ar ir 794 » It ffi'l Cornell Htjivi^t^itg JilrtMg THE GIFT OF :^..£_...^^Ma^.. i Q.a^<£^d^a^y:i^/UA^^^ _tkL^>,A.ii^,j_ \ .K.3..:m.i ,-„■■■■■ ^AflLlZ^. I Cornell University Library arV17946 College requirements In algebra. 3 1924 031 264 041 olln,anx The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031264041 COLLEGE EEQUIEEMENTS A L G- E B R A A FINAL REVIEW BY GEORGE PARSONS TIBBETS, A.M. INSTRUCTOB IN MATHEMATICS, WiLLISTON 8KMINART o>«o BOSTON, U.S.A. PUBLISHED BY GINN & COMPANY 1892 COPTBIGHT, 1892, By GEORGE parsons TIBBETS, A.M. Ttpographt bt J. S. CusHiHS Sc Co., Boston, U.S.A. Pbbsswobk et Gihn & Co., Boston, U.S.A. PKEFACE. The Williston students have found these reviews so serviceable that a more convenient form has become nec- essary. From a wide collection of college papers about four hundred examples, illustrating nearly every prin- ciple in Algebra, were selected and carefully arranged by subjects. Whenever a suitable one could not be so obtained, an original problem or one from foreign texts was inserted. The parallel sections are for the use of two divisions and for recitation-room drill. Colleges will find the work useful as an initial review ; while college candidates may be assured of entering if they perform all the examples without aid. Suggestions in regard to the work will be gladly received. G. P. T. Williston Seminaey, Easthampton, Mass., Jan., 1892. COI^TENTS. Sight Pboblems Parentheses and Evaluation Faotoeing ..... H.C.F., L.O. M., Evolution Fractions . ... Simple EauATioNS . Simultaneous Equations Theory op Exponents . Radicals .... ... Quadratics . . Simultaneous Quadratics Inequalities, Proportion, Variation . Progressions Binomial Theorem, Permutations Undeter. Coee., Limits, Logarithms Specimen Paper, Advanced Problems. Harvard, Yale . . . Vassar, Wellesley Cornell, Bryn Mawr . . . . tedhnologt, sheffield . . . Princeton Section L IL, IIL Page 1 8,9 IV., V. 10, 11 VL, VII. 12, 13 VIII., IX. 14, 15 X., XI. 16,17 XII., XIIL 18,19 XIV., XV. 20,21 XVI., XVII. 22,23 XVIII., XIX. 24,25 XX., XXI. 26,27 XXII., XXIII. 28, 29 XXIV., XXV. 30, 31 XXVI., XXVII. 32,33 XXVIII., XXIX. 34,35 XXX,, XXXI. -36, 37 XXXII., XXXIII. 38,39 XXXIV., XXXV. 40,41 XXXVI., XXXVII. 42,43 XXXVIII., XXXIX. 44,45 XL. 46 COLLEGE REQUIREMENTS IN ALGEBRA. 7 SECTION I. SIGHT PROBLEMS. 1. Of 452 students who tried the examinations last June, X were men, the rest were women. In all, z students failed. How many of the failures were men ? 2. The sum of 3 consecutive numbers exceeds the middle number by 10. What are the numbers? 3. In how many weeks will x horses eat 50 bushels of oats, if one horse eats y bushels in a week ? 4. A is 20 years old, and B is — 2 years older ; what is the age of B ? [Harvard.'] 5. Two men working separately can do a piece of work in X days and y days respectively ; find an expression for the time in which both can do it working together. [Harvard.'] 6. A pole 100 feet high, standing on the side of a hill, breaks off so as to form a right angle at the break, with the top resting on the hill 75 feet from the foot of the pole ; where did the pole break? [Cornell.] 7. Two steamers ply between the same two ports a distance of 420 miles. One travels half a mile an hour faster than the other, and is two hours less on the journey ; find the rates. [Vassar.] 8 COLLEGE REQUIREMENTS IN ALaEBRA. SECTION II. PARENTHESES AND EVALUATION. 1. ?>x — (bx-[^x — {y — x)'\)—{—x-2>y). 2. a- (26 + [3c - 3a - (a + b)] + 2a- (6-|- 3c)). 3. ax + b{x + c) + c'- [(-a -'b)x— {b~c)(b + c)]. 4. ab — c(x— b) — \(x + c){x — e) —c(b — \c — x\) — oc''\. 5. -[13a;-14y-3|2a;-2(-y + 2a;)-73/|]. 6. a + 6 {a; - 3a(y - 2a;) + 3a; [4a + 2(46 + 3)]}. -IhK-^)]}- 8. Evaluate a + 2a;— {S + y — [a — a; — (6 — 22/)]|. X— Va; 9. Evaluate a; - (Vx+l + 2) - ^ — ^^ when a- = 8. a; — 4 10. Substitute a; + 8 for y in y' + 2y'' - 15y — 86. COLLEGE EEQVIBEMENT8 IN ALGEBRA. 9 SECTION III. PARENTHESES AND EVALUATION. 1. 2a-[56 + {3c-(a + [26-3a + 4c])j]. 2. 3a-f3a— [3a-(3a-3a-3a)-3a] — 3a}-3a. 3. {a -{- b)x — Q) — c) c — \{h — x)b — (b — c){a + c)] — ax. 4. 2,x - '^y + b[- 4:x -\Zy - (2+1 x-2y) - m 5. 3c' + c(2a- [5c— ^3a + c-4a|]). 6. oi?y'' -(-xy'' + x?- - j xy - a;'' [- \f - y (a-y - a;'') | ]. 7. (a^ - 6')c - (a - h){a{h + c]-b{a~ c]). 8. Evaluate Va + Va6 + V6^ when a = 8, 6 = 64. 9. Evaluate -V^{b'+ ^ a' + 2ab + b-' J ^j^^^ „ _ g, 6 = 1. 10. Substitute y — 3 for a; in a;' + 2x'^ — Ibx — 36. 10 COLLEGE MEQUIItllMSNTS IN ALGEBRA. SECTION IV. FACTORING. 1 . Difference of Squares : {x' + y" - z^f - 4 xy. 2. Cubes: a^ — l. 3. Simple Trinomial : a;* - I2>x^y'' + 36/. 4. Complex Trinomial : Qx' — bxy — &y''. 5 . Literal Trinomial : x^ +(a-] — \xy + y'^. 6. Four Terms : a* + a^V — V& - c\ 7. Five Terms: 6a'' + 5ai — 66'' — 6ac + 4&c. 8. Six Terms: '2,a^ - ah - ac -^V -hhc - c\ 9. Perfect Square : 16 a;' + 16 a;' - 4a;* - 4a;' + a;^" 10. Imperfect Square : a;* — 7 aj^y^ + y*. 11. Eadicals : a;* + l. 12. Separation : a? — 6a;* + 11a; — 6. 13. Literal Exponents : a?"^ ■\-\3^ + -^. 14. Parentheses: fl +2/)' - Sa-'fl - y*) + a;*(l - y)'. COLLEGE BEQUIBEMENTS IN ALOEBMA. H SECTION V. FACTORING. 1. Difference of Squares : 1 — {x' -\- y^) + '2xy. 2. Cubes: a}^ -¥\ 3. Simple Trinomial : x^ — 2x — 2>. 4. Complex Trinomial : 8a'^ + 2a — 3. 5. Literal Trinomial : 1 — {rn? + n')3? + m'n^x*. 6. Four Terms : m?x + m^y — -dx — n'y. 7. Five Terms : aaf — 3 ax* + 2 ax^ + aa;'' — aa;. 8. Six Terms: Gcj' - 5a5 + 14ac + 5' - 56c + 4c'. 9. Perfect Square : 67a;^ + 49 + 9a;*- 70a;- 30a;l 10. Imperfect Square : 9 a* - 40a'^>' + 166*. 11. Radicals: a-\-b. 12. Separation: 3;= + 10a;^ + 29a; + 20. 13. Literal Exponents : a;" — ?/"- 14. Parentheses: a{a — l)x^ — {a — b — V)xy — b{h +l)y''. 12 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION VI. H.C. F., LC. M., EVOLUTION. Express both H. 0. F. and L. C. M. in factors : — 1. 6a;^+13a;-.5, 2,3? + 2x'+2x-\. [Frinceto7i.] 2. h*-6b'-8b~S, 4J3_i2J_8. [Yale.] 3. 6a;* - 13a;' + 3a;' + 2a;, 6 a;* -9 a;' + 15 a;' -27a;- 9. [Brown.] 4. 4a;*— 9a;' + 6a; — 1, 6 a;' - 7 a;' + 1 . [Johns Hopkins.] 5. 6a;* - 5a;'- 10a;' + 3a; -10, 4a;'-4a;'-9a; + 5. [narvard.] 6. 8a;' + 27, 16a;* + 36a;' + 81, 6 a;' + 5 a;— 6. [Vassar.] 7. Square Ecot of 4a* + 12a' + 5a'- 6a+ 1. 8. Cube Eoot of 8rt*+ 12a* — 6a*- lla' + 8a' + 3a -1. COLLEGE REQUIREMENTS IN ALGEBRA. 13 SECTION VII. H.C. F., L C. M., EVOLUTION. Express both H. 0. F. and L. C. M. in factors : — 4 a^ + 11 a;* + 81. {Johns SopUns.] 2. 8a;' + 2a; -3, 12 a;' + 10 a;" — 4. [Amherst] 3. 15aV-20aV-65a'a;-30a^ 12 6a;' + 20 5a;'- 16 &a;- 16 S. [Brown.] 4. a;' -3a; -2, a;^ — 2a;* — a; + 2. [Bartmo^dh.] 5. a;^-l, a;'' -2a; -3, 6 a;' — a; — 20. [ Technology.] 6. 2a;* + a;' -8a;' -a; + 6, 4 a;* +12 a;' -a;' -27 a; -18, 4a;* + 4a;' -17a;' -9a; +18. [Harvard.] 7. Square Root of 8a;H -i- - 8a;* + 2a;' - ^ + 2. [Bryn Mawr.] 8. Cube Eoot of a;« - 6a;» + 8 x* + 28a;' - 9a;' - 54 a; - 27. 14 COLLEGE REQUIBEMENTS IN ALGEBRA. SECTION VIII. FRACTIONS. 4a;^ + 3a;-10 1 . Eeduction : 4a;' + 7a;' -3a; -15 2. Addition and Subtraction : 0? — he , V -\- ca I _ c' + ab 3. Multiplication : r^±^_^^__i^l by 1"^+^' {a-b){a-o) {b + c){h-a) {c-a){c + b) 4. Division: \-2_±f^-^l by T^ + ^+l' 5. Complex : 12- 14 1 ^4 6. Evaluation : ^c iab £ J ,1 1 f a; + 2a , a; + 25 If a; = -, find tne value ot — '——- -\ '— pT7" a+o a; — 2a a; — 26 1 — a; V a; 1 + V \1 + a; 7. Miscellaneous: ^— ^ 1-y + .V'' - a; + y COLLEGE REQUIREMENTS IN ALGEBRA. 15 SECTION IX. FRACTIONS. 1 . Keduction : 1 2a;«+lla;'-43a.--24 X 14a;'-31a;^-31a;-6 2. Addition and Subtraction : 1 , 1 ,. 1 x{x — a){x — h) a{a — x){a — b) b(b — x){b — a) 3. Multiplication: ^^ x f±f X |±^^ ^. a^ + y' o" + by ¥ + y c 4. Davxsion : [^^^^ - ^ by ^^— , + ^—^ 5. Complex: 1. ^-2 2a;+l 6. Evaluation : If = a, " = 5, find value of 1- - — - + 7. Miscellaneous: 1+a 1+5 1+c ("■+„-qbi)(°'-'') b lA a-6\ a + 6 2^^ a + V 16 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION X. SIMPLE EQUATIONS. 2 5a; + 2 /g 3a; - 1 \ _ 3a;+ 19 ( x + l . gN 3 V 2^ 2 ve/ 8a; + 23 5a; + 2 ^ 2a; + 3 , 20 3a.' + 4 5 4- ^^^g^-A(y-4) = |(y-6) + A. 5. 12{3a;.-.25(a;-4)-.3(5a;+14)|=47. X a—J)cx X ac — A.bx 6. 2 2ic 6c 36c 7. -^-3+-^ = -^-2a(2-8a). 2a 4a' 3a' ^ -^ 8. At what time between 7 and 8 o'clock are the hands of a watch (a) together T (b) opposite? (c) at right angles ? COLLEGE REQUIREMENTS IN ALGEBRA. 17 SECTION XI. SIMPLE EQUATIONS. 1. 22; '" "~"' ^"^ e^^> 4 2. 