The student may try to resolve a358 +a%c3 + b3c3 — abc(a + b3 + c3) and a%b3 + b3c8 +a%c3 + abc(as + b3 + c3 + 2abc), each into three factors. or 3. ac 1 , Y-'= 16(a+b)222 _ XXIV. 3(x - a)(x-7)(x —c) 1. The difference is (.1-a)(x - 5)(x - C) 1 2. Here xyz=1 and y== and x2= y 1+x+xz =1. 1+x+;*1+3+1+1+y+"1+x+uz+1+x+r+1+x+3x1+x+rz 3. The substitutions for x, y, z being made, the expression becomes ab bc 2abc (a+c)(6+c)(a +c)(a +0) 16+c)(a +b) (a+b)(6+c)(c+a)' which when reduced gives the required result. 4. Only requires the substitutions to be made for x and y. 5. The truth of this may be directly shewn by simply substituting for x and y their values in terms of a and b. Or indirectly, by assuming the result as true, namely, 13+y=y3 +®, then 33 – y=x – y and a2 + xy + y2=1, and if for x and y their values be substituted, the sum be equal to unity, and the truth is proved. 6. 3-a= a? + 2ab + 512 a: + 2ab - 362 4(a+b) 4(a - b) then (x – a)? +(y - 3)2 = 16(a + b)2 7. The given expression is reducible to (x+y+z)(xy+xz+y=)=0. 8. First, су whence ad.c(d+y)=bcy(b +x) 0(b + x) d(d+y)' and bacy – adx== xy(ad – bc), then dividing these equals by bdcy bc ad xylad – bc) dx by bury 6 a 9. If ab+cd, ac+bd, ad + bc be denoted by p, q, r respectively, it may be easily shewn from the equality of the first and second fractions, that p+q+r=0. The same condition may be found from the equality of the second and third fractions, whence P+9_P+r_9+r. -1. 9 9 10. Let ab +ac - bc=p, bc + ac – ab=, ab + bc - ac=r, then Pl_Pre_9, whence p=q and q=r, 9 P bc+ac - ab=ab+bc - ac, ..c=b, hence a=b=c. + (a - b)(a - c) (6 - c)(6-a) (c- a)(c-b) and their sum is – 2ab +2(a + b)c - 2c2 a a с с or a b十 bd+y or - 2 a constant negative quantity. (c-a)(c-b) XXV. 1 +a? 1+a 1. is greater or less than according as 1+a? is greater or less 1-a? 1-a than (1 +a). 2. Shew that (a - b)? > 0. 3. See Elementary Arithmetic, p. 13, Sect. IX., Exercises VIII. 9–12. 5. The former is greater or less than the latter, according as x2 is greater or less than y?. 3 a a с с a 1 26 6. is less than if (a + b)(b+c) be less than 26(a + c) or if ac +12 a+c (a + b)(b + c) be less than ab + bc, or if (a - b,c be less than (a - b)b, that is if c be less than b, and a-6 be positive or a greater than b. 7. x may have any integral value between 1 and 7. 2 2 3 2 8. If x+=>3, then x? +2> 3x and 1+ -1, x2 22 2 1 and 2? > 3x – 2, .. 2? +.. > 3(x+ -1), but x+ > 2. 2? + -> 3. x2 9. If a>b then a-6> 0 and a2 - 2ab +63 >0,.'. a? +62 > 2ab, and+">2. 10. Let the expression a(6+c) +62(a +c) + c2(a+b) or its equivalent ab(a + b) + ac(a + c) + bc 6+c) be divided by abc, 6 6 b b (a - b)(b – c)(a–c)>0, abc > 0, and “+ b 12. Let b+c-a=X, C + a-b=y, a +b-c 1 1 1 by substitution y [ + y + But x2 + y2 > 2xy, y2 + x2 > 2yr, 22 +22 > 2xz; and x?z + y2z> 2ry-, y2x+z2x > 2.xyz, x2y + z2 Y > 2xyz, also 3xyz = 3xyz. Hence by addition x?y + xy2 + x*: +xz2 + y2 + yz2 + 3xyz > 9xyz, 9 x+y+z' ญ x + y +7 and 2:1 = y +2, 2b =x+z, 2c=x + y. 7 Hence ily+, 2+2, x + y Y Suppose x, y, z in order of magnitude, and shew the aggregate to be greater than 3. 