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 A B 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. 9F Join A C, AE, BC, 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 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 AB, in the points G, g, the points F and G being towards the same parts, as also f and g: then, A G is equal to the sum, and Ag to the difference of A B, BC. But, because A B C is a triangle I. 10.), the sum of the sides A B, BC is greater, and their difference is less, than the side A C, that is, than AF or Af. Therefore, A G is greater than AF, and Ag is less than Af. because (7.) the circumferences cannot have more than the two points C, E in common, it is evident that if the arcs CGE, CFE do not coincide, the one must be wholly without or wholly within the other: and the same may be said Conseof the arcs Cg E, CƒE. quently, the arc C GE of the second circle is without the first, and the are CgE of the same circle within the first, another in the points C and E. that is, the circumferences cut Therefore, &c. But, one Cor. 1. (Euc. iii. 11, 12, and 13.) Circles that touch one another meet in one point only, which point is in the straight line that joins their centres, or in that straight line produced. For, should they meet in two points, they would have a common chord, which common chord would (3. Cor. 3.) be bisected by the straight line joining their centres; and, therefore, the points of meeting being upon either side of this straight line, the circles, as in the proposition, would not only meet, but cut one another. Cor. 2. Hence, if two circles touch one another, the distance between their centres must be equal to the sum or to the difference of their radii; the sum if they touch externally, the difference if they touch internally. Scholium. We may remark that from the second part of the demonstration of this proposition, it likewise follows that 1. If a point A be taken in the dia meter of a circle C GE, which is not the centre (see the lower figure), of all the straight lines which can be drawn from that point to the circumference, the greatest is that which passes through the centre, viz. A G, and the other part Ag of that diameter is the least: also 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 circumference, 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 H g K convex. PROP. 9. g K ference of the circle which has the centre A; and join A D, D B. Then, because AD B is a triangle, the side D B (I. 10.) is greater than the difference of AB, A D, that is, greater than B C: but BC 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 EC, 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 than the sum, or less than the difference of their radii, 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 that line produced, cut the circumferences in the points C, D. Then, it is B AB D 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 of the radii and if it be equal to the difference, then, for the same reason, the 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 2o, 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 2°, 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, 1o, 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. 3o. 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. SECTION 2.-Of Angles in a Circle. PROP. 11. 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. Let C be the centre of a circle A B D, and let A B, 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 DE. For, the radii A C, CB being equal to the radii DC, 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, 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 AB greater than the base DE: therefore the angle A C B (I. 11.) is likewise greater than the angle DCE. The same demonstration may be applied to the case of equal circles. Therefore, &c. 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.) In the same or in equal circles, equal angles at the centre stand upon equal arcs; and conversely. Let C, c be the centres of two equal circles, and let A CB, acb be equal E B angles at the centres; the arc AB shall be equal to the arc a b. 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 ab, which is absurd; therefore, the angle a cb cannot but be equal to A CB. In the next place, let A C B, DCE be equal angles in the same circle: then, if c 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 A C B, a cb, be any an gles at the centre: the angle ACB shall be to the angle a cb as the arc A B to the arc a b. Let the angle acb be divided into any number of equal angles by the radii cd, ce, cƒ, 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 a d, 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 "a b. 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 demonstration. Therefore, &c. Scholium. Hence the angle at the centre of a circle is said to be measured by the arc 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 (1. 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. This complete definition of angular magnitude is of the greatest importance in the higher parts of the mathematics, and may be well illustrated by help of the measuring circumference. With the centre C, and radius CA, let there be described a circle A Q Q2 Q3, and let the diameters A Q2, QQ be drawn at right angles to one another, 3 Q3 Q2 dividing the whole angular space about the centre into 4 equal angles, each of which will be measured by a quadrant,* or fourth part of the circumference. Let us now suppose that the radius of the circle, being made to revolve about its centre from the original position CA, is brought successively into the positions C Q, C Q 2, C Q,, and thence again, continuing its revolution, a second time into the same positions CA, CQ, CQ2, CQ3, and so on. Then it is evident that the angular space through which the radius will have revolved, will be, in these successive positions, one, two, three right angles; upon returning to A four right angles, which is the whole angular space about the point C: and thence again, coming a second time to the same positions CQ, C Q2, C Q3, five, six, seven right angles, and so on: which angular spaces will be measured respectively by one, two, three quadrants, a whole circumference; five, six, seven quadrants, and so on: and any angular spaces intermediate to these will be measured by corresponding arcs intermediate, that is, of magnitudes between one and two, two and three, three and four, &c. quadrants. PROP. 14. (Euc. iii. 20.) The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is, upon the same part of the circumference. Let A C B be any angle at the centre C of the circle A BD, and let AD B be E E From the Latin word quadrans, a fourth part, an angle at the circumference upon the same base A B: the angle ACB shall be double of the angle A D B. Join D C, and produce it to E. Then, because C A is equal to CD (I. 6.), the angle CAD is equal to CDA: therefore the angle ACE, which is equal to CAD, CDA together (I. 19.) is double of CD A. In like manner it may be shown that the angle B C E is double of CD B. Therefore the sum or difference of the angles E C A, E C B is also double of the sum or difference of the angles CD A, C D B, that is, the angle AC B is double of the angle A D B. PROP. 15. (EUc. iii. 21.) Angles in the same segment of a circle are equal to one another. For they are halves of the same angle, viz. the angle at the centre which stands upon their common base; or, which is the same thing, they are measured by the same arc, viz. the half of their common base. Cor. 1. (Euc. iii. 31., first part of.) The angle which is in a semicircle is a right angle, for it is measured by half the semi-circumference, that is, by a quadrant. Cor. 2. (Euc. iii. 31. second part of.) The angle, which is in a segment greater than a semicircle, is less than a right angle; and the angle, which is in a segment less than a semicircle, is greater than a right angle; for the one is measured by |