3^-^:^-4 = ^Hili. 4 3 3. ^±10_2(3^_4)+(3^^2X2£:^=^._A. 3 5 6 15 leD-K'-Mei)-- S. 3.3a;- -'^^^r-^^ =.la^ + 9.9. .5 „ax—b \ — X . r^ 6. — :; h m = 0. 7. 3aa; — 2hx — ^c ~ \mx = f c + f ma; — w — Sa; + 2aa;. fi. A man bought a horse, and expected to sell it at 10 % profit ; but had to sell it for $ 50 less than he expected, and then found he had lost 15 % on what it cost him. What did he pay for the horse ? 18 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XII. SIMULTANEOUS EQUATIONS. 1. Simple : ax +by = c 1 a'a; + Vy = c' J 2. Reciprocals : 3a; 52/ 9 bx By 4 3. Parentheses : (a + b)x-{a- b) {a-b)x-{a+b) 4. Three Quantities : ax -\-by=l cy -\- dz ^m ex -\- fz = w 5. Fractions : — + -=« X y X y J ' 6. Numerical : a;-2y + 3z = 2 2a;-3y+ z=l %x- y + \ Iz = 9 j/ = 4aS"» 7. Problem. A firm has Java coffee at a cents per pound and Mocha at b cents per pound. How much of each is there in a mixture of a — J pounds if it can be sold at c cents a pound without loss ? COLLEGE REQUIREMENTS IN ALGEBRA. 19 SECTION XIII. SIMULTANEOUS EQUATIONS. 1. Simple : 2( a 4:ab - 2. Reciprocals : -Vx -\fy 1 + ' -1 Va; Vy 3. Parentheses : 7(a; + y) + 3(a;-y) = 4. Three Quantities : y + z =a- a; + z = 6 ■ 5. Fractions : a h a c 6 c 6. Numerical : 1x— Zy=\hz= 3 9a;-10y = = 33 z = 4 801 32) Problem. A and B walk in a circle whose circumference is C. If they start from the same point and go in opposite directions they meet in 4 hours. If they start from opposite points and walk around the circle in the same direction they meet in 8 hours. Find the rate at which each one walks. 20 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XIV. THEORY OF EXPONENTS. a" 3. 1. Discussion; 7-; x*. 2. Proof: (»")" = a"". 3. Removal of negative exponents : ^ — -— ^ — -■ 1 — X"'')/"^ + x~^ 4. Co„b.n...n: ($)Hf)'< IT 5. Multiplication : (a* — a'S* + ah'^ -ab + ah^ — b^ by (J + bi). 6. Division : x^i/"^ — 2 + x~''y^ by a;*y~* — x~^y^. 7. Involution : -VG ^^-\/('|i)j 8. Evolution : 1 + 4y"* - 2y"^ - 4y-' + 25y"^ - 24?/"^ + IGy" 9. Reduction : /^ + 4 + 4a;"^ 10. Problem. Find the number whose cube root is one-fifth of its square root. COLLEGE REQUIREMENTS IN ALGEBRA. 21 SECTION XV. THEORY OF EXPONENTS. 1. Discussion : I — — ^ — - — — r(a— ")'^"(a")' |_ (a"")*"-" 2. Proof: a''=l, a-» = -- a" 3. Removal of negative exponents : — . , , ■ c~* — a~' 4. Combination : (vy^ ^ { i ] ~^ i ' 5. Multiplication ; (a^ - a* + 1 - a~* + a"^) by (a* + l + a"^). 6. Division: ^"'"'^" ^ f^)'^'" 7. Involution ; 8. Evolution: a;' + 2a;^-3a;''-4a;^ + 4a; 9. Reduction ; x^ — ViX'^y 0? + 2-Vxy -\- 4:7/^ 10. Equation : '^ = 2V2 ; find a; 22 COLLEQE BEQUIBEMENTS IN ALGEBBA. SECTION XVI. RADICALS. x+1 j x-1 1. Reduction " " ■ + : 2. Addition and Subtraction : 3. Arrangement: \/2^, VS, -v/Sl. 4. Multiplication: [a;-i(l - A/^)][a; -^(1 + V^)]. 5. Division : 6^^<^ by :^, Vc-' Vet' 6. Involution and Evolution : [2a/3 + SVa + V6]l 7. Eationalization : Va;-V^+^ Va; + -y/x + y 8. Imaginaries : V— 9a;* + V— 16a;* - V- (a; - 1)V. 9. Binomial Surds : -\/41 + 12 V5. 10. Radical Equation : Va; + 2 — Va; - 2 = 2a;. COLLEGE BEQUIBEMENTS IN ALGEBRA. 23' SECTION XVII. RADICALS. 1. Reduction: ^(g? —i){^a-l)\ 2. Addition and Subtraction : 2 Vl25 - A/fl + ■v'ST - (- 512)J + Vm,. 3. Arrangement: VS, ^6, -v/IO. 4. Multiplication: (x- ^ ~ ^ Ya:- ^ + ^^ Ya:+ J-\ V 2V2 y^ 2V2 A viy 5. Division: ^2 \/2 -^ VV2A/i. 6. Involution and Evolution : 7. Rationalization : ( V a;y/ V3 2- V^^ 2 - V3 2 + V- 2 8. Imaginaries: [2V^^ - 5 V-^J. 9. Binomial Surds : ^14 + 4V6. 10. Radical Equations : Va; + 5 + Va; — 8 = VS. 24 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XVIII. QUADRATICS. 1. Simple: l2x' + x-l = Q. 2. Literal: a(s? - x) -\- b {x^ -\- x) = "^ a + b 3. Fractional: (Mzi^!!!^ = 2.. 4. Parentheses; (a;-2)(a: + 3)(a;H 3 a;- 4) = 0. 5. Eationalization : = x -\- -^/x' — 8. x-^/x^'-8 6. Fractional Exponents : x^ — x^ = 256. 7 . Quadratic Form : {ax — by—ia {ax — S) = | al 10a; 8. Eadicals: ^"-^ -Vinr + ^ = Vl0a;-9 Vl0a;-9 9. Cubic Equation : a;' — a;' — a; + 1 = 0. 10. Formation : Find equation ■with roots {a-{-b — c) and {a — b-\-c). 11. Problem. Tristram is ten years younger than Launcelot ; and the product of the ages they attained in 1890 is 96, Find the ages they attain in 1908. [Sarvard.] COLLEGE REQUIJREMENTS IN ALOEBRA. 25 SECTION XIX. QUADRATICS. 1. Simple: 91a;^- 2a;-45 = 0. 2. Literal: {a + 2b). ^_^^^_ a— 2,0 a — 26 x 3. Fractional : x + 1 2_ a; + 2 c ex ax — bx 4. Parentheses: (x -lXx-2Xx^ -6x+9) = 0. 5. Binomial Surds : (7 - 4 VS) x' + (2- VS) x=2. 6. Fractional Esponents : 4 Va; + Va; = 21. 7. Quadratic Form : x^ — 2x + 6-\/x^ — 2x + b = ll. 8. Radicals: Va; + a+ Va;+ Va; — a = 0. 9. Bi-quadratic : a;* — 2ar' + a; — 2 = 0. 10. Formation : Find equation with the roots i — / and (b — a). a — 11. Problem,. A cask P is filled with 50 gallons of water, and a cask Q with 40 gallons of brandy, x gallons are drawn from each cask, mixed and replaced, and the same operation is repeated. After the second replacement there are 8-|- gallons of brandy in P. Find X. 26 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XX. SIMULTANEOUS QUADRATICS. 1. Substitute: 2a; + 3^ =12 2. Obtain a;3/ ; ^ — y = 4") x' + y'= 106 I 3. Let y = vx: x''-\-\Qxy =11 bxy — 2>y'= 2 } } } 4. Divide : a? — •/ = 56 ' x^ + a;2/ + y' = 28 . 5 . Eetain Fractions : ^0 x y x' f 6. Yale: x'-y'' = % 1^:?1::3:7 X y 7. Wellesley : x'^ + xy=lb' xy-f = 15-1 '= 2J 8. Michigan University : x-\-y — ^xy = 7 ") a;H2/' + ^y =133] 9. Columbia ; Mines : a;' + ?/ + 3a;= 73 — 2a;yl y' + a; =44-32/1 COLLEGE REQUIREMENTS IN ALGEBRA. 27 SECTION XXI. SIMULTANEOUS QUADRATICS. ' = 17/ = 58 I 1. Substitute: a; — 3y = l X? — 2 xy -\- 'd y' 2. Obtain a;y: ^ — J/ =10 x' + y 3. Let y = vz: x' -\-?>xy = — \ y^ — xy = 12 4. Divide : a;'^ — a;y + 3/^ = 9 | ;c* + xY + y' = 243 I 5. Retain Fraction : - + - =7 ^ y i-l=25 6. Vassar: a;' + t/' = 56 a; + y = 5b| = 2 I 7. Princeton : «' + a;y + y' = 52 xy — x^ } 8. Univ. Penn. : a;' - ary + y" = 7 | a:' + 3?f + y* = 133 J 9. Harvard : _y a: + 6a\ ^ q a; + y V«(^-y) a^' - y'V 2/:(7a;-22/) = (S-a):(2a-9J) 28 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XXII. INEQUALITIES, PROPORTION, VARIATION. 1. Prove that the square of half the sum of any two quan- tities < half the sum of their squares. 2. Prove a'>a + -— 1 if a>l. a 3. What two numbers whose difference is d are to each other as a is to S ? 4. Ifa; — yisamean proportional between y and (y + 2 — 2 a;) , show that a; is a mean proportional between y and z. 5. Prove that if 2x:y::a:b, \a — x:\a-\- x: -.h — y.h +y. 6. Prove that a proportion taken by inversion is a true pro- portion. 7. A varies jointly as B and C; and A = & when 5=3, C=2. Find^ when5 = 5, (7=7. 8. A varies as the square of B, and inversely as the square of 0, and J. =4 when .2=1, 0=2. What is the value of '?' + ^' when B = 2, 0=21 B"- A' 9. The volume of a sphere varies as the cube of the radius. If 3 spheres with radii 9, 12, and 15 inches are melted into a single sphere, find its radius. aOLLEQE REQUIREMENTS IN ALGEBRA. 29 SECTION XXIII. INEQUALITIES, VARIATION, PROPORTION. 1. Prove that ( ■r\<:ab. [2 + bJ^ 2. Find the limit of x in (3x + 2)(x-S) > {z + 4:){3x-l)-S. 3. What number added to 2, 20, 9, 34, will make the results proportional ? 4. Find two numbers such that their sum, difference, and the sum of their squares are to each other as 4, 1, 17. 5. If a:b::c:d show that a:b:: VS^+b? : V3F+5^. 6. Prove that a proportion taken by division is a true pro- portion. 7. The area of a circle varies as the square of its radius and the area of a circle is 154 sq. ft. when the radius is 7 ft. Find the area of the circle whose radius is 10 ft. 6 in. 8. The offing at sea varies as the square root of the height of the eye above sea-level, and the distance is 3 miles when the height is 6 ft. Find the distance when the height is 50 yds. 30 COLLEGE BEQUIBEMENTS IN ALGEBRA. SECTION XXIV. PROGRESSIONS. 1. Given a, d, and n in an arithmetical progression, to find I. 2. Given o, I, and r in a geometrical progression, to find the sum s. [Note. — The above problems are given frequently.] 3. The first and ninth terms of an arithmetical progression are 5 and 22. Find the sum of 21 terms. 4. Find the wth term of the series 2, 2|-, 2|. 5. Find d and I when a = 3, n=l5, s = — 165. 6. Insert 3 arithmetical means betv?een — 9 and 18. 