one Prop. 7. (Euc. iii. 10.) fore, the two circumferences meet one The circumferences of two circles another in the point E. cannot intersect one another in more Again, let the circumference of the than two points. first circle cut the line A B produced in For if they should have three points of the other circle meet the same line the points F, f, and let the circumference 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 sum, and Ag to the difference of A B, BC. But, because A B C is a triangle each of the circles; that is, there would (1. 10.), the sum of the sides A B, 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 AF or sible. Therefore, &c. A f. Therefore, A G is greater than 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 arcs 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 CgE, Cf E. Consein a second point upon the other side quently, the arc CGE of the second of that straight line, and shail 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 the circumferences of which meet one another in the point C, which is not in Therefore, &c. 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 For, should they meet in two points, in each of the points C, E. 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. Join A C, A E, B C, B E. Then be- We may remark that from the second cause the triangles A DC, A DE have part of the demonstration of this pro, two 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 с a IT 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 A D, D B. Then, beremote. (Euc. iii. 7. part of.) cause A D B is a triangle, the side D B 2. If a point A be taken without (I. 10.) is greater than the difference of a circle CĜE (see the upper figure), A B, A D, that is, greater than B C: 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 EC, 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 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 a 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 A G 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. (I, 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 straight line joining their centres, greater than the sum, or less than the or in that straight line produced, they difference of their radii, according as shall meet in no other point; the cir- each of the circles is without the other, cumference of that which has the greater or one of them within 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. thetwo circles, Let A, B be and let the the centres of line AB, or two circles, the that line procircumferences duced, cut of which meet the circumone another in ferences in the AB Dj the point C, points C, D. which is in the Then, it is line AB, or in evident that A B is equal to the sum, or A B produced : to the difference of A C, BC, accordand 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 of shall touch one another in C. the radii: and if it be equal to the difLet D be any point in the circum- ference, then, for the same reason, the B E D E D difference of À C, 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 DE, 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, 1o, 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 difference of their radii; and conversely, parts of.) 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 A CB, 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. 39. 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 SECTION 2.- Of Angles in a Circle. upon c, and the radius CA upon ca, the 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, C B 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 ACB, a cb shall be likewise equal. A B D E be two chords For, if not, let any other angle a c Ô' be in the same circle, of taken equal to A CB; 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 ACB. For, the radii A C, C B being equal In the next place, let A CB,DCE be to the radii DC, CE respectively, equal angles in the same circle: then, if c 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 cb be made equal to of the other, each to each, but the angle ACB or DCE, the arc a b will be equal ACB greater than DCE: therefore, to AB or DE; therefore, the arcs A B, A 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 angle 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 obtuse 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 same 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 a b. 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. B where arc ab 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 acd; and if the arc AD be angle, in general, cannot be estimated contained in AB 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 Å C B the same number of times is supposed to have been generated, with a remainder less than A CD: 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 a b is divided. There and passed by the other; for the same fore, (II. def. 7.) the angle ACB is to apparent opening is the result of difthe angle a c b as the arc A B to the ferent angular revolutions: just as the 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 A CB is to the whether it has made fifty and a quarter, sector a cb as the arc AB to the arc ab or a hundred and a quarter, or a hundred more circuits. The traversed space (12. Cor. 2.) The case of arcs or sectors occurring being made up of parts which coincide, and which do not therefore distinctly in the same circle has a similar demonstration. 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 an angle is sometimes said to be supplementary, viz. circle is said to be measured by the when it is considered as the supplement of the adarc upon which it stands: and gene- jacent angle to two right angles : in like manner, an angle takes the name of an explementary angle, rally, any angle in a circle is said to when together with the adjoining and opposite angle be measured by that part of the çir- it fills up the whole space about the angular point. a arc ab. 2 Q2 B E This complete definition of angular an angle at the circumference upon the magnitude is of the greatest importance same base A B: the angle ACB shall in the higher parts of the mathematics, be double of the angle A DB. and may be well illustrated by help of Join D C, and produce it to E. Then, the measuring circumference. because C A is equal to CD (I. 6.), the With the cen angle CAD is equal to CDA: theretre C, and radius fore the angle ACE, which is equal to CA, let there be CAD, CD A together (I. 19.) is doudescribed a cir ble of C D A. In like manner it may cle AQ Q, Q3, be shown that the angle B C E is double and let the dia of C D B. Therefore the sum or differmeters A Q2, ence of the angles E C A, E C B is also Q Q, be drawn double of the sum or difference of the at right angles angles C DA, C D B, that is, the angle to one another, AC B is double of the angle A D B. dividing the whole angular space about the centre into 4 equal angles, each of The adjoined figure which will be measured by a quadrant,* shows that this proof or fourth part of the circumference. is equally applicable Let us now suppose that the radius when the angle ACB of the circle, being made to revolve is re-entering. about its centre from the original position CA, is brought successively into Therefore, &c. the positions CQ, C., CQ,, and Cor. 1. Any angle at the circumferthence again, continuing its revolution, ence is measured by half the arc upon a second time into the same positions which it stands. CA, CQ, C Q2, CQ3, and so on. Cor. 2. (Euc. iii. 26 and 27, second Then it is evident that the angular space parts of, and Euc. vi. 33 part of). In through which the radius will have re- the same or in equal circles, equal angles volved, will be, in these successive po- at the circumference stand upon equal sitions, one, two, three right angles; arcs, and conversely; and, generally, any upon returning to A four right angles, angles at the circumference are as the which is the whole angular space about arcs upon which they stand. (13. and the point C: and thence again, coming II. 17.) a second time to the same positions CQ, CQ2, CQ3, five, six, seven right angles, and so on : which angular spaces PROP. 15. (Euc. iii. 21.) will be measured respectively by one, Angles in the same segment of a cirtwo, three quadrants, a whole circum- cle are equal to one another. ference; five, six, seven quadrants, and For they are halves of the same anso on: and any angular spaces interme- gle, viz. the angle at the centre which diate to these will be measured by cor- stands upon their common base; or, responding arcs intermediate, that is, of which is the same thing, they are meamagnitudes between one and two, two sured by the same arc, viz. the half of and three, three and four, &c. quadrants. their common base. Cor. 1. (Euc. iii. 31., first part of.) PROP. 14. (Euc. iii. 20.) The angle which is in a The angle at the centre of a circle is semicircle is a right andouble of the angle at the circumference gle, for it is measured upon the same base, that is, upon the by half the semi-circumsame part of the circumference. •ference, that is, by a Cor. 2. (Euc. iii. 31. circle, is greater than a * From the Latin word quadrans, a fourth part, right angle; for the one is measured by D E E |