INCLUDITO SQUAEE BOOT, CUBE BOOT, AND OTHER BOOTS. A HIGHLY PRACTICAL, BRIEF AND UNIQUE METHOD FOR THE EXTRACTION OF ALL ARITHMETICAL ROOTS. A SCIENTIFIC PROCESS NOT HERETOFORE PRESENTED IN ANY PUBLISHED WORK ON ARITHMETIC, AND SAVING NINE-TENTHS OF THE LABOR USUALLY NECESSARY FOR THE EXTRACTION OF ROOTS, AND ESPECIALLY OF CUBE ROOT, UNDER THE RULES NOW EMPLOYED. FOR THE USE OF ALL GRADES AND ALL SCHOOLS ABOVE THE PRIMARY, AND FOR TEACHERS IN PARTICULAR. ^^ OF THB***^ " OF THE lUKIVEESITY] — BY— G. D. HINES, A. M. w CLEVELAND. 0.: J. R. HOLCOMB & CO., PUBLISHERS, 1886. COPYBIQHT, BY J. E, HOLCOMB & Co., 1886. ,1^ -Copies of this work will be sent to any address, postpaid, on receipt of price. Active Agents wanted. J. R- HOLCOMB & CO., Pdblishbes, Cleveland, Ohio. ^7-9^ DEDICATION. TO MY WIFE, THE SYMPATHETIC SHARER WITH ME OF THE MIXED CUP OF FORTUNE INCIDENT TO A LONG SCHOOL LIFE, THIS LITTLE VOLUME IS AFFECTIONATELY INSCRIBED. G. D. HiNES. PREFACE. -^HIS little work needs but a short introduction. It is brevity itself, and whatever of merit it may possess, is largely due to its brevity and practicability. The methods of treating difficult roots, herein contained, and especially of cube root, were first suggested to the Author in the winter of '8i and '82, while teaching the Lincoln School, Plumas County, California. The pupils of that school, as is usual with most pupils, on first meeting the subject, were having trouble with cube root. This caused the teacher to put his wits to work in the almost hopeless effort to devise, if possible, some easier and shorter way of solution than the prolix processes extant in the arithmetics ; and, after some days of close scrutiny of the meaning and relation of roots and powers, the Author detected, for the first time, a new method of teaching cube root. The treatment of other roots, and of surds, naturally followed ; and the result has been what my fellow-teachers and others will find in the following pages of this work. It is believed that the addenda of Interest, with its unique, brief, and simple treatment, will prove an attractive feature of the book. This work is distinctly original. It details our own discoveries and is the product of our own thought respecting the treatment of that difficult subject, cube root. We send it forth on its mission, conscious that it must stand or fall on its own merits. We are fully persuaded, at the same time, that it needs only to be seen and understood to be appreciated ; and that, if generally introduced, it must supplant the perplexing and unsatisfactory rules in the text-books on cube root and the higher roots, and on surds. We ask an impartial examination by our fellow-teachers everywhere, and believe they will find the little book helpful, if not indispensable, to them. Selma, Cal., May 28, 1886. cube"root AND OTHER ROOTS UPROOTED. Def. — A root is one of the equal factors that has been repeated in multiplication a given number of times, to produce a given pov/er. Ex. — 3 is the cube root of 27, having been twice repeated in multiplication, or thrice used as a factor, to produce the power. 27 is the cube root of 19683, having been twice repeated in multiplication, or thrice used as a factor, to produce the power. 15 is the fourth root of 50625, having been three times repeated in multiplication, or four times used as a factor, to produce the power. And so on for any other roots, integral or fractional, positive or negative. To find the roots of all powers that legitimately belong to arithmetic, is the chief object of this work. Obs. — As this treatise deals chiefly with cube root and other higher roots, no extended notice of square root will be taken. Only an incidental usage will be made of it. THE BASIS. It is a well-known fact that the principles underlying Arithmetical evolution are derived from the mother science of Algebra, and that the arithmetical rules have been formulated out of the algebraic formulce. No other rules for the extraction of roots have been presented in the arithmetics, and per- haps, substantially, no others can be framed than those which depend on algebraic principles. But certain rules can, nevertheless, be formulated, which, while they have reference to algebraic principles, completely revolutionize the old mammoth rules, and, in brevity, almost annul them. THE NEW TREATMENT. The methods about to be illustrated neted have no reference to algebra, nor do they require any knowledge of that science. They annihilate the "cubic block" system, which clearly presents the principles of evolution to only the maturer scholars ; and then only in cube root, and are of no real advantage in the actual work of even the cube root, in very large numbers, and certainly of no advant- age in extracting any other than the cube root. CUBE ROOT. We will now present our method for the extraction of the cube root. Powers are either perfect or imperfect. 