6-1 1 0-1 1 1. Here a= andre from which the required expres. b 1-C 6 1-C sion may be found. bc 2. Let a =b- -P; then a? - bc=pa and b? – ac=pb, b .. (a? – 62) + c(a - b)=(a - b)p and p=a+b+c. ab ah p= 3. Equate the values of 6 from each of the given expeessions. hd 4. Here ax + a'=bx+u, bx+1=cx+é; from the first x= a-t' a-6 (L a' - b 6 d-c. 5. Since + =0, then and (a - b)(1+cd)=(1+a))(d – c), 1+ ab '1+cd 1+ull+c' ..a+abc - d. – abc=b+bad - c-ced, or asl + bc) – ((1+bc)=b(1 + ad) - c(1 + ad), b-c a-d or (a - d)(1 + bc)=(6 - c)(1+ad), .. and 1+ bc ital' 1+ bc1+ad 6. 1 1 and b-c + с 2 с 2 a C c с ay: 1 1. alc e+ 1-6-+cz _ (1 – e)(1 - -); 7. First 1-=11+ez 1+ez 1+ ez 1+e+:+e: (1+e)(1+ ) 1+ez 1-2 1-elThirdly, by dividing the former by the latter, 1+x 1+el+7 a+by 8. If the value of y be substituted for y in := and the expression c+dy' reducedl, and divided by b+c, the following result will be obtained : ad +1,2 ad +62 (c? +ad): dz c2 + ad -b and C, ito the provision will hold good. 9. Here a2(h—c)<b>(a ---), clearing from fractions by multiplying each side a-d b-d by (a - d) {-d), then a’ {b2 – (c+d)b+cd}=bz{a? – (c+d)a+cd}, whence (c+d)(b-a)ab=cd(12 - a2), (c+d)ab=cil(b+a), dividing by b - a, (c+d)ab_cd(b + a) dividing by alcu, 十二三二十一, 6 12 73 b + abic and 1 + ax a - alac and and 1- buc 1- buc 7,3 =(a + b)(a? +1%). 22 and X _ Y __(a+)(b.x - ay) b y b, and 2: _Y_(2+b)(aur - by); b ху ?y? . . ab(x2 + y2)-(a? +62)xy=abxy – abxy=0. ху ab x(y* – 22)=0-c, then =a-C or { b-c + C-a a-b C-a + + C-a = 12. z(x2 - y2) + x(y2 – –-) xy y2 y*(in2 – --) 2 – 22_202 =a - C, whence XY 13. Put each fraction equal to p, then 2p=a+b, and 2p=c+d. 14. Let each expression be denoted by P, then 22+x=y8xzp, y2+x=x8y=P, 32+y-=zexyp, 2 and 2(x2 + y2 +-2)=pxyz(x2 + y2 +-2).. p= xy 15. X==- – 1 s(a - b)y+ (a - b)=) f(b-c)x (b-cz) Y $(c-a)x, (c-a)y? la- -c I? Ş(a-6)} + (a –c)2+b-a)}=&c a-b similarly by=b(c-a)p, cz=c(a - b)p. =0. Y 1 1 Hence ab + ac + bc -0. a? - bc 12 - ac C2 - ab a+b+c a+b+c abc 19. Here abc= - abc= ayz – x(x+y)(x+2) (x+y)(y + 2)(2+2) (x+y)(x + 2)(y + 2) 1 (x+y)(x+2)(y+:) 22 and (x + y)(x + 3)(y+). asl - bc) x?(+y+z) all – bc) x+y+By a similar process and may be shewn equal to the same (1-ac) c(1 - ab) quantity. 20. x+yy+2-3 x + y - 2 ax+by - cz add 1 to each and reduce ax +by - by +62 - 6 Y+:-* by + cz – ax 1 y+:-X to improper fractions, then and 1 1 and &+y-2 ax +cz - by a ax+by - cz Otherwise let the reciprocals of each = ==P, then ax +by – cz=P(x + y - x)) then adding the first and second ; next the first and by + cz – ax=p(y+--~) third ; and, lastly, the second and third, and the ax+cz – by=p(x+- y)) results are, p =b, pra, p=c, and a=b=c. C-a C-a C-a C-a EDITED BY ROBERT POTTS, M.A., TRINITY COLLEGE, CAMBRIDGE, HON. LL.D., WILLIAM AND MARY COLLEGE, VA., U.S. EUCLID'S ELEMENTS OF GEOMETRY. 1. Euclid's Elements of Geometry, the University Edition, with Notes, Questions, and Geometrical Exercises, selected from the Cambridge Senate House and College Examination Papers, with Hints for Solution of the Exercises. Demy 8vo., pp. 520, 10s. 2. The School Edition, with_Notes, Geometrical Exercises, &c. 12mo., pp. 418, 48. 6d. Books I.