If any point be taken in the diameter of a circle which is not the centre, of all the straight lines which can be drawn from it to the circumference the greatest is that in which the centre is, and the other part of that diameter is the least; and, of any others, that which is nearer to the line which passes through the centre is always greater than one more remote: and from the same point there can be drawn two, and only two, straight lines that are equal to one another, one upon each side of the diameter. Let ABCD be a circle, and AD its diameter, in which let any point F be taken which is not the centre: let the centre be E: of all the straight lines FA, FB, FC, &c. that can be drawn from F to the circumference, FA shall be the greatest, and FD, the other part of the diameter AD, shall be the least: and of the others, FB the nearer to that through the centre shall be greater than FC, the more remote. *20. 1. B A G Draw BE. CE, GE: and because two sides of a triangle are greater than the third, therefore FE, EB are greater than FB: but EA equal to EB; there+12 Def. 1. fore FE, EA, that is, FA is greater than FB, therefore the line which passes through the centre is the greatest. Again, because EB is equal to EC, and FE common to the triangles BEF, CEF, the two sides FE, EB are equal to the two FE, EC each to each; but the angle BEF is DH greater than the angle CEF; therefore the base FB is greater than the base FC: therefore the line which is nearer to that through the centre is greater than one more remote. Again, because FG, EF are greater than EG, and EG is equal to ED; therefore FG, EF are greater than ED: take away the common part EF, and the remainder FG is greater than the remainder FD. Therefore, FD is the least of all the straight lines from F to the circumference; and BF is greater than CF, and CF than GF. 19 Ax. 1. *24. 1. *20. 1. +5+Ax. *23. 1. Also, there can be drawn two, but only two, equal straight lines from the point F to the circumference, one upon each side of the diameter AD. For draw from F any straight line FG, and at the point E, in the straight line EF, make the angle FEH equal to the angle FEG, and join F, H: then, because GE is equal to EH, +12 Def. 1. and EF common to the two triangles GEF, HEF; the two sides GE, EF are equal to the two HE, +Const. EF, each to each; and the angle GEF is equal to the angle HEF; therefore, the base FG is equal* to the base FH: but, besides FH, no other straight line can be drawn from F to the circumference equal to FG: for, if there can, let it be FK: and, because FK is equal to FG, and FG to FH, FK is equal to FH, but one of these must be nearer to that through the centre than the other, so that a line nearer to that which passes through the centre, is equal to one which is more remote; which has been proved to be impossible. Therefore, if any point be taken, &c. #4. 1. +1 Ax. Q. E. D. PROP. VIII. THEOR. If any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one passes through the centre; of those which fall upon the concave part of the circumference, the greatest is that which passes through the centre, 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 part of the circumference, the least is that, which, when prolonged, passes through the centre; and of the rest, that which is nearer to the least is always less than one more remote: also two, and only two, equal straight lines can be drawn from the same point to the circumference, one upon each side of the line through the centre. Let ABF be a circle, and D any point without it, from which let straight lines DA, DE, DF, be drawn to the cir cumference, whereof DA passes through the centre. Of those which fall upon the concave part of the circumference AEF, the greatest shall be DA, which passes through the centre; and the nearer to it shall be greater than the more remote, viz. DE greater than DF: but of those which fall upon the convex part of the circumference LKG, the least shall be DG between the point D and the diameter AG; and the nearer to it shall be less than the more remote, viz. DK less than DL. #1. 3. +2 Ax. #20. 1. +9 Ax. #24. 1. LEGEN M Take M the centre of the circle F, and join ME, MF, MK, ML. And because AM is equal to ME, add MD to each, therefore AD is equalt to EM, MD: but EM, MD are greater than ED; therefore also AD is greater than ED, that is, the line through the centre is the longest line. Again, because ME is equal to MF, and MD common to the triangles EMD, FMD; EM, MD, are equal to FM, MD, each to each: but the angle EMD is greatert than the angle FMD; therefore the base ED is greater than the base FD, that is, the line nearer to that through the centre is greater than one more remote. And, because MK, KD are greater than MD, and MK is F +12 Def. 1. equal to MG, the remainder KD is greater than the remainder GD, that is, DG is less than DK: and DK is any straight line from D to the convex part of the circumference, therefore DG is the least, and because MLD is a triangle, and from the points M, D, the extremities of its side MD, the straight lines MK, DK are drawn to the point K within the triangle, therefore MK, KD are less than ML, LD: but MK is equal to +12 Def. 1. ML; therefore, the remainder DK is less than the remainder DL, that is, a line nearer to the least is less than one more remote. Also, there can be drawn two, and only two, equal straight lines from the point D to the circumference, one upon each side of the line through the centre. For draw any straight line DK, and #20. 1. #5 Ax. - *#21. 1. +5 Ax. at the point M, in the straight line MD, maket the 123. 1. angle DMB equal to the angle DMK, and join DB: and because MK is equal to MB, and MD common to the triangles KMD, BMD, the two sides KM, MD are equal to the two BM, MD, +Const. each to each; and the angle KMD is equalf to the angle BMD; therefore the base DK is equal❤ to the base DB: but, besides DB, there can be no straight line drawn from D to the circumference equal to DK: for, if there can, let it be DN: and because DK is equal to DN, and also to DB, therefore DB is equal to DN; but one of these must be nearer to that through the centre than the other, so that a line nearer to the least is equal to one more remote, which has been proved to be impossible. If, therefore, any point, &c. #4. 1. Q. E. D. PROP. IX. THEOR. If any point be taken, from which there fall more than two equal straight lines to the circumference of a circle, that point is the centre of the circle. For if the point were not the centre there could be but two #7.3. and equal straight lines drawn from it to the circumference* 8.3. NOTE. The above is not the demonstration usually inserted in the Elements, although it is substituted for it by De Chales, Elrington, and some other editors, on account of its conciseness.+ In the demonstration selected by Simson, Playfair, and most other modern editors the concluding sentence, "In like manner it may be demonstrated," &c. is open to objection; since by assuming different positions for the fictitious centre the reasoning, will become somewhat modified as was remarked by Williamson in his edition of the Elements, 1781. But two different demonstrations of this very simple proposition occur in the earlier copies of Euclid, and they are PROP. X. THEOR. One circumference of a circle cannot cut another in more than two points. If it be possible, let the circumference t1. 3. FAB cut the circumference DEF in B D #9.3. +12 Def. 1. KG, KF are all equal† to each other: and because from the point K to the circumference DEF there fall more than two equal straight lines namely, KB, KG, KF, therefore the point K is the centre +Const. of the circle DEF: but K is also the centret of the circle ABC; therefore the same point is the centre of two circles that cut one another, which is impossible. Therefore, one circumference of a circle cannot cut another in more than two points. #5.3. Q. E. D. PROP. XI. THEOR. If one circle touch another internally, the straight line which joins their centres being produced shall pass through the point of contact. Let the circle ADE touch the circle ABC internally both preserved in several modern editions, as for instance in those of Gregory, Stone, Williamson, and Bonnycastle. The second of these demonstrations, which is the one selected by Simson, is thought to be the addition of some succeeding Geometer; and that the first only is Euclid's. Austin, however, in his " Examination of Euclid" suspects that both these operose and superfluous demonstrations are interpolations; for the demonstration given above is so easy and obvious, "that," as he justly remarks, "we cannot suppose Euclid would have overlooked it." See the NOTE on this proposition at the end. |