15625 is a perfect cube, while 18740 is an imperfect cube, or third power, and is called a surd. We will present a rule for the extraction of the cube root of perfect third powers, and one also for that of surds. Obs. — It may be remarked that a surd may be considered an imperfect power of any degree what- ever. Thus, 18740 may be considered an imjaerfect square, cube, fourth power, or any other power ; for we may require the approximate square root, cube root, fourth root, or any other root, of 18740. But it sometimes happens that a perfect power of one degree is a surd of another degree, and vice versa. Ex. — 25 is a perfect square, but an imperfect cube ; while 27 is an imperfect square, but a perfect cube, EXACT CUBES OF TWO PERIODS. Let us extract the cube root of the following numbers, viz.: -^ 13824 74088 185 193 A little observation and practice enable us to determine by inspection the root 250047 figure of the first period of the pow^r. Thus, the root figure of 13, in the first of 262144 91 125 the preceding numbers, is 2. And by the new method, the root figure of the last 166375 ■ 97336 period is 4, when the period ends in 4. Hence, the cube root of 13824 is 24. The 704969 226981 root figure of the first period of 74088 is 4, and the root figure of the last perlod'is 2, when the period ends in 8, (It is 8 when the periocl ends in 2.) So the cube i-oot, of the last number is 42. The cube root of 262144 is obtained in the same way. The root figure of 262 is 6, and of 144 it is 4. So the ^262144 is 64. Again, the root figure of the first period of 166375 is 5; and the root figure of the last period is 5, when the period ends in 5. So the 1^166375 is 55. 704969 gives, for the first root figure, 8 ; and the last is 9, when the period ends in 9. So, the ^'704969 is 89. In like man- ner, 185193 gives, for the first root figure, 5 ; and last figure is 7, when the period ends in 3. (It is 3, when the period ends in 7.) Thus, the 1^185193 is 57. The cube root of 250047, for reasons already MATHEMATICAL ROOTS UPROOTED. Stated, is 63. The cube root of 9"25, for similar reasons, is 45. The ^97336 is to be written out impromptu, just as the p revious roots have been ; making the last root figure 6, when the final period ends in ^. Also, the ^226981 is 61, the last root figure being i, when the concluding period ends in I. Observe that, in all perfect cubes, the final root figure is simply chosen, according to the character of the terminating figure of the power. This is a great saving of time and work, as will be shown hereafter. EXPLANATION. The reasons for the foregoing selection of the final root figure, depend on a very plain principle. In all perfect third powers, it is evident that the final figure of the pnver arises from the multiplication of the final figure of the root tzuice into itself. Now, if we multiply the nine digits, respectively, twice into themselves, we shall have this result, viz.: Observing the final figures of these powers, we see that the final root figure i pro- duces a 1, on being twice multiplied into itself. We see that the final root figure 2 produces an 8, and an 8 a 2 ; that 3 produces a 7, and a 7 a 3 ; that a 4 produces a 4; that a 5 produces a 5 ; that a 6 produces a 6, and that a 9 produces a 9. Obs. — The cubes of the several digits, viz.: i, 8, 27, 64, 125, 216, 343, 512, 729, must be so thoroughly familiar to the student that he can select the root figure of the first period impromptu. If the first period is, in magnitude, between i and 8, the root figure is i ; if the first period is, in magnitude, between 8 and 27, the root figure is 2 ; if the first period is embraced between 27 and 64, the root figure is 3 ; if it is between 64 and 125, the root figure is 4; and so on. 1. What is the root figure of the first period, if the period is comprised between 512 and 729? 