-IV., 38. Books I.-III., 28. 6d. , . Books I., II., 1s. 6d. Book I., 18. The University Edition of Euclid's Elements was first published in 1845, and the first School Edition in 1846. Both Editions have been enlarged and improved from time to time, and the total sales of copies of the work up to the present year amount to a number very considerably above half-a-million. In the year 1853, the Council of Education at Calcutta were pleased to order the introduction of these Editions of Euclid's Elements into the Schools and Colleges under their control in Bengal. In the year 1860, a Translation of the Geometrical Exercises was made into the German Language, hy Hans H. Von Aller, with a Preface by Dr. Wittstein, and published at Hanover. At the International Exhibition of 1862, in London, a Medal was awarded to R. Potts, “For the Excellence of his Works on Geometry.” Jury Awards, Class xxix., p. 313. A Medal, as an award of merit, was conferred on Mír. Potts's Educational Works by the Judges at the International Exhibition, Philadelphia, 1876. Critical Remarks on the Editions of Euclid. “In my opinion Mr. Potts has made a valuable addition to Geometrical literature by his Editions of Euclid's Elements."—11'. Whewell, D.D., Master of Trinity College, Cambridge. (1848.). “Mr. Potts has done great service by his published works in promoting the study of Geometrical Science."--H. Philpott, DL., Master of St. Catharine's College. (1818.) “Mr. Potts' Editions of Euclid's Geometry are characterized by a due appreciation of the spirit and exactness of the Greek Geometry, and an acquaintance with its history, as wel! as by a knowledge of the modern extensions of the Science. The Elements are given in such a form as to preserve entirely the spirit of the ancient reasoning, and having been extensively used in Colleges and Public Schools, cannot fail to have the effect of keeping up the study of Geometry in its original purity.”—J, Challis, M.A., Plumian Professor of Astronomy and Experimental Philosophy in the University of Cambridge.. (1818.) "Mr. Potts' Edition of Euclid is very generally used in both our Universities and in our Public Schools; the notes which are appended to it shew great research, and are admirably calculated to introduce a student to a thorough knowledge of Geometrical principles and methods.”—George Peacock, D.D., Lowndean Professor of Mathematics in the University of Cambridge, and Dean of Ely. (1848.), "By the publication of these works, Mr. Potts has done very great service to the cause of Geometrical Science. I have adopted Mr. Potts' work as the text-book for my own Lectures in Geometry, and I believe that it is recommended by all the Mathematical Tutors and Professors in this University:"-R. Walker, M..4.,,F.K.S., Reader in Experimental Philosophy in the University, and Tutor of Wadham College, Oxford. (1848.) greater Portion of this part the Course was printed, and had for some time been in use in the Academy, a new Edition of Euclid's Elements, by Mr. Robert Potts, M.A., of Trinity College, Cambridge, which is likely to supersedé most others, to " When |