7. Find the twelfth term of V2, - 2, 2V2, - 4, etc. 8. Sum the infinite series „ +5^+ k; + ^*''- 000 9. The sum of 3 numbers in arithmetical progression is 12, and the sum of their squares is 50. Find the num- bers. 10. If a clock is constructed so as to strike up to 24, how many strokes will it make in the revolution of the index ? COLLEGE REQUIREMENTS IN ALGEBRA. 2,1 SECTION XXV. PROGRESSIONS. 1. Given a, I, and n, in an arithmetical progression, to find s. 2. Given a, n, and r, in a geometrical progression, to find I. [Note. — The above problems are given frequently.] 3. In an arithmetical progression, s = — -^i w = 20, a = ^. Find d. 4. Find the (2w)th term of 1, 3, 5, 7, etc. 5. Find a and w when I = — 4:7, d = — l, s = — 1118. 6. Insert 3 geometrical means between -J and 128. 7. Find the seventh term of — i. i, — f , etc. 8 . Sum the infinite series -g- + ^5" + tts 9. A traveller has a journey of 132 miles. He goes 27 miles the first day, 24 the second, and so on, travelling 3 miles less each day. In how many days will he complete his journey ? 10. Find the sum of all the numbers which are less than 500 and are divisible by 11 without a remainder. [YaZe.] 32 COLLEGE BEQUIBEMENT8 IN ALGEBRA. SECTION XXVI. BINOMIAL THEOREM. PERMUTATIONS AND COMBINATIONS. 1. Expand by the binomial theorem {^d? — f 6/. 2. Expand to 4 terms — • \Yale.\ 3. Expand (2a -3 6)-' 4. Obtain 4 terms ( a^ -{ ) 5. Find the fifth term of (a;- - 22/*)". 6. Second term of f Va' — ——^ ] {Harvard.'] 7 . Find the term independent ofa;in [3a; )• 8. Expand (1 + 2 a;')" to 4 terms. 9. Expand (a' + 1 + a-y. 10. How many different amounts can be made up from 5 different coins ? 11. In how many ways can 7 children form a ring? 12. I have 5 single volumes and a set of 3 volumes. In how many ways can I arrange these 8 books on a shelf, keeping the set together and in order ? COLLEGE BEQUIBEMENT8 IN ALGEBRA. 33 SECTION XXVII. BINOMIAL THEOREM. PERMUTATIONS AND COMBINATIONS. 1. Expand by the binomial theorem [ a j • 2. Expand to 4 terms (a + a;)~*. 3. Expand (VS — 3\/a)*. [TechTiology .] 4. Obtain the first 3 and last 3 terms of (x — y)"". 5. Find the fourth term of {2x- 2>y)-^. 6. Fourth term of (^-\ah-^^ ■ [Harvard.] 7. Find the terms without radicals (2Va — -v/-) ■ 8. Expand to 5 terms (1 + a)'*. 9. Expand (e" - e-"')'. 10. How many difi'erent signals can be made with 12 diflferent flags by hoisting 4 at a time above each other? 11. How many combinations can be made from the word Payson taken 3 letters at a time ? 12. At a whist party there are 6 ladies and 6 gentlemen. The host is to play with the most honored guest, and the hostess with the poorest player. In how many ways can the players be arranged if each man has a lady partner ? 34 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XXVIII. UNDETERMINED COEFFICIENTS, LIMITS, LOGARITHMS. 1. Expand ~ into a series of four terms by unde- termined coefficients. 2. Develop ^ ,^ into a series. 3. Separate — -— — into partial fractions. 4. Separate ■ into partial fractions. 5. Find the limit, when x increases without limit, of {x+\){x'-Z) x*-3x 6. Prove log5m = -:r-^^ loe„6 7. Simplify 2.372 X 7232 X .003722 (- 22.37)(72230000) 2 8. Find tlie value of x in the equation 5* = 30. COLLEGE REQUIREMENTS IN ALGEBRA. 36 SECTION XXIX. UNDETERMINED COEFFICIENTS, LIMITS, LOGARITHMS. 1. Expand into a series by undetermined coefficients l + 2x l—x — x^ 3 + a; 2. Develop to four terms 2-x-x'' 3. Separate — ^ into partial fractions. x" — a % ap -i- X 2 4. Separate ■^^- -— into partial fractions. iX — Zij (i — /iXj 5. Evaluate ^— — P'^^' + ^^ for « = ( 6. Prove that logjflsX log»6 = 1. 7. Simplify -^126 vT08 ^ ^i008A/T62. 8. Given the amount of a given principal for a given num- ber of years, to find the rate per cent. 36 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XXX. SPECIMEN PAPER. 1. Evaluation: The score of the Amherst-Technology game on Saturday was l±fl+£ to g+\f-[h-{e + g)-]\ + ^g. Find numerical value if e =f — g = A= 4. 2. Expression : A man's monthly salary is %x.. His weekly expenses are f w, besides annual tax of %t, and semi- annual insurance of $s. How much does he save yearly ? 3. Parentheses: c-[2a-J-(3a-2&-4a-36)]. [Sheffield:] 4. Multiplication: {\(r + ^0^-' + \){\a-~ \). 5. Division: {a^ -2^" -Qc' + ab ~ ac + 1bc) by (a-J + 2c). [Wor.TecL] 6. Formulas : Square [(& — 2) .-c + (1 — 6)]. 7. Inspection: Divide (a* jj by (a J [Mines.] 8. Factoring: a? - y^ — z^ + 2yz. [Yale.] 9. G.G. Measure: \^^f~^l'''^]'^' [Harvard.] (18a;^+ 3 a;— 10. 10. Miscellaneous : H. 0. F. by factoring or division ■x'-{a' + ¥)x' + a'b\ x''-{a+byx' + 2ab{a + h) x — a'^.^ L Golleetwn French COLLEGE REQUIREMENTS IN ALGEBRA. 37 SECTION XXXI. ADVANCED PROBLEMS. 1. A certain librarian spends every year a fixed sum for books. In 1886, the cost of bis purchases averaged two dollars per volume ; in 1887, he bought 300 more volumes than in 1886 ; and in 1888, 300 more volumes than in 1887. The average cost per volume was thirty cents lower in 1888 than in 1887. Find the number of volumes bought each year, and the fixed price paid for them. (Obtain two solutions.) \_IIarvard.'\ 2. A and B start at the same time from two towns and travel towards each other. When they meet B has travelled a miles more than A ; it will take A b days longer to reach the town B left, and B c days longer to reach the town A left. Find the distance between the towns. [Harvard.'] 3. Three students A, B, and 0, agree to work out a series of difficult problems in preparation for an examination ; and each student determines to solve a fixed number every day. A solves 9 problems per day, and finishes the series 4 days before B ; B solves 2 more problems per day than 0, and finishes the series 6 days before 0. Find the number of problems and the number of days given to them by each student. \^IIarvard.'\ 38 COLLEGE REQUIREMENTS IN ALGEBBA. SECTION XXXII. HARVARD. [Write legibly and without crowding ; give the work clearly, and find all possible answers. The shortest methods of work are preferred. Abridged processes of work may be used, but should be distinctly indi- cated.] 1. Solve the equation l(x__ b\2a 2 b a ia U_b 11 _ X a X b_ = 0. 2. Two tanks A and B are discharging water; A at the rate of x barrels per hour, and B at the rate of (x + 100) barrels per hour. At (1 + y) hours after noon, A contains 470 barrels less than at noon ; and at a time (1 — y) hours after noon, B contains 400 barrels less than at noon. Find the rate at which each tank is discharging water, and the times (1 + y) and (1 — y) hours after noon. Obtain two sets of answers, and interpret negative results. 3. Find the Greatest Common Measure and the Least Com- mon Multiple of 123;" - 29a; + 14 and 18 a;'' + 3 a; -10. 4. Find the fourth term of ( a/w ] 5. Find a mean proportional between 4a;' — 3a;— 1 and a; — 1. COLLEGE REQUIREMENTS IN ALGEBRA. 39 SECTION XXXIII. YALE. Examination for Admission. 2-v/3 3 . 1. Express ^r = in an equivalent fraction having 1 + V3 - V5 a rational denominator. 2. Divide 81a;- 16 by 3a;i + 2. 3. Find the cube of 1 -■ 1 + a;* 4. Find the square root of 27 — 12V5. 5. Solve the equation I — 7 x' = 2ax — bx'. ( xy ^ a, 6. Solve the equations ^ , „ , (. x' + y^ = b. 7. Expand into a series, giving five terms. (a — 2xy 40 COLLEQE REQUIREMENTS IN ALGEBRA. SEOTION XXXIV. VASSAR. 1. Prove that the product of two quantities is equal to the product of their G. 0. D. and L. C. M. Find G. 0. D. and L. C. M. of 8x' + 21; 16 a;* + 36a;' + 81; 6a;^ + 5a;-6. 2. Simplify J^J^Xxl^-J^. 3. Solve the equations : (1) = b. a+ -Vd' — x' (2) rK' + y' = 56, X +y =2. 4. Define root of an equation. How many roots has a sim- ple equation? Form the equation whose roots are a and — b. 5. Define a"', m being a positive integer, and prove (»")"=«"". Multiply together ^ abc, a^b~'^c^, a~^b^c~\ Write down the values of 3*; 5°; O'' ; 16"^; 2-"; #. 6. There are three numbers in arithmetical progression whose sum is 15. If 1, 3, 9 are added to them respec- tively, they are then in geometrical progression. Find the numbers. 7. Explain the meaning of " mean proportional," "fourth proportional." li a : b : : c : d, prove that each of their ratios is equal to (a + mc) : (b -f- md). COLLEGE REQUIREMENTS IN ALGEBRA. 41 SECTION XXXV. WELLESLEY. 1. Factor a; — 3 aVa;- 28 a'; m* + mV + w* and a^b'-'^c-' + a^b-'^'^c'' — 2. 2. Simplify 'a"-'-' -v/J-to-*)' ^a"-" VoJ^^b'- 3. Add 2 Vl25 - A/ff + a/sI - ( - 512)* + -^192 and [7^ + 2 Vl20 + 3(YP + 3 Vi+ 7(f )^](V^)^ 4. Define simultaneous equations. 5. Sum the series 1, ^, —\ to 9 terms, and derive the formula for the sum of a series. 