Let A B be a dia D shall be the centre of the circle meter of the circle ABC. AB D, the centre of Join A B and B C, and bisect them in which is C: let DE, the points E and Frespectively; and join FG be any two chords DE, DF. Then, because D A B is an to which perpendi isosceles triangle, the straight line DE, culars CH, CK are which is drawn from the vertex D to drawn; and let the the bisection of the base A B, is at right distance CH be less than CK: the angles to AB (I. 6. Cor. 3.); and, bediameter AB shall be greater than the cause D E bisects the chord A B at right chord D E, and the chord D E shall be angles (3. Cor. 2.), it passes through the greater than the chord F G. centre of the circle. In the same manJoin CD, CE, CF. Then, because ner it may be shown that the straight CA is equal to CD, and C B to CE, line DF passes through the centre of the whole A B is equal to C D and CE the circle. But the only point through together: but CD and D E together which each of the straight lines D E, (1.10.) are greater than DE: therefore D F passes, is their point of intersecA B is greater than D E. Again, be- tion D. Therefore D is the centre of cause C HD, CKF are right-angled the circle. triangles, and that C D-square is equal Therefore, &c. to C F-square, the squares of CH, À D Cor. l. From any other point than together (I. 36.) are equal to the squares the centre there cannot be drawn to of CK, KF together: but the square the circumference of a circle more than of C H is less than the square of Ć K; two straight lines that are equal to one therefore the square of H D is greater another, whether the point be within or than the square of KF, that is, H D is without the circle. (Euc. iii. 7 and 8 greater than KF. And D E, F G are parts of.) double of HD, K F respectively, be- Cor. 2. It appears from the demonstracause the perpendiculars cHC K tion that if three points A, B and C be pass through the centre (3.): therefore given which are not in the same straight DE is greater than F G. line, a circle may be found, the circumNext, let the chord D E be greater ference of which shall pass through the than FG; it shall also be nearer to the three points A, B and C; the circle, centre. For, C H and C K being drawn namely, which has for its centre the inas before, the squares of CH, HD tersection of the two lines which bisect together are equal to the squares of A B and B C at right angles. CK, KF together ; but the square of HD, which is half of D E, is greater Prop. 6. than the square of KF, which is half If two circles have the same centre, of FG: therefore the square of CH is either they shall coincide, or one of them less than the square of CK, and CH is shall fall wholly within the other, less than C K, that is, D E is nearer to For if the radii of two the centre than F G is. concentric circles be Therefore, &c. equal to one another, it Cor. (Euc. iii. 14.) Equal straight is manifest that every lines in a circle are equally distant from point in the circumthe centre; and those which are equally ference of the one must distant from the centre are equal to one be at the same distance another. from their common cen tre, with every point in the circumProp. 5. (Euc. iii. 9.) ference of the other; and therefore the If a point be taken, from which to the two circumferences cannot but coincide, circumference of a circle there fall more But if the radii be unequal, every point than two equal straight lines, that point in the circumference of that which has shall be the centre of the circle. the lesser radius is at a less distance Let ABC be a circle, from the common centre, and therefore and let D be a point ta must fall within the circumference of ken, such that the three the greater circle. straight lines D A, D B, Therefore, &c. DC drawn from the Cor. (Euc. iii. 5 and 6.) If two cir point D to the circum cles cut or touch one another, they canference, are equal to not have the same centre. one another : the point G Prop. 7. (Euc. iii. 10.) fore, the two circumferences meet one The circumferences of two circles another in the point E. Again, let the circumference of the cannot intersect one another in more than two points. first circle cut the line A B produced in the points F, f, and let the circumference For if they should have three points of the other circle meet the same line in common, those three points could A B, in the points G, g, the points F not be (1.) in the same straight line, and G being towards the same parts, as Therefore a point might be found also f and g: then, A G is equal to the (5. Cor.) equally distant from the three, which point would (5.) be the centre of BC. But, because A B C is a triangle sum, and Ag to the difference of AB, each of the circles; that is, there would (I. 10.), the sum of the sides AB, BC be two circles cutting one another and is greater, and their difference is less, having the same centre, a thing impos- than the side A C, that is, than A For sible. A f. Therefore, A G is greater than Therefore, &c. AF, and A g is less than Af. But, because (7.) the circumferences cannot PROP. 8.' have more than the two points C, E in If the circumferences of two circles common, it is evident that if the ares meet one another in a point which is CGE, C FE do not coincide, the one not in the straight line joining their must be wholly without or wholly within centres, or in that straight line pro the other; and the same may be said duced ; they shall meet one another of the arcs Cg E, CFE. Consein a second point upon the other side quently, the arc CGE of the second of that straight line, and shuil cut one circle is without the first, and the arc another. CgE of the same circle within the first, Let A, B be the centres of two circles, another in the points C and E. that is, the circumferences cut one the circumferences of which meet one Therefore, &c. another in the point C, which is not in Cor. 1. (Euc. iii. 11, 12, and 13.) A B, nor in A B produced: from C Circles that touch one another meet in draw CD perpendicular to A B or to A B produced, and produce C D to E one point only, which point is in the so that D E may be equal to DC: the straight line that joins their centres, or circumferences shall meet one another in that straight line produced, in the point E, and shall cut one another they would have a common chord, which For, should they meet in two points, in each of the points C, E. 