Let A B be a diameter of the circle AB D, the centre of which is C: let DE, FG be any two chords to which perpendiculars CH, CK are drawn; and let the distance CH be less than CK: the diameter AB shall be greater than the chord D E, and the chord D I shall be greater than the chord F G. Join C D, CE, CF. Then, because CA is equal to CD, and CB to CE, the whole A B is equal to CD and CE together: but C D and D E together (I. 10.) are greater than DE: therefore A B is greater than DE. Again, because CHD, CKF are right-angled triangles, and that CD-square is equal to C F-square, the squares of CH, HD together (I. 36.) are equal to the squares of CK, KF together: but the square of C H is less than the square of CK; therefore the square of HD is greater than the square of K F, that is, HD is greater than KF. And DE, FG are double of H D, K F respectively, because the perpendiculars C H, C K pass through the centre (3.): therefore DE is greater than F G. Next, let the chord DE be greater than F G; it shall also be nearer to the centre. For, CH and C K being drawn as before, the squares of CH, HD together are equal to the squares of CK, KF together; but the square of HD, which is half of D E, is greater than the square of K F, which is half of FG: therefore the square of CH is less than the square of C K, and CHis less than CK, that is, DE is nearer to the centre than F G is. Therefore, &c. Cor. (Euc. iii. 14.) Equal straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre are equal to one another. PROP. 5. (EUc. iii. 9.) If a point be taken, from which to the circumference of a circle there fall more than two equal straight lines, that point shall be the centre of the circle. Let ABC be a circle, and let D be a point taken, such that the three straight lines Ꭰ Ꭺ, Ꭰ Ᏼ, DC drawn from the point D to the circumference, are equal to one another: the point D shall be the centre of the circle Join A B and B C, and bisect them in the points E and Frespectively; and join DE, DF. Then, because D A B is an isosceles triangle, the straight line DE, which is drawn from the vertex D to the bisection of the base A B, is at right angles to AB (I. 6. Cor. 3.); and, because D E bisects the chord A B at right angles (3. Cor. 2.), it passes through the centre of the circle. In the same manner it may be shown that the straight line DF passes through the centre of the circle. But the only point through which each of the straight lines DE, D F passes, is their point of intersection D. Therefore D is the centre of the circle. Therefore, &c. Cor. 1. From any other point than the centre there cannot be drawn to the circumference of a circle more than two straight lines that are equal to one another, whether the point be within or without the circle. (Euc. iii. 7 and 8 parts of.) Cor. 2. It appears from the demonstration that if three points A, B and C be given which are not in the same straight line, a circle may be found, the circumference of which shall pass through the three points A, B C; the circle, namely, which has for its centre the intersection of the two lines which bisect A B and B C at right angles. PROP. 6. If two circles have the same centre, either they shall coincide, or one of them shall fall wholly within the other. For if the radii of two concentric circles be equal to one another, it is manifest that every point in the circumference of the one must be at the same distance from their common centre, with every point in the circumference of the other; and therefore the two circumferences cannot but coincide, But if the radii be unequal, every point in the circumference of that which has the lesser radius is at a less distance from the common centre, and therefore must fall within the circumference of the greater circle. Therefore, &c. Cor. (Euc. iii. 5 and 6.) If two cir. cles cut or touch one another, they cannot have the same centre. G PROP. 7. (EUc. iii. 10.) The circumferences of two circles cannot intersect one another in more than two points. For if they should have three points in common, those three points could not be (1.) in the same straight line. Therefore a point might be found (5. Cor.) equally distant from the three, which point would (5.) be the centre of each of the circles; that is, there would be two circles cutting one another and having the same centre, a thing impossible. Therefore, &c. PROP. 8. If the circumferences of two circles meet one another in a point which is not in the straight line joining their centres, or in that straight line produced; they shall meet one another in a second point upon the other side of that straight line, and shail cut one another. Let A, B be the centres of two circles, the circumferences of which meet one another in the point C, which is not in A B, nor in A B produced: from C draw CD perpendicular to AB or to A B produced, and produce CD to E so that D E may be equal to DC: the circumferences shall meet one another in the point E, and shall cut one another in each of the points C, E. A 91 JA BD fore, the two circumferences meet one another in the point E. Again, let the circumference of the first circle cut the line A B produced in the points F, f, and let the circumference of the other circle meet the same line A B, in the points G, g, the points F and G being towards the same parts, as also f and g: then, AG is equal to the sum, and Ag to the difference of AB, BC. But, because A B C is a triangle (I. 10.), the sum of the sides AB, BC is greater, and their difference is less, than the side A C, that is, than A For Af. Therefore, A G is greater than AF, and Ag is less than Af. But, because (7.) the circumferences cannot have more than the two points C, E in common, it is evident that if the ares CGE, CFE do not coincide, the one must be wholly without or wholly within the other and the same may be said of the arcs Cg E, CƒE. Consequently, the arc CGE of the second circle is without the first, and the are Cg E of the same circle within the first, another in the points C and E. that is, the circumferences cut