6. Solve for x in the following equations. 1. C^ + - 2 X + V2 - a;' a; - V2 ■ 2. 3a;"A/? + 2-^ = 16.' Va;" 7. Find a; and y from the equations xy = ^ ) using the principles of proportion as far as possible. Prove the most important of the principles of proportion used. 8. A distributes $180 in equal sums among a certain num- ber of people. B distributes the same sum, but gives to each person $6 more than A, and gives to 40 per- sons less than A does. How much does A give to each person ? 42 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XXXVI. CORNELL 1. Define the H. 0. F. and L. 0. M. of two polynomials, and explain fully the theory of the method of finding them. 2. Determine the common factors of l2a?-Qx'-?,x + 2 and 16a;' + 12a;'' -4a;- 3, and hence find 3 values of x, which when substituted in the expression [(12a;' - 8a;'' - 3a; + 2)] + [(16a;' + 12a;^ - 4a; - 3)] will give zero for the result. 3. If the roots of the quadratic a;'— ^a;+$' = are two consecutive integers, prove p^ — 4 g' — 1 = 0. 4. Extract the square root of 47 — 12Vl5, and find the value, when :c= V3, of 2a;-l _j^+l {x-l)\ {x+lf 5. For what values of m will the equation a;* - 2 (1 + 3 wij a; + 7 (3 + 2 m) = have equal roots. 6. Solve the equations : 1 5Va--l 3/ 10(a;-l) (ax-by = 2ah, f y^ + a:y = 4, \2bx-2ay = ?,b^~a\ ^"^ Xx' + 2^ -xy = Q. 7. Simplify tI^X;^!1"xtK \a» \a^ Ma" 8. A line 6 in. long is produced. Find the length of pro- duced part so that the rectangle contained by half the line and the line made up of the half and the part produced may be equal to the square on the produced part. COLLEGE REQUIREMENTS IN ALGEBRA. 43 SECTION XXXVII. BRYN MAWR. (Students are expected to answer questions in each of the divisions B and C of the paper. Division A is in Arithmetic] B. 5. Prove the identity : (a? + V-\- c'Xd' + e' +f) - {ah + he + cfj = {bf- cef + \cd - aff + [ae - bdf. 6. Solve the equations for all values of x : fax — cy -\- cz ^^ a r S5 x'' -\- 4:9 xy = x bx + cy =b also (2) \ z' = 4y' ex — cz = c l _a 7. Prove that A = ^^^^-r — is a root of 2a ax' + bx-\-c = 0. Show that if B is the other root, AB must = — a c. 8. Explain as clearly as you can why a~^, a", a~*, are said to be equal to -, 1, — • « a" 9. A varies as the square of B and inversely as the square of C. Also A =4: when B=l, C= 2. What is the value of ^^^4! ^lien B = 2, 0=2? E' — A^ 10. What is the ^th term in the arithmetical progression whose first term is a and nth term is Z? Sum the series - + — + rr+ to infinity. 3 ?>^ 2> How many terms of the series 1 + 3 + 5 + '?+ must be taken in order that the sum may be (r — 1)^ where r is a positive integer? 44 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XXXVIII. TECHNOLOGY (BOSTON). 1. Reduce to their lowest terms x^-l&x , a' + 2a(a + 1)4-4 x'-d.x-'k ^° a* -4 2. Show that 3. Show that 2 4. Solve the simultaneous equations ^(2/+l) = 8; 2/(2; + 2) = 12. 5. Find the square root of x^ - 4cx' + 20a;' + 16a; + 16. 6. Solve the equation A/28-a;+\/i = 4. 7. Form the quadratic equation whose roots are f, f , and f,. 8. The sum of three numbers in arithmetical progression is 18. If the first be increased by 1, and the third by 2, they will be in geometrical progression. What are the numbers ? COLLEGE REQUIREMENTS IN ALGEBRA. 45 SECTION XXXIX. SHEFFIELD SCIENTIFIC SCHOOL. Entrance Examination, algebra from quadratics. [Note. — State at the head of your paper what text-book you have studied on the subject and to what extent.] 1. Solve the equation aa? + 62; + c = 0, and point out what relation must exist between the coefficients in order that the roots may be equal. 2. Determine by inspection the roots of the equation 3. Solve the equation x'-x + b-^2x'-bx + e> = ^'^^^^ - A 4. There are 20 things of one kind and 10 of another; how many different sets can be made, each containing 3 of the first kind and 2 of the second ? 5. Insert 3 arithmetical means between 4 and 20. 6. Expand into a series by the method of unde- ^ \-\-x^x' •' termined coefficients. 7. Find the limit of ^ — — — '-^ '- when x is increased x^ —Zx without limit. 46 COLLEGE REQUIREMENTS IN ALGEBRA. SECTION XL. PRINCETON. [State what text-books you have read.] 1. (a) Find G. CD. of 6a;' + 13a; -5 and 3a;' + 2a;' + 2a; - 1. (5) Factor {nx — x + y — ny); also (9a'5' — j/'). 2. Find the square root of X* + 4a;y' + y"' + 4a;% - 2x^y-'- - 4a;*. 3. Solve the equations CL — X ■ — OC 1 G X p. be ca ah 4. Solve the equations 4 4 — a; (6) (a; - 1)~* -b{x- 1)"* + 1 = 0. 5. Solve the simultaneous equations 7 — 2,x 3 . — — — - = -; y — a; = 4. 5-3y 2' ■^ 6. Solve the simultaneous equations 3^+y=2; .-2/ = 4. y + 5 a; 7. What number must be added to m and to n, in order that the sums may be in the ratio ofp to g'? (D O) CO CL (Ji CQ Tl (Q CD Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page Missing Page MATHEMATICS. WENTWORTH'S SERIES. n^HE United States Bureau of Education, in Circular of Infor- mation, No. 3, 1890, says, on the subject of mathematical text-books, "If we were called upon to name the writer whose books have met with more wide-spread circulation during the past decennium that those of any other author, we should answer, Wentworth. ... In point of scientific rigor, Wentworth's books are superior to the popular works of preceding decades." These two points indicate the two leading characteristics of these remarkable books. They combine mathematical scholarship with all the elements of school-room popularity. The author's aim was to be helpful. He has made a study of his subjects with the psychology and probable capacity of the students constantly in mind. His books enable the average student not merely to learn something, but to master the study, and at the same time they give the brighter ones. a plenty to exercise their faculties upon. There is no attempt to make a parade of learning. On the other hand, there is no absolution from solid work. Everything is made simple, practical, direct, and thorough. Mental energy is econo- mized. The teacher has to use his strength only in the necessary work of teaching, and the pupil his only in the necessary work of acquiring. To learn by doing, and to learn one step thoroughly before the next is attempted, are the chief elements of the method, and yet no books give pupil or teacher less of the tread-mill feeling. The consciousness of mastery constantly cheers and invigorates the student, while the teacher has the satisfaction of wielding an in- strument fitted to the hand. All will recognize in these characteristics the marks of ideal text-books. No one will claim that any series could be suited to every particular case. The merit of Wentworth's is that it suits in all but now and then a peculiar one. The books are found everywhere in the United States, and here and there all around the world. The testimony of use may be summed up in one word, Satisfaction. What more can one desire than to be satisfied? 66 MATHEMATICS. Introductory Boohs. Wentworth's Primary Arithmetic. Introduction price, 30 cents; allowance in exchange, 10. cents. Wentwortlt^s Grammar School Arithmetic. Introduction price, 65 cents; allowance in exchange, 20 cents. These two books form a complete course in Arithmetic foK- Common Schools, and are believed to present the best known methods in the most attractive, available, and practical form. Wentworth and Reed''s First Steps in Number. For prices, see List at the beginning of this Catalogue ; for full descrip- tion, see Common School Catalogue. A High School Arithmetic. (^Wentworth & Hill's Practical Arithmetic.') By G. A. Wentworth, Professor of Mathematics, Phillips Exetei Academy, and Dr. Thomas Hill, Portland, Me. 12mo. Roan back, 367 pages. Mailing Price, $1.10; Introduction, $1.00; Allowance, 30 cents; Answers free, on teacher's order. Same. Abridged Edition. For Grammar Schools. 