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. Join A C, AE, B C, B E. Then be We may remark that from the second cause the triangles A DC, ADE have part of the demonstration of this protwo sides of the one equal to two sides of position, it likewise follows thatthe other, and have also the included 1. If a point A be taken in the diaangles ADC, ADE equal to one meter of a circle CGE, which is not another, the base A C (I. 4.) is equal to the centre (see the lower figure), of all AE: therefore, the point E is in the the straight lines which can be drawn circumference of the circle which has the from that point to the circumference, centre A. In the same manner it may the greatest is that which passes through be shown, that the same point is in the the centre, viz. AG, and the other part circumference of the other circle. There- Ag of that diameter is the least: also 9 15 of any others that which is nearer to ference of the circle which has the the greatest is greater than the more centre A ; and join AD, D B. Then, beremote. (Euc. iii. 7. part of.) cause A D B is a triangle, the side DB 2. If a point A be taken without (I. 10.) is greater than the difference of a circle CGE (see the upper figure), A B, A D, that is, greater than BC: but and straight lines be drawn from it to BC is the radius of the circle which has the circumference, whereof one AG the centre B : therefore, the point D lies passes through the centre; of those without the latter circle. And the same which fall upon the concave circum- may be demonstrated of every point in ference, the greatest is that which passes the circumference of the greater circle. through the centre, viz. AG; and of Also, because the arcs E C, eC of the the rest, that which is nearer to the one one circle, lie upon the same side of the passing through the centre is always arcs DC, dC of the other, the circles greater than one more remote: but of meet, but do not cut one another in the those which fall upon the convex cir- point C; that is, they touch one another. cumference, the least is that between the Therefore, &c. point without the circle and the dia Cor. 1. Circles that cut one another meter; and of the rest, that which is meet in two points, one upon either side nearer to the least is always less than of the straight line which joins their the more remote. (Euc. iii. 8. part of.) centres. For circles meeting in a point The parts of the circumference which which is in that straight line do not cut, are here termed concave and convex but touch one another, as is shown in towards the point the proposition: and such as meet in 2 A, are determined point which is not in that straight line, by the points H meet also (8.) in a second point upon and K, in which the other side of it. tangents drawn As Cor. 2. Hence, if two circles cut one from A meet the another, the straight line which joins circumference, their centres must be less than the sum, the part HGK K and greater than the difference of their being concave, radii. (1. 10.) and H g K convex. PROP. 10. If the circumferences of two circles do If the circumferences of two circles not meet one another in any point, the meet one another in a point which is in distance between their centres shall be the strught line joining their centres, greater thar the sum, or less than the op in that straighi line produced, they difference of their radii, according as shall meet in no other point ; the cir- each of the circles iɛ without the other cumference of that which has the greater or one of them wiihin the other. radius shall fall wholly without the cir Let A, B be cumference of the other; and the two the centres of circles shall touch one another. the two circles, Let A, B be and let the the centres of line AB, or two circles, the that line proeircumferences duced, cut of which meet the circumone another in ferences in the the point points C, D. which is in the Then, it is line A B, or in evident that A B is equal to the sum, or A B produced : to the difference of A C, BC, accord. and let the ra ing as each of the circles is without the dius of the first other, or one of them within the other. circle be greater If it be equal to the sum, then, bethan the radius cause B C is greater than BD, the of the other : the circumference of the sum of A C, B C is greater than the sum first shall fall wholly without the cir- of A C, BD; that is, the distance of cumference of the other, and the circles the centres is greater than the sum o shall touch one another in C. the radii : and if it be equal to the dif. the Let D be any point in the circum- ference, then, for the same reason, G 2 G 8. c difference of AC, B C is less than the the base A B (I. 11.) is likewise greater difference of A C, BD, that is, the than the base D E. distance of the centres is less than And, conversely, if AB be greater the difference of the radii than D Е, it shall subtend a greater Therefore, &c. angle at C: for C A B and C D E are, in Cor. 1. Hence it appears, conversely, this case, two triangles having two sides that two circles will, 1°, cut one another; of the one equal to two sides of the or 2°, touch one another ; or 3°, one of other, each to each, but the base AB them fall wholly without the other; ac- greater than the base DE: therecording as the distance between their fore the angle ACB (I. 11.) is likewise centres is, 1°, less than the sum, and greater than the angle DCE. greater than the difference of their radii ; The same demonstration may be apor 2°, equal to the sum, or to the differ- plied to the case of equal circles. ence of their radii; or 3°, greater than the Therefore, &c. sum or less than the difference of their Cor. In the same or in equal circles, radii. equal chords subtend equal angles at Cor. 2. Therefore, 1°, if two circles the centre; and conversely. cut one another, the distance of their centres must be at the same time less Prop. 12. (Euc. iii. 26 and 27, first than the sum and greater than the dif parts of.) ference of their radii; and conversely, if this be the case, the circles will cut In the same or in equal circles, equal one another. angles at the centre stand upon equal 2o. If two circles touch one another, arcs ; and conversely. the distance of their centres must be Let C, c be the centres of two equal equal to the sum or to the difference of circles, and let ACB, acb be equal 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 con- angles at the centres; the arc AB shall versely, if either of these be the case, be equal to the arc ab. the circles will not meet one another, 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 SECTION 2.