one Therefore, &c. Join A C, AE, B C, BE. Then because the triangles A DC, ADE have two sides of the one equal to two sides of the other, and have also the included angles ADC, ADE equal to one another, the base A C (I. 4.) is equal to AE: therefore, the point E is in the circumference of the circle which has the centre A. In the same manner it may be shown, that the same point is in the circumference of the other circle. There of any others that which is nearer to the greatest is greater than the more remote. (Euc. iii. 7. part of.) 2. If a point A be taken without a circle CGE (see the upper figure), and straight lines be drawn from it to the circumference, whereof one AG passes through the centre; of those which fall upon the concave circumference, the greatest is that which passes through the centre, viz. AG; and of the rest, that which is nearer to the one passing through the centre is always greater than one more remote: but of those which fall upon the convex cireumference, the least is that between the point without the circle and the diameter; and of the rest, that which is nearer to the least is always less than the more remote. (Euc. iii. 8. part of.) The parts of the circumference which are here termed concave and convex towards the point A, are determined by the points H and K, in which tangents drawn A from A meet the circumference,the part HGK being concave, and Hg K convex. g circle be greater than the radius H PROP. 9. If the circumferences of two circles meet one another in a point which is in the straight line joining their centres, or in that straight line produced, they shall meet in no other point; the circumference of that which has the greater radius shall fall wholly without the circumference of the other; and the two circles shall touch one Let A, B be the centres of two circles, the circumferences of which meet one another in the point C, which is in the line AB, or in A B produced: and let the radius of the first another. K of the other: the circumference of the first shall fall wholly without the circumference of the other, and the circles shall touch one another in C. Let D be any point in the circum ference of the circle which has the centre A; and join A D, D B. Then, because A D B is a triangle, the side D B (I. 10.) is greater than the difference of A B, A D, that is, greater than B C : but B C is the radius of the circle which has the centre B: therefore, the point D lies without the latter circle. And the same may be demonstrated of every point in the circumference of the greater circle. Also, because the arcs E C, e C of the one circle, lie upon the same side of the arcs DC, dC of the other, the circles meet, but do not cut one another in the point C; that is, they touch one another. Therefore, &c. Cor. 1. Circles that cut one another meet in two points, one upon either side of the straight line which joins their centres. For circles meeting in a point which is in that straight line do not cut, but touch one another, as is shown in the proposition: and such as meet in a point which is not in that straight line, meet also (8.) in a second point upon the other side of it. Cor. 2. Hence, if two circles cut one another, the straight line which joins their centres must be less than the sum, and greater than the difference of their radii. (I. 10.) PROP. 10. If the circumferences of two circles do not meet one another in any point, the distance between their centres shall be greater thar the sum, or less than the difference of their radit, according as each of the circles is without the other or one of them within the other. Let A, B be the centres of the two circles, and let the line AB, or cut points Then, C, D. it is evident that A B is equal to the sum, or to the difference of A C, B C, according as each of the circles is without the other, or one of them within the other. If it be equal to the sum, then, because BC is greater than BD, the sum of A C, B C is greater than the sum of AC, BD; that is, the distance of the centres is greater than the sum the radii: and if it be equal to the dif ference, then, for the same reason, the G 2 ABDO 84 the base A B (I. 11.) is likewise greater than the base D E. And, conversely, if AB be greater than DE, it shall subtend a greater angle at C: for C A B and C D E are, in this case, two triangles having two sides of the one equal to two sides of the other, each to each, but the base A B greater than the base DE: therefore the angle ACB (I. 11.) is likewise be apgreater than the angle D C E. The same demonstration may plied to the case of equal circles. Therefore, &c. difference of A C, B C is less than the difference of A C, B D, that is, the distance of the centres is less than the difference of the radii Therefore, &c. Cor. 1. Hence it appears, conversely, that two circles will, 1o, cut one another; or 2°, touch one another; or 3o, one of them fall wholly without the other; according as the distance between their centres is, 1°, less than the sum, and greater than the difference of their radii ; or 2o, equal to the sum, or to the difference of their radii; or 3°, greater than the sum or less than the difference of their radii. Cor. 2. Therefore, 1°, if two circles cut one another, the distance of their centres must be at the same time less than the sum and greater than the difference of their radii; and conversely, if this be the case, the circles will cut one another. 2o. If two circles touch one another, the distance of their centres must be equal to the sum or to the difference of their radii, according as the contact_is external or internal; and conversely, if either of these be the case, the circles will touch one another. 3. If two circles do not meet one another, the distance of their centres must be greater than the sum or less than the difference of their radii, according as each is without the other, or one of them within the other; and conversely, if either of these be the case, the circles will not meet one another, Cor. In the same or in equal circles, equal chords subtend equal angles at the centre; and conversely. PROP. 12. (Euc. iii. 26 and 27, first parts of.) SECTION 2.