288 pages {includ- ing Answers). Mailing Price, $1.00; for introduction, 75 cents; Allow- ance, 25 cents. rpmS book is intended for high and normal schools and acade- mies. It assumes that the pupil has some knowledge of the simple processes of Arithmetic, and aims to develop his power over practical questions, as well as to increase his facility in computing. Such problems are selected as are calculated to interest the pupil, and lead him to independent thought and discovery. The prob- lems cover a wide range of subjects, and are particularly adapted to general mental discipline, to preparation for higher studies, mechanical work, business, or professional life. J. M. Peirce, Prof, of Mathe- matics, Harvard College: It is clear, straightforward, sound, care- ful, and abundantly supplied with weU-chosen examples. I would es- pecially commend the pains that have been taken to set forth intelli- gibly the shortest and easiest methods of work, such as are actually used by the best computers, and the inclusion of various principles and rules, ofteu neglected by mere text-book com- pilers, which go far to form a really scientific habit of thought in the young student. H. Carr Pritohett, Prof, of Mathe- matics, Sam Houston Normal Insti- tute, Tex. : Wentworth and Hill's Arithmetic is exceedingly practical. All useless matter has been discarded, and the fundamental principles sa clearly set forth that it is a model of simplicity and brevity. MATHEMATICS. 67 Wentworth's School Algebra. Bt G. a. Wentworth, Professor of Mathematics in Phillips Exeter Academy. Half morocco, v + 362 pages. Mailing Price, ^1.25; for introduction, $1.12; allowance for an old book in exchange, 35 cents. Answers in pamphlet form, free, on teachers' orders. TN this book the author has availed himself of his own expe- rience in writing and teaching the elements of Algebra, and of the experience of hundreds of others. Where improvement was possible, it has been made. The School Algebra is offered as exactly right for the usual high school and academic courses. It gives a thorough and practical treatment of the principles of Algebra up to and including the binomial theorem, and is strictly in line throughout with the author's College Algebra. For college preparation it is particu- larly well suited. The problems in this book are nearly all new, either original or selected from recent examination papers, and are graded with the utmost care. They are sufficient in number to illustrate and fix all the principles, and interesting and varied enough to hold the student's attention through the book. The passage from Arith- metic to Algebra is made easy, and the advantages of using letters clearly pointed out. The treatment of fractions has been further simplified. Radicals precede quadratics. There is at the end a carefully made collection of miscellaneous examples, covering nearly every principle of Algebra. This, with the author's College Algebra, makes a complete and consistent course. J. B. Colt, Prof, of Mathematics, Boston University : In the hands of intelligent teachers, it should lead the young student to pursue Algebra without feeling that it is character- ized by arbitrary laws and mysteri- ous processes. J. J. Hardy, Prof, of Mathematics, Lafayette College, Pa. : Here is an at- tempt by a good teacher, who is also familiar with the work of great schol- ars, to make the advances worked out by them tell for the improve- ment of elementary teaching. The result is a most excellent book. It is simple, yet scientific; scholarly, yet an excellent drill book. Oscar Scluuiedel, Prof, of Mathe- matics, Bethany College, W. Va. : A book for beginners, written by a teacher whose methods are clear and concise, who understands the diffi- culties encountered by his pupil, and who knows how to clear away these difficulties. 68 MATHEMATICS. David Eugene Smith, Teacher of Mathematics, State Normal and Training School, Cortland, N.Y.: I have examined it with a good deal of care. It seems well adapted to the needs of our schools, — even bet- ter adapted than the author's former work, which I have used and recom- mended. The improvements to be found in this work are such as will meet the approval of all teachers. ErastusTest, Prin.of Purdue Uni- versity, Prep. Dept. : After a trial of three months, I am more than satis- fied with the book. Barring a few mistakes unavoidable in a first edi- tion, I do not see how it can be easily improved, and am about ready to regard it as the ne plus ultra in the line of a school Algebra. Frank E. Thompson, Prin. of High School, Newport, E.I.: I am pleased with it, especially the intro- ductory chapter and the interpreta- tion of negative answers. A thorough knowledge of the contents of the book will enable a pupil to pass an entrance examination to any college. J. S. Slocum, Prin. of South Divis- ion High School, Chicago : I have used it in connection with the prep- aration of a class for college, and have been pleased with its clear defi- nitions, logical arrangement, and happy selection of both examples and problems. Bichard T. Lewis, Pres. of Judson College, N.C. : It is fresh, strong, really invigorating, and requires thought to master. E. D. Sherman, Prin. of High School, Bay City, Mich. : We think it an excellent text-book, or we should not at present be using it. 0. S. Westcott, Prin. of North Division High School, Chicago : The student who finishes it will be splen- didly prepared to grapple with the beautiful discussions of higher Al- gebra. Newton B. Hohart, Prin. of West- ern Reserve Academy, Hudson, 0. : I know of nothing better, and in some respects I consider it superior to any- thing of the kind I have seen. W. P. Durfee, Prof, of Mathemat- ics, Hohart College, N.Y. : An admi- rable hook for college preparation. The arrangement of topics is good, their presentation clear and logical, and the illustrative examples all that could be desired. Cooper D. Schmitt, Prof, of Math- ematics, University of Tennessee ■■ For the work intended, I do not think it can be surpassed. W. N. Hailmann, Supt. of Schools, La Porte, Ind. : For a high school using but one book in Algebra, we consider it decidedly the best in the market. C. L. Sawyer, Prin. of High School, Watikegan, III. . It has what so many books lack, — method. George E. Gay, Prin. of High School, Maiden, Mass.: It is better adapted to use in high schools than any other. J. E. Smith, Supt. of Schools, San Antonio, Tex. : I think I can con- scientiously say that it is the best high school Algebra that I have ex- amined. E. E. Stratton, Prin. of High School, Davenport, la. : The author evidently has the right conception of what is needed for a high school Algebra. MATHEMATICS. 69 Wentworth's College Algebra. By G. A. Wentworth, Professor of Mathematics in Phillips Exeter Academy. Half morocco. 500 pages. Mailing Price, $1.65; for in- troduction, ^1.50; allowance for an old book in exchange, 40 cents. Ansioers in pamphlet form, free, on teachers' orders. rPHIS is what its name indicates, a text-book for colleges and scientific schools. The first part is simply a concise review of the principles of Algebra preceding quadratics, with enough examples to illustrate and enforce the principles. Room is thus left for a full discussion of the higher topics. The endeavor has been to give in matter and methods the best training in algebraic analysis at present attainable. The work covers a full year, but by omitting starred sections and problems, the instructor can arrange a half-year course. Many professors helped with sug- gestions to make the College Algebra fit prevailing requirements. William Beebe, Assistant Prof, of Mathematics and Astronomy, Yale Colleye : I find it characterized by the clearness and method of all Pro- fessor Wentworth's books, and am particularly struck with the amount of matter in the Algebra. Dascom Greene, Prof, of Mathe- matics and Astronomy, Rensselaer Polytechnic Institute, Troy, N. Y. : The methods of treatment seem to be generally judicious, and the style attractive, and to the student well grounded in the elements it furnishes an excellent practical course in the higher branches of the subject. E. S. Crawley, Instructor in Mathematics, University of Penn- sylvania : It is a great improvement over — — for college use, and in its way seems to me to leave nothing to be desired. T. C. Leonard, S. J., Prof, of Math- ematics, St. Ignatius Colleye, San Francisco : The book, in my opin- ion, is a model Algebra, distinguished for its clearness of explanation and the eminently practical nature of its matter. J. C. Glashan, Inspector of Public Schools, Ottawa, Canada : I am satisfied I can unqualifiedly recom- mend it. Henry Bay Warner, Prof, of Mathematics, Mt. Union Colleye, 0. : Both as to subject-matter and style, I regard it as the most perfect Alge- bra I have ever examined. Henry C. King, Prof, of Mathe- matics, Oberlin Colleye, 0.: The chapter on Determinants I think a specially valuable feature, which alone would tell strongly in favor of the book for college use. E. P. Thompson, Prof, of Mathe- matics, Geneva College, Pa. : This is such a work as the college student ought to use. 70 MATHEMATICS. Wentworth's Higher Algebra. By 6. A. Wentworth, Professor of Mathematics in Phillips Exeter Academy. HaK morocco. 