coincide, the figures described by their revolution about the same point will +1.Nom.14. coincide also, and coinciding will bet equal. CoR. 5. A straight line from the centre of a circle to a point outside, coincides with the circumference only in a point. {INTERC.10. For if about the centre of the circle bef described a sphere with Cor. 6. a radius equal to the radius of the circle, the straight line from *INTERc.10. the centre coincides" with the surface only in a point. And ** because the circle (as being described with the same radius) lies in the surface of the sphere and no where else; the straight line which coincides with the surface only in one point, can coincide with the circle only in that same point. PROPOSITION XIV. THEOREM.–Any three points are in the same plane. [That is to say, one plane may be made to pass through them all.] First Case. If the three points are in one straight line, any +Intenc.13. planet surface that is made to pass through the two extreme, will Nom. also pass through the other; for the whole of the straight line will lie in that surface. *INTERC, 2. Second Case ; where the three points are A If then a plane be made to pass through two of the points as B and C, it may be" turned about them till it also passes through the other point A. And by parity of reasoning, the like may be done in every other instance. Wherefore, universally, through any three points, one plane may be made to pass. Which was to be demonstrated. CoR. 1. Any three points which are not in the same straight *INTERC-12. line being joined, the straight lines which are the sides of thet Cor. 7. three-sided figure that is formed lie all in one plane. For (by Prop. XIV. above) a plane may be made to pass through the three points. And the straight lines joining them will lie wholly in that plane. CoR. 2. Any two straight lines which proceed from the same point, lie wholly in one plane. For (by Cor. 1 above) a plane may be made to pass through the point in which they meet, and also through their two other extreme points. And the straight lines between these points, will lie wholly in such plane. CoR. 3. If three points in one plane (which are not in the same straight line) are made to coincide with three points in another plane, the planes shall coincide throughout, to any extent to which they may be prolonged. For because the three points are in each of the two planes, the straight lines which join them and formit a three-sided figure will lie wholly in each of the planes; and consequently the planes will coincide throughout the straight lines which form the sides of such figure. But if the planes do not also coincide in all that is nvithin the figure, then from two points [among the points in which the planes coincide], may be drawn two straight lines [one in each of the planes, in the parts where they do not coincidel, which shall inclose a space. And if they do not further coincide in all that is nithout the figure, then two straight lines may be drawn [one in each of the planes, in the parts where they do not coincide], which shall have a common segment [in the part wherein the two planes coincide]. Each of which is impossible. f INTERc.12. Cor. 7. END OF THE INTERCALARY BOOK. INTER CALARY Book. Prop. 1. Prop. 1. Cor. 1. Prop. 1. Cor. 2. Prop. 1. Cor. 3. Prop. 1. Cor.4. Prop. 1. Cor. 5. Prop. 1. Cor. 7. Prop. 1. Cor. 8. RECAPITULATION of matters and propositions which have been usually received without proof under the title of Arioms, Postulates, &c., or have not been proved where required. With references to the places where they are demonstrated in the Intercalary Book. (Intended to enable such as postpone the reading of the Intercalary Book, to proceed as if the same had been prefixed under the title of Azioms, Postulates, &c.) Magnitudes which are equal to the same, are equal to one another. If of equals, one be equal to some thing else, the rest are severally equal to the same. If of equals, one be greater, or less, than some thing else; the rest are severally greater, or less, than the same. Or if some thing be greater, or less, than one; it is greater, or less, than each of the others also. Magnitudes which are equal to equals, are equal to one another. Of magnitudes, if to equals be added the same, the sums are equal. If equals be added to equals, the sums are equal. If unequals be added to equals, the sums are unequal. And that sum is greatest, in which the unequal was greatest. If equals be taken from equals, the remainders are equal. If equals be taken from unequals, the remainders are unequal. And that remainder is greatest, in which the unequal was greatest. Book. Prop. 1. Cor. 9. Prop. 1. Cor.10. Prop. 1. Cor. 11. Prop. 1. Cor.12. Prop.1. Cor.13. If unequals be taken from equals, the remainders are unequal. And that remainder is least, in which the unequal was greatest, Magnitudes which are double of the same or of equal magnitudes, are equal to one another. And so if, instead of the double, they are the treble, quadruple, or any other equimultiples. The double of a greater magnitude is greater than the double of a less. And so of any other equimultiples. Magnitudes which are half of the same or of equal magnitudes, are equal to one another. And so if, instead of the half, they are the third, fourth, or any other equipartites. The half of a greater magnitude is greater than the half of a less. And so of any other equipartites. If the doubles of two or more magnitudes be added together, the amount is double of the sum of the magnitudes. And so of any other equimultiples. If there be two unequal magnitudes, and from the double of the greater be taken the double of the less; the remainder is double of the difference of the magnitudes. And so of any other equimultiples. Any given magnitude may be multiplied, [that is to say, magnitudes equal to it may be added one to another], so as at length to become greater than any other given magnitude of the same kind which shall have been specified. Any solid, surface, line, or figure, may be turned about any one point or about any two points, in it ; such point or points remaining unmoved. From one of two assigned points to the other, to describe a line, which being turned about its extreme points, every point in it shall be without change of place. Such a line is called a straight line. A body or figure which is turned about two points in it that are also the eatremities of a straight line, (inasmuch as the whole of the straight line remains without change of place) is said to be turned round such straight line. When from any point to any other point, a straight line is described or made to pass ; the two points are said to be joined. If to a straight line addition is made at either end, in such manner that Prop. 1. Cor. 14. See Note. a Prop. 2. Cor. ... Prop. 9. See Note. the whole continues to be a straight line, the original straight line is said to be prolonged, and the part added is called its prolongation. 48 RECAPITULATION. A straight line may be described or made to pass from any one point to any other point. Between two points there cannot be more than one straight line. Two straight lines, which are not in one and the same line, cannot have a common segment. If any two points in one straight line coincide with two points in another, the two straight lines shall coincide with one another to the extent of the length that is common to both, and be one and the same straight line throughout. If two straight lines cut one another, they coincide only in a point. Any straight line may be applied to any other, so that they shall coincide to the extent of the length common to both. A terminated straight line may be prolonged to any length in a straight line. JWhen a straight line is said to be of unlimited length, the meaning is, that no point is assigned at which it shall be held to be terminated, but on the contrary it shall without further notice be considered as prolonged to any extent a motive may ever arise for desiring. Any three points (which are not in the same straight line) being joined, there shall be formed a three-sided figure ; and no point in any one side, shall coincide with any point in either of the other sides, except the points which were joined. To describe a surface in which any two points being taken, the straight line between them lies wholly in that surface. Such a surface is called a plane surface; or when no particular boundaries to it are intended to be specified, a plane. If to a plane surface addition is made in any direction, in such manner that the whole continues to be a plane surface, the original plane surface is said to be prolonged, and the part added is called its prolongation. A figure which lies wholly in one plane, is called a plane figure. The whole plane surface within the boundaries of a plane figure which is bounded on all sides, is called the area of the figure ; and its whole linear boundary, of whatever kind or composition, is called the perimeter. |