-Of Angles in a Circle. In the same, or in equal circles. the greater chord subtends the greater angle at the centre: and conversely, the greater angle at the centre is subtended by the greater chord. In the same or in equal circles, equal angles at the centre stand upon equal arcs; and conversely. Let C be the centre of a circle A B D, and let AB, D E be two chords in the same circle, of which AB subtends a greater angle at C than DE does: A B shall be greater than D E. For, the radii A C, C B being equal to the radii D C, CE respectively, CAB and CDE are triangles having two sides of the one equal to two sides of the other, each to each, but the angle ACB greater than DCE: therefore, Let C, c be the centres of two equal circles, and let ACB, acb be equal angles at the centres; the arc AB shall be equal to the arc ab. For if the circles be applied one to the other, so that the centre C may be upon c, and the radius CA upon ca, the radius CB will coincide with cb, because the angle ACB is equal to a cb. Also the points A B will coincide with the points a, b respectively, because the radii CA, CB are equal to the radii ca, cb. Therefore the arc AB coincides with the arc a b, and is equal to it. And conversely, if the arcs A B, a b be equal to one another, the angles A CB, a cb shall be likewise equal. For, if not, let any other angle a cb' be taken equal to ACB; then, by the former part of the proposition, the arc a b' is equal to A B, that is, to a b, which is absurd; therefore, the angle a cb cannot but be equal to A C B. In the next place, let A C B, DCE be equal angles in the same circle: then, ifc be the centre of a second circle equal to it, and if the angle a c b be made equal to ACB or DCE, the arc a b will be equal to AB or DE; therefore, the arcs Á B, DE are equal to one another. And, in cumference which measures an equal like manner, the converse, angle at the centre. Therefore, &c. Cor. 1. (Euc. iii. 28 and 29.) In the same or in equal circles, equal arcs are subtended by equal chords; and conversely (11. Cor.). Cor. 2. By a similar demonstration it may be shown that in the same or in equal circles, equal sectors stand upon equal arcs; and conversely. PROP. 13. (Euc. vi. 33, part of.) In the same or in equal circles, any angles at the centre are as the arcs upon which they stand; so also are the sectors. Let C, c be the centres of two equal circles; and let AC B, a cb, be any an gles at the centre: the angle ACB shall be to the angle acb as the arc AB to the arc a b. Let the angle acb be divided into any number of equal angles by the radii cd, ce, cf, cg, and therefore the arc ab into as many equal parts (12.) by the points d, e, f, g. Then, if the arc AD be taken equal to ad, and if CD be joined, the angle ACD will be (12.) equal to a cd; and if the arc AD be contained in AB a certain number of times with a remainder less than AD, the angle AC D will be found in the angle AC B the same number of times with a remainder less than A CD: and this, whatsoever be the number of parts into which the arc a b is divided. Therefore, (II. def. 7.) the angle ACB is to the angle a cb as the arc AB to the arc ab. And in the same manner it may be shown that the sector ACB is to the sector a cb as the arc AB to the arc ab (12. Cor. 2.) The case of arcs or sectors occurring in the same circle has a similar demon stration. Therefore, &c. Scholium. Hence the angle at the centre of a circle is said to be measured by the are upon which it stands: and generally, any angle in a circle is said to be measured by that part of the cir In Book I. def. 9. an angle was stated to have its origin in the meeting of two straight lines in a point, and to be greater or less according to the extent of the opening between those lines; a right angle was then defined; an angle "less" than which was said to be an acute angle, and an angle "greater" an obtuse angle. We ought rather however to have defined the obtuse angle to be greater than one, and less than two right angles: for if the opening between the legs of such an angle be increased to a still further degree, it becomes equal to two right angles-greater than two-equal to three-greater than three-and, by a still increasing separation of one leg from the other in the same direction, equal to four right angles—greater than four-and so on. An angle which is greater than two and less than four right angles is frequently called a reverse or re-entering angle.* These angles (right, acute, obtuse, and re-entering) are all that have place in elementary Geometry, or in the subjects to which it is commonly applied; the angles spoken of being understood never to exceed four right angles. But where no such limitation, confining the magnitude of the angle, is supposed, it is plain from the considerations abovementioned, that the magnitude of any angle, in general, cannot be estimated from the apparent opening between the legs. Besides this opening, there is to be considered the direction in which it is supposed to have been generated, and yet further, the number of times the revolving leg may have coincided with and passed by the other; for the same apparent opening is the result of different angular revolutions: just as the hand of a watch is at the same apparent distance from any given position, whether it has made fifty and a quarter, or a hundred and a quarter, or a hundred more circuits. The traversed space being made up of parts which coincide, and which do not therefore distinctly appear, the number of these parts must be specified if we would form an estimate of the whole. It has been already observed (I. 2. note) that an angle is sometimes said to be supplementary, viz. when it is considered as the supplement of the ad. jacent angle to two right angles: in like manner, an angle takes the name of an explementary angle, when together with the adjoining and opposite angle it fills up the whole space about the angular point. |