528 pages. Mailing price, $1.55; for intro- duction, $1.40; allowance for an old book in exchange, 40 cents. Answers in pamphlet form, free, on teachers' orders. rplilS work is designed to prepare thoroughly for colleges and scientific schools, and to furnish in addition what is needed for the general student in such institutions. The preparatory schools and academies for which it is particu- larly recommended are those of high grade, especially such as give the pupils a thorough preliminary drill in Arithmetic. Normal schools, seminaries, and a large percentage of the col- leges, particularly in the West and South, will find it specially suited to their requirements. It is substantially equivalent to the author's Complete Algebra, but is believed to be in many respects better. The arrangement and the treatment of the topics have been revised, and fuUer explanations have been given. As compared with the School Algebra, this is more complete, inasmuch as it takes up the topics usually included in higher Algebra. As compared with the College Algebra, it gives a fuller treat- m.ent of the simple elements of Algebra and not so elaborate a treatment of the more advanced portions. In a word, the Higher Algebra provides in a single book a com- plete course parallel to the course provided by the School and College Algebras together. It is not necessary to add that it is characterized by the special excellences of Professor Wentworth's other works on this subject. As this book is published while the catalogue is printing, it is impossible to present testimonials. WENTV^ORTH'S ALGEBRAS. Full Coukse ; School Algebra, followed by College Algebra. Condensed Coukse : Higher Algebra. Special Coukses : Shorter Course; Complete Algebra. MATHEMATICS. 71 Wentworth's Elements of Algebra. By Geokgb a. Wbntworth, Professor of Mathematics, Phillips Exeter Academy, and author of Geometry, Trigonometry ,_ etc. Half morocco. X + 325 pages. Mailing price, $1.25 ; for introduction, $1.12 ; allowance, 35 cents. Answers bound separately in pamphlet form. rPHIS book is designed for high schools and academies, and con- tains an ample amount for admission to any college. The single aim in writing this volume has been to make an algebra which the beginner would read with increasmg interest, intelligence, and power. The fact has been kept constantly in mind that, to accomplish this object, the several parts must be presented so distinctly that the pupil will be led to feel that he is mastering the subject. The chapter on Choice, heretofore included in the Elements of Algebra, has been omitted in accordance with the wishes of many teachers. Wentworth's Complete Algebra. Includes the subjects usually taught in Colleges. Half morocco. 525 pages. Mailing price, $1.55; for introduction, $1.40; allowance, 40 cents. Answers hound separately in pamphlet form. rpHIS work consists of the author's Elementary Algebra, with about one hundred and eighty-iive pages additional. The additions are chapters on Choice, Chance, Interest Formulas, Con- tinued Fractions, Theory of Limits, Indeterminate Coefficients, the Exponential Theorem, the Differential Method, the Theory of Numbers, Imaginary Numbers, Loci of Equations, Equations in General, Higher Numerical Equations. T. H. Safford, Prof, of Mathe- matics, Williams College : I am using Wentworth's Complete Alge- bra to my entire satisfaction. A. S. Hardy, Prof, of Mathematics, Dartmouth College, in New York " Critic " : It is no easy matter to attain the true mean between a dif- fuseness which enervates and a brevity which discourages. It is not too much to say that the author of the Complete Algebra has been emi- nently successful in this respect, and it is safe to predict for this work the favorable reception which has been already awarded to his Geometry. Joseph Hall, Prin. of High School, Bartford, Conn. : This is the ouly Algebra that we use in preparing students to enter the academic or scientific department of any college or university. 72 MATHEMATICS. Wentworth's Shorter Course in Algebra. By George A.Wentwobth. Half morocco. 258 pages. By mail, $1.10 ; for introduction, 5fl.OO; allowance, 35 cents. Answers in pamphlet form, free, on teachers' orders. nPHIS book was prepared and is recommended only for a special use. It was based upon the author's Elements of Algebra, and contains all the essential subjects treated in that book, but with fewer examples, so as to make a one year's course. Care has been taken to exclude all exacting problems, and yet the problems are not so easy as to deprive the student of the satisfaction of mastery. It is a significant recommendation of the Shorter Course in Algebra that it was recently adopted for exclusive use in all the schools of the State of Washington. C. K. Wells, Svpt. of Schools, Marietta, 0.: It has given good sat- isfaction. It is full enough for a high school course. I think it will hold its ground in our high school for a long time. 0. T. Snow, Prin. of High School, Batavia, III. : It is the best treatise on the subject I have ever seen. I shall recommend its adoption. Alfred S. Bowe, late Prin. of High School, Worcester, Mass. : I am sat- isfied that it will meet a popular demand. It is not so difficult as to discourage, nor so easy as to give a misimpression of the character of the study. B. T. McCord, Prof, of Mathematics, Lincoln University, III. : I think it is admirably adapted for a short course. Algebraic Analysis. By G. a. Wentwokth, A.M., Professor of Mathematics in Phillips Exe- ter Academy; J. A. McLellan, LL.D., Inspector of Normal Schools, and Conductor of Teachers' Institutes for Ontario, Canada ; and J. C. Glashan, Inspector of Public Schools, Ottawa, Canada. Part I. con- cluding with Determinants. 12mo. Half leather, x + 418 pages. By mail, f 1.60 ; to teachers and for introduction, f 1.50. HIS work, which has been previously announced as Wentwortk If McLellan's University Algebra, is intended to supply students of Mathematics with a well-filled storehouse of solved examples and unsolved exercises in the application of the fundamental theo- rems and processes of pure Algebra, and to exhibit to them the highest and most important results of modern algebraic analysis. J. M. Taylor, Prof, of Mathemat- examples, and clearly presents the ics, Colgate University, N.T. : It best methods of their solution, contains an admirable collection of T MATHEMATICS. 73 Wentworth's New Plane Geometry. By Geokge a. Wentwoeth, Teacher of Mathematics, Phillips Exeter Academy, 'SIR. 12mo. x + 242 pages. Mailing Price, 85 cents ; Intro- duction, 76 cents ; Allowance lor old book, 25 cents. Wentworth's New Plane and Solid Geometry. By George A. Wentworth, Phillips Academy, Exeter, N.H. 12mo. Hall morocco. xi + 386 pages. Mailing Price, f 1.40; Introduction, 11.25 ; Allowance for old book, 40 cents. The book now includes a treatise on Conic Sections (Book IX.). A LL the distinguisHng characteristics of the first edition have been retained. The subject is treated as a branch of practical logic, the object of which is to detect and state with precision the successive steps from premise to conclusion. In each proposition a concise statement of what is given is printed in one kind of type, of what is required in another, and the demonstration in still another. The reason for each step is indicated in small type between that step and the one following ; and the author thus avoids the necessity of interrupting the process of demonstration to cite a previous proposition. The number of the section on which the reason depends, is, however, placed at the side of the page ; and the pupil should be prepared, when called upon, to give the proof of each reason. Each distinct assertion in the demonstrations and each particular direction in the construc- tion of the figures begins a new line, and in no case is it necessaiy to turn the page in reading a demonstration. In the new edition will be found a few changes in the order of the subject-matter. Some of the demonstrations have been given in a more concise and simple form. The diagrams, with which especial care was taken originally, have been re-engraved and mate- rially improved. The shading, which has been added to many of the figures, has proved a great help to the constructive imagination of pupils. The theory of limits — the value of which the author emphasizes — has been presented in the simplest possible way, and its application made easy of comprehension. But the great feature of this edition is the introduction of nearly seven hundred original exercises, consisting of theorems, problems of construction, and problems of computation, carefully graded and adapted to beginners in Geometry. 