–Of Angles in a Circle. radius CB will coincide with cb, because PROP. 11. the angle ACB is equal to acb. Also the points A B will coincide with the In the same, or in equal circles. the points a, b respectively, because the ragreater chord subtends the greater dii CA, CB are equal to the radii ca, angle at the centre: and conversely, the cb. Therefore the arc A B coincides greater angle at the centre is subtended with the arc ab, and is equal to it. by the greater chord. And conversely, if the arcs A B, a b Let C be the centre of be equal to one another, the angles a circle ABD, and let A CB, a cb shall be likewise equal. AB, D E be two chords For, if not, let any other angle a cb' be in the same circle, of taken equal to ACB; then, by the forwhich AB subtends a mer part of the proposition, the arc a b' greater angle at C than is equal to AB, that is, to ab, which is D E does : AB shall be absurd; therefore, the angle a cb cangreater than D E. not but be equal to A C B. For, the radii A C, C B being equal In the next place, let ACB, DCE be to the radii DC, CE respectively, equal angles in the same circle: then, ifc CAB and CDE are triangles having be the centre of a second circle equal to two sides of the one equal to two sides it, and if the angle a c b be made equal to of the other, each to each, but the angle ACB or DCE, the arc a b will be equal CB greater than DCE: therefore, to AB or D E; therefore, the arcs A B, B в C A DE are equal to one another. And, in cumference which measures an equal like manner, the converse, angle at the centre. Therefore, &c. In Book I. def. 9. an angle was stated Cor. 1. (Euc. iii. 28 and 29.) In to have its origin in the meeting of two the same or in equal circles, equal arcs straight lines in a point, and to be are subtended by equal chords; and greater or less according to the extent conversely (11. Cor.). of the opening between those lines: a Cor. 2.' By a similar demonstration right angle was then defined; an angla it may be shown that in the same or in "less" than which was said to be an acute equal circles, equal sectors stand upon angle, and an angle“ greater" an ohtuse equal arcs; and conversely. angle. We ought rather however to have defined the obtuse angle to be greater Prop. 13. (Euc. vi. 33, part of.) than one, and less than two right angles : In the sume or in equal circles, any for if the opening between the legs of angles at the centre are as the arcs upon such an angle be increased to a still which they stand ; so also are the further degree, it becomes equal to two sectors. right angles-greater than two-equal Let C, c be the centres of two equal to three-greater than three-and, by a circles; and let ACB, acb, be any an 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. gles at the centre: the angle ACB shall These angles (right, acute, obtuse, and be to the angle acb as the arc A B to re-entering) are all that have place in the arc ab. elementary Geometry, or in the subjects Let the angle acb be divided into to which it is commonly applied ; the anany number of equal angles by the gles spoken of being understood never radii cd, ce, cf, cg, and therefore the to exceed four right angles. But where arc a b into as many equal parts (12.) no such limitation, confining the magby the points d, e, f, g. Then, if the arc nitude of the angle, is supposed, it is AD be taken equal to ad, and if CD be plain from the considerations abovejoined, the angle ACD will be (12.) mentioned, that the magnitude of any equal to a cd, and if the arc AD bé angle, in general, cannot be estimated contained in A B a certain number of from the apparent opening between the times with a remainder less than AD, legs. Besides this opening, there is to the angle AC D will be found in the be considered the direction in which it angle ÅC B the same number of times is supposed to have been generated, with a remainder less than ACD: and and yet further, the number of times the this, whatsoever be the number of parts revolving leg may have coincided with into which the arc ab is divided. There, and passed by the other; for the same fore, (II. def. 7.) the angle A CB is to apparent opening is the result of difthe angle a c b as the arc AB to the ferent angular revolutions: just as the arc ab. hand of a watch is at the same appaAnd in the same manner it may be rent distance from any given position, shown that the sector ACB is to the whether it has made fifty and a quarter, sector acb as the arc AB to the arc ab or a hundred and a quarter, or a hundred (12. Cor. 2.) more circuits. The traversed space The case of arcs or sectors occurring being made up of parts which coincide, in the same circle has a similar demon- and which do not therefore distinctly stration. appear, the number of these parts must Therefore, &c. be specified if we would form an esti mate of the whole. Scholium. Hence the angle at the centre of a • It has been already observed (1. 2. note) that circle is said to be measured by the an angle is sometimes said to be supplementary, viz. when it is considered as the supplement of the ad. arc upon which it stands: and gene- jacent angle to two right angles: in like manner, an rally, any angle in a circle is said to angle takes the name of an explementary angle, when together with the adjoining and opposite angle be measured by that part of the cir- it fills up the wbolo space abous the angular poips. |