A COR. 11. The double of a greater magnitude is greater than с B D. equal to B. If to C and D severally, be added B, the sum of B and to the double of B a third magnitude equal to B. And so on. Cor. 12. 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. For, if this be disputed, let it be assumed that one is greater than the other. But if it be greater, its double must (by Cor. 11) be greater than the double of the other; which is impossible, for it is *I.Nom.26. equal. The assumption", therefore, cannot be true ; or the one magnitude, that was doubled, is not greater than the other; and because one is not greater than the other, they are equal. And in like manner in respect of magnitudes which are the third, fourth, &c. CoR. 13. The half of a greater magnitude is greater than the half of a less. And so of any other equipartites. For, if it be not greater, it must either be equal or less. It cannot be equal, for then (by Cor. 10) its double would be equal to the double of the other; and its double is not equal, for it is greater. And it cannot be less, for then (by Cor. 11) its double would be less than the double of the other; and its double is not less, for it is greater. But because it is neither equal nor less, it is greater. And in like manner in respect of the third, fourth, &c. COR. 14. If the doubles of two or more magnitudes be added together, the amount is double of the sum of the magniSee Note. tudes. And so of any other equimultiples. Let A and B be two magnitudes. The double A с of A, added to the double of B, is double of B D the sum of A and B. For, let C be another magnitude equal to A, and D to B. Because A is equal to C, the sum of A and C is equal to the double of A; and for the like reason, the sum of B and D is equal to the double of B. Therefore (by Cor. 5), the double of A, added to the double See Note. of B, is equal to the sum of A, B, C, and D. But (by Cor. 5), the sum of A and B is equal to the sum of C and D; therefore the sum of A, B, C, and D, is equal to double the sum of A and B. Where. fore (by Cor. 1), the double of A, added to the double of B, is equal to double the sum of A and B. If the magnitudes are more than two, then to each of them is to be taken an equal, as before. If the same is to be proved of the trebles, quadruples, &c., then to each magnitude are to be taken two, three, &c. magnitudes, equal to it. COR. 15. If there be two unequal magnitudes, and from the double of the greater he taken the double of the less; the re- A B of B, the remainder is double the difference of A and B. For A the greater, is equal to the sum of B the less, and of the dif ference between A and B. Therefore (by Cor. 14) the double of A, B, is equal to the double of the difference between A and B. quadruples, &c. to any number however great, cannot surpass a given magnitude; these magnitudes cannot be all equal to one another. See Note. M P N For, let A, B, C, D, E, F, G, H, I, &c. be magnitudes which being added one to another to any number however great, cannot surpass a given magnitude MN. But if so, either MN is the smallest magnitude which they cannot pass, or it is not. And if it is not, then there is some magnitude which may be cut off from it, and the remainder be a magnitude which they cannot pass. Wherefore there will be some remainder MO such, that it is a limit which they cannot pass, but if any smaller magnitude be substituted they shall pass it; for if they would not pass such smaller magnitude, the difference might be cut off. And because A, B, C, D, E, F, G, H, I, &c. may be taken till they surpass any magnitude that is smaller than MO, they may be taken till they surpass the half of MO. Let, then, a certain number (as for instance A, B, C, D, and E) be the magnitudes which are together greater than half MO; or of which the sum MP is greater than PO the remainder of MO. Because A, B, C, D, and E are together greater than PO, they are together greater than the sum of all the remaining magnitudes F, G, H, I, &c. however many, that may be taken afterwards ; still more are they greater than the sum of only an equal number of those magnitudes. But if the magnitudes were all equal to one another, the magnitudes A, B, C, D, and E would (by Cor. 10) be together equal to the sum of an equal number of the magnitudes which should be taken afterwards; and they are not equal to this, *I.Nom.26. for they are greater. The assumption* therefore cannot be true; or the magnitudes A, B, C, D, E, F, G, H, I, &c. cannot be all equal to one another. say, magnitudes equal to it may be added one to another], so as consequently (by Cor. 16) the magnitudes would not be all equal to +1.Nom. 26. one another. Which cannot be, for they are equal. The assumption therefore cannot be true ; or there is not a magnitude which they cannot pass. That is, they 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. COR. 18. If from the greater of two proposed magnitudes be taken not less than its half, and from the remainder not less than its half, and so on; there shall at length remain a magnitude less than the least of the proposed magnitudes. See Note. For if AB and C be the two proposed magnitudes, of which AB is the greater, then because (by Cor. 17) C may be multiplied (that is to say, magnitudes equal to it may be added one to another) so as at length 10 become greater than AB, C may be doubled successively (that is to say, over and over] so as at length to become greater than AB; for on each doubling is added a portion as great or greater than the portion added by the simple addition of C. Let C be doubled successively till the result, as DG, is greater than AB. If then from AB be taken its half BH, and from the remainder HA be taken its half HI, and so on till the half be taken successively as many times as C was doubled successively to make DG ; the last remainder AJ will be less than C. For as many times as C is contained in DG, so many times is Al contained in AB; but AB is less than DG; therefore AI (by Cor. 13) is less than C. And if from AB or any of the remainders were taken more than the half; only the more would the last remainder AI be less than C. N M PROPOSITION II. or about any two points, in it ; such point or points remaining First Case; the body A may be turned For the body may be placed in another situation, as M, such that the point B occupies the same place in fixed space as it did when the body was in the situation A. Also between the situations A and M, the body may be placed in other situations as N and O, in all of which the point B shall occupy the same place as when the body was in the situation A. And because the number of such situations, with their nearness to one another, may be increased without limit; it may be increased till the body is moved continuously from the situation A to the situation M, the point B preserving ever the same place, that is to say, remaining unmoved. And in like manner, the body may be moved continuously to any other situation which is such that the point B preserves the same place; or about any other point in it than B. Second Case ; the body A may be turned about any two points in it, as B and C, both these points remaining unmoved. * By the For because the body is* a hardt Hypothesis. body, the distance from the point B in t I. Nom. 3. it to the point C will be unaltered in N situation, as M, such that the points B and C respectively occupy the same places in fixed space as they did when the body was in the situation A. And for the same reason, between the situations A and M the body may be placed in other situations as N and O, in all of which the points B and C shall respectively occupy the same places as when the body was in the situation A. And because the number of such situations, with their nearness to one another, may be increased without limit; it may be increased till the body is moved continuously from the situation A to the situation M, the points B and C preserving ever the same places, that is to say, remaining unmoved. And in like manner, the body may be moved continuously to any other situation which is such that the points B and C preserve the same places respectively ; or about any other two points in it than B and C. And by parity of reasoning, the like may be proved of every other hard body. Wherefore, universally, a hard body may be See Note. turned &c. Which was to be demonstrated. Cor. 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. For it is supposed to be represented on a hard body which may be so turned. PROPOSITION III. be equidistant from an assigned point within, and at a distance Let A and B be the two assigned points, in a hard body of any kind; D and let B be the point from which all the points in the surface of the required solid are to be equidistant and at a distance equal to the distance of A and B. The point B remaining at rest, let A describe a line of any kind AC by the hard body in *INTERC. 2. which are the points A and B being turned* about B; and let the line AC be expressed on a hard body in which the point B is also situate. By the turning of this last-mentioned hard body about B, let the line AC be made to turn about the fixed point B; and |