74 MATHEMATICS. George W. Sawin, late Instr. in Mathematics in Harvard College : The old edition of Wentworth's Geometry, at least the part devoted to Plane Geometry, I always re- garded as, on the whole, the best text-book of its kind in English. The new edition is, in my opinion, a far better book than the old. The books on Solid Geometry have been raised to the excellence of the first five books. Even with a teacher of very moderate abilities, this text- book ought to render the study of Geometry attractive, and, I may add, fascinating, to the student of average talents. It may interest you to learn that I found last year that out of a class of one hundred and ten in Solid Geometry, one huiv- dred and one had been prepared in Wentworth's Plane Geometry, (.Nov. 13, 1888.) W. A. Moody, Prof, of Mathemat- ics, Boiodoin College : I have exam- ined it with considerable care, and consider it an improvement on the old edition, noticeably in the better figures and general finer appearance of the book and in the numerous " original " exercises. J. R. rrench. Prof, of Mathemat- ics, Syracuse University: It seems to be a great improvement upon the former edition. We shall doubtless continue to use it. W. C. Bartol, Prof, of Mathe- matics, Buchnell University; I be- lieve it is a decided improvement on the old edition, and shall use It in my classes. Lyman Hall, Prof, of Mathematics, Georgia School of Technology : I find the new edition of Wentworth's Ge- ometry admirably adapted to our needs here. I think the author has accomplished the great end of giving clear and concise proofs in the mini- mum of space, and has materially improved, without rendering too diffi- cult, the original exercises and nu- merical examples. E. H. Stanley, Instructor of Math- ematics, Oberlin College: Quite a careful examination of the Geometry has led me to regard it one of the very best of its kind. W. Hoover, Prof, of Mathematics, Ohio University : I have introduced into my classes the new edition of Wentworth's Geometry, and flnd it very much of an improvement upon the old. ... I never had such good results in my geometry classes. E. Miller, Prof, of Mathematics, University of Kansas : The book is a very superior one, and grows in ou' estimation because it is exactly suited to our needs. It is a book that com- pels a student to think and invent new methods and demonstrations. L. S. Hulburt, Prof, of Mathe- matics, University of Dakota : The revised edition is the best text-book ou the subject of geometry that I have ever used. J. M. Taylor, Prof, of Mathematics, University of Washington Territory: The best book on geometry has been made better by the revision. A. S. Eoe, lately Prin. of High School, Worcester, Mass. : I wish to express my unqualified approval of it. S. Weimer, Teacher of Mathemat- ics, High School, Cleveland, Ohio: The new book is a decided improve- ment over the old, although his first book, which we are now using, is superior to any text-book for class work that I have any knowledge o£ MATHEMATICS. 75 Wentworth's Trigonometries. 'pHE aim has been to furnish just so much of Trigonometry as is actually taught in our best schools and colleges. The prin- ciples have been unfolded "with the utmost brevity consistent with simplicity and clearness, and interesting problems have been selected with a view to awaken a real love for the study. Much time and labor have been spent in idevising the simplest proofs f oi the propositions, and in exhibiting the best methods of arranging the logarithmic work. Answers are included. Plane and Solid Geometry, and Plane Tr/gonometry. 12mo. Half morocco. 490 pages. Mailing Price, $1.55; Introduction, $1.40; Allowance for old book, 40 cents. Plane Trigonometry. 12mo. Paper. SOpages. MailingPrioe, 35 cents; Introduction, 30 cents. Plane Trigonometry Formulas. Two charts (30 x 40 inches each) lor hanging on the walls of the class- room. Introduction Price, $1.00 per set. Plane Trigonometry and Logarithms. 8vo. Cloth. 160 pages. Mailing price, 8.5 cents ; introduction, 80 cents. Plane and Spherical Trigonometry. 12mo. Hall morocco, iv + 151 pages. Mailing Price, 80 cents; for introduction, 75 cents ; allowance, 20 cents. Plane and Spherical Trigonometry, with Tables. 8vo. Hall morocco, vi + 259 pages. Mailing Price, $1.25 ; for Intro- duction, $1.12. Allowance for old book, 35 cents. Wentujorth's Plane and Spherical Trigonometry and Surveying, with Tables. Svo. Half morocco. 307 pages. Mailing Price, $1.40; Introduction, $1.25 ; Allowance for old book, 40 cents. Surveying. 8vo. 80 pages. Paper. Mailing Price, 35 cents; for introduction, 30 cents. Wentworth's Plane and Spherical Trigonometry, Surveying, and Navigation. 12mo. Half morocco. 359 pages. Mailing Price, $1.25; Introduction. $1.12; Allowance for old book, 35 cents. 76 MATHEMATICS. rPHE object of the work on Surveying and Navigation is to pre- sent these subjects in a clear and intelligible way, according to the best methods in actual use ; and also to present them in so small a compass, that students in general may find the time to acquire a competent knowledge of these very interesting and important studies. Answers are included. A. H. Pierce, Instructor in Mathe- matics, Amherst College : I consider Wentworth's Trigonometry a perfect book for the class-room. All unnec- essary matter is omitted, and the ar- rangement of the work is such as to help a student to a clear outline of the whole subject, . . . and the plen- tiful supply of exercises and practical problems relieves the teacher of the necessity of constantly consulting other text-books. Wentworth & Hill's Fiue-Place Logarithmic and Trigonometric Tables. By G. A. Wentworth, A.M., and G. A. HrLL, A.M. Seven Tables (for Trigonometry and Surveying): Cloth. 8vo. 79pagea Mailing Price, 55 cents; Introduction, 50 cents. Complete (for Trigonometry, Surveying, and Navigation) : Half mo» rocco. 8vo. 158 pages. Mailing Price, $1.10; Introduction, $1.00. rpHESE Tables have been prepared mainly from Gauss's Tables, and are designed for the use of schools and colleges. They are preceded by an introduction, in which the nature and use of logarithms are explained, and all necessary instruction given for using the tables. They are printed in large type with very open spacing. Compactness, simple arrangement, and figures Jarge enough not to strain the eyes, are among the points in their favor. Wentworth & Hill's Exercises in Arithmetio. I. Exercise Manual. 12mo. Boards: vi + 282 pages. Mailing Price, 55 cents ; for introduction, 50 cents. II. Examination Manual. 12mo. Boards. 148 pages. Mailing Price, 40 cents ; Introduction Price, 35 cents. £oth in one volume, 80 cents. Answers to both parts together, 10 cents. rpHE first part (Exercise Manual) contains 3869 examples and problems for daily practice, classified and arranged in the common order; and the second part (Examination Manual) con tains 300 examination-papers, progressive in character. MATHEMATICS. 77 Wentworth & Hill's Exercises in Algebra. I. Exercise Manual. 12mo. Boards. 232 pages. Mailing Price, 40 cents; Introduction Price, 35 cents. II. Examination Manual. 12ino. Boards. 159 pages. Mailing Price, 40 cents; Introduction Price, 35 cents. Both in one volume, 70 cents. Answers to both parts together, 25 cents. rpHE first part (Exercise Manual) contains about 4500 problems classified and arranged according to the usual order of text- books iu Algebra; and the second part (Examination Manual) contains nearly 300 examination-papers, progressive in charac- ter, and well adapted to cultivate skill and rapidity in solving problems. British Mail: All engaged in the much precious time and labor, and practical woik of education will to give his students the benefit ol appreciate these Manuals, as they progressive and carefully thought- are calculated to save the master out exercises. Wentworth & Hill's Exercises in Geometry. 12mo. Cloth. 255 pages. 70 cents. Mailing Price, 80 cents ; Introduction Price, rpHE exercises consist of a great number of easy problems for beginners, and enough harder ones for more advanced pupils. The problems of each section ai'e carefully graded, and some of the more difficult sections can be omitted without destroying the unitv of the work. The book can be used in connection with any text- book on Geometry as soon as the geometrical processes of reason- ing are well imderstood. ■select propositions from It to supple- ment every stage of our work. Amelia W. Platter, High School, Indianapolis, Ind. : I find the sub- {iect so carefully graded, that I can Analytic Geometry. By 6. A. Wkntwoeth. Revised edition. 12mo. Half morocco, xii + 301 pages. Mailing Price, $1.35; for introduction, $1.25; allowance in exchange, 30 cents. rpHE aim of this work is to present the elementary parts of tha subject in the best form for class-room use. The connection between a locus and its equation is made per feetly clear in the opening chapter. 78 MATHEMATICS. The exercises are well graded, and designed to secure the best mental training. By adding a supplement to each chapter, the author has made provision for a shorter or more extended course, as the time given to the subject will permit. Sascom Greene, Prof, of Mathe- matics and Astronomy, liensselaer Polytechnic Institute, Troy, N.Y.: It appears to be admirably adapted to the use of beginners, and is espe- cially rich in examples for practical application of the principles of each chapter. The full and clear explana- tion of first principles given in the opening chapter is a new and highly commendable feature of the work. E. Miller, Prof, of Mathematics, University of Kansas, Lawrence : As a book for beginners, it is admi- rable in all its arrangements and features. The great number of problems scattered through it will largely relieve it of that abstract analysis which is so often a terror to students. The book is, like the other works of Professor Wentworth, a good thing. Elementary Mathematical Tables. By Alexander Macfaklane, D.Sc, LL.D., Professor of Physics in the University of Texas. 8to. Cloth. iv + 106 pages. Mailing price, i!5 cents ; for introduction, 75 cents. rpmS collection of tables contains logarithms, antilogarithms, addition and subtraction logarithms, logarithmic sines and cosines, tangents and cotangents, natural sines and cosines, tan- gents and cotangents, secants and cosecants ; arcs, reciprocals, squares, cubes, square roots, cube roots, circumferences, circular areas, spherical contents, powers, constants, hyperbolic logarithms, exponentials, divisors, least divisors, interest tables, and a large number of auxiliary tables. The tables are mostly four-place; they have a uniform decimal arrangement, and have been made every way convenient and adequate. elegant arrangement. The book is excellently adapted for use in every form of computation where the more elaborate tables are not demanded. {Jan. IS, 1890.) J. B. Coit, Prof, of Math., Boston Univ., Boston, Mass. : They are surely worthy of very high commen- dation. I am impressed with the amount of valuable material and the The Algebra of Logic. With examples. By Alexander Macparlane, Professor of Physics in the University of Texas. 12mo. Cloth. Illustrated with diagrams. X + 155 pages. By mail, f 1.35 ; for introduction, $1.25. The principles of the Algebra of Quality investigated and compared with the principles of the Algebra of Quantity. Westminster Beview, London : It will find eager and attentive readers. MATHEMATICS. 79 A Treatise on Plane Surveying. By Daniel Carhaet, C.E., Professor of Civil Engineering in the West- ern University of Pennsylvania, Allegheny. Illustrated. 8vo. Half leather, xvii + 498 pages. Mailing Price, $2.00; for introduction, $1.80. rriHIS work covers the whole ground of Plane Surveying. It illustrates and describes the instruments employed, their ad- justments and uses ; it exemplifies the best methods of solving the ordinary problems occurring in practice, and furnishes solutions for many special cases which not infrequently present themselves. It is the result of twenty years' experience in the field and technical schools, and the aim has been to make it extremely practical, having in mind always that to become a reliable surveyor the student needs frequently to manipulate the various surveying instruments in the field, to solve many examples in the class-room, and to exercise good judgment in all these operations. Not only, therefore, are the different methods of surveying treated, and directions for using the instruments given, but these are supplemented by various field exercises to be performed, by numerous examples to be wrought, and by many queries to be answered. The judicial functions of surveyors, as given by Chief Justice Cooley, are set forth in an appendix. As a practica,l and complete treatise, Carhart's Surveying has received a cordial welcome. W. A. Moody, Prof, of Mathemat- ics, Bowdoin College : I consider the hook exceptionally fine in execution, suhject-matter, and arrangement. D. W. Hering, formerly Prof, of Math., Univ. of City of New York: The Surveying is, I think, superior as a text-hook to any book on the subject with which I am acquainted. It is compendious without being too voluminous, and the skilful treat- ment of the subject accords perfectly with the methods of the author, both as a teacher and a practical engi- neer. Oren Boot, Prof, of Mathematics, Hamilton College : I have looked it through with great interest. The mechanical execution is, in the first place, elegant; the arrangement is admirable. . . . The work seems ad- mirably adapted to student use and, the class-room. C^ Wm. Hoover, Prof, of Mathemat- ics, Ohio University: It is indeed a\ superior work, and merits the widest \ adoption. ' Colmau Bancroft, Prof, of Mathe- matics, Hiram College : I find in it several important matters not con- tained in other text-books with which I am acquainted, — matter which I have felt obliged to give my classes by lectures. 80 MATHEMATICS. Hill's Geometry for Beginners. T By G. A. Hill, A.M. 12mo. Cloth. 320 pages. Mailing price, ^l-lOj for introduction, fl.OO; allowance, 30 cents. Answers, in pamphlet form, can be had by teachers. HIS book presents the subject in the natural method as distin- guished from the formal method of Euclid, Legendre, and the common text-books. The central purpose is intellectual training, or, teaching by practice how to think correctly and continuously. been taught geometry in the manner set forth by Mr. Hill, I should have been saved at least six years of blun- dering effort in endeavoring to grasp the subject of geometry from set propositions. "W. E. Byerly, Prof, of Mathe- matics, Harvard University : I do not see how the part devoted to plane geometry could be improved. John Trowbridge, Prof, of Mathe- matics, Harvard University: If I had Hill's Lessons in Geometry. For the Use of Beginners. By G. A. Hill, A.M., author of the Geometry for Beginners. 12mo. ' Cloth. 190 pages. Mailing price, 75 cents ; for introduction, 70 cents; allowance for an old book in exchange, 24 cents. Answers, in pamphlet form,, can be had by teachers. fymS is a course similar to that given in the Geometry for Begin- ners, but it is shorter and easier, and does not require a knowl- edge of the metric system. Like the Geometry for Beginners this is especially commended to those who cannot pursue the study far but desire the discipline of geometry. C. C. Bounds, Prin. State Normal School, Plymouth, N.H.: For giving to students who have never studied geometry a real and living knowl- edge of the subject and a commaud of its more important applications, I know of no book equal to this. Corwin F. Palmer, Supt. of Schools, Dresden, Ohio : It is a delightful little work, and full of inspiration. A teacher who gets the spirit of that hook into him cannot fail to teach well. In the hands of the pupil I know of nothing that approaches it. Hill's Drawing Case. Prepared expressly to accompany Hill's Lessons in Geometry, and con- taining, in a neat wooden box, a seven-inch rule with a scale of milli- meters ; pencil compasses, with pencil and rubber ; a triangle ; and a protractor. Retail price, 40 cents ; for introduction, 30 cents. A specimen copy of the Lessons in Geometry with the Drawing Case will be sent, postpaid, to any teacher on receipt of $1.00. m kJ SmS mtm [inll-f f-'-^i •"'» ■SP "■M*