2. What is the root figure of the first period, if the period is larger than 729 ? EXAMPLES. iXiX 1= I 2x2x2= 8 3X3X3= 27 4X4X4= 64 5x5x5 = 125 6X6X6 = 216 7 X 7 X 7 = 343 8x8x8=512 9 X 9 X 9 = 729 Let it be required to extract the cube root of the following, by the foregoing principles, viz.: Thus, we may write out, impromptu, according to the foregoing principles, the'roots of these, and of all perfect cubes involving only two periods. Let the student find the roots of the following, writing out the answers, off-hand, with- out any figuring or formal extraction, and choosing, at sight, the final root figure, ac- cording to the character of the terminal figure of the power, viz.: 39304 #" ■091125 f 46656 f 405224 f 389017 f 474582 lA ] ^. 0000021744 ^.000024389 f^ . 000079507 ^ .000614125 f 2.19 7000 f^ 941. 192000 0l>s. — The last six preceding numbers consist of three periods, but may be solved without premed- itation by the same rules, respecting the terminal figure, as are given for powers of two periods. EXACT CUBES OF THREE PERIODS. Let us extract the cube root of the following numbers: : .- 7. 153990656 V'^^. I. 44361864 4- 12.812904 8. 387420489 \ Q^^ 2. 134217728 5. 5545233 9- 10077.696 \ , ',f 3. 12812904 6. 2000376 10, 36.926037 ^/u ' •' Taking the first of the preceding numbers, and selecting the first root figure, 3, we take its cube 44361864 I 354. I from the period, and to the remainder attach the next period. Find the second /^ . 27 I Ans. root figure by dividing the partial dividend, save the two right-hand figures, by triple the square of the first root figure. Choose the last figure according to the 17.361 I character of the terminal figure of the power. 27 75 In the same manner, the cube root of 134)217728 is 1 512, 125 I Ans 92,17 So, the. cube root of For a partial divisor we take three times the square of the first root figure, and again choose the last figure. 1 28 1 2004 is I 234, found thus: Partial divisor 12 48,12 The last figure is simply chosen. The cube root of 12:812904 is 2,34, the same in form but different in value. MATHEMATICAL ROOTS UPROOTED. { I 3 I 45-45 Dividing and allowing for a completed divisor, we get 7 for the second root figure, and choose the last, which must be 7. Why? The ^2000376= 126. Ans. / -^> •- f ' Partial divisor 3 !4- 10.00 Why is the last root figure 6? No other figuring than the above is necessary. Divisor 52x3=75 The ^153990656=536. 125 Always triple the square of first root figure for ^pktt&l divisor of all 289.90 but the two right-hand figures of the partial dividend. The ^387420489 = 729. I 343 Divisor 147 | 444.20 Having obtained the second root figure, we choose the last unerringly. Why is it 9 ? How is 147 obtained ? 21.6. The ^10077.696: I 8 Divisor 12 | 20.77 O"^ the same principles the cube root of 36.926037 27 99.26 3-33. and is found thus : 27 On the law of the terminal figure of the power depends the secret of this new method with cube root. It is worth much to the student. Let him verify all of the above answers, by going through the actual work in every case, and thus acquire the needed familiarity. THIRD POWERS. K ^^K Involving periods of noughts at the left or right of the significant figures, and extracted by the ^^B. foregoing rule. i The ^.000520476129 = .0809. 512 Partial divisor 192 174-75 The partial divisor is not contained in the brought-down dividend shorn of its two right-hand figures, and we place a nought for the third figure of the root, and arbitrarily choose the last figure which is 9. The ^.048228544000 = .3640. 27 Divisor 27 212.28 Dividing, we find the partial divisor is contained in 212 only 6 ^ times, allowing for the effect of a finished divisor. ' We choose the remainder of the root figures. .0192. The ^.000,007,077, I Divisor 3 60.77 Dividing by the partial divisor, we find it will go into the par- tial dividend the largest possible number of times. (No divisor can go more than 9 times.) The last figure of the root is 2. Why ? The ^.000618470208: 512 Divisor 192 .0852. Ans. 1064.70 I. How is the partial divisor, 192, obtained? We select, unerringly, the last root figure. MATHEMATICAL ROOTS UPROOTED. ADDITIONAL EXAMPLES FOR THE STUDENT. Find the cube root of the following, viz.: Find the value, also, of these : I. 84604519. 7. f 13481272 2. 2803221. ^ 8. 1^8615.125000 3- 3176523. 9- f 738.763264 4. 382657176. 10. f 561.515625 5. 40.353607. II. -^ 21024576 6. 1520875. 12. if 67917312 In solving these examples, let it be understood that our only rule for cubes of three periods is, to take out by inspection the root figure of the first period, and, having taken its cube from that period and attached the next peiiod to the remainder for a new dividend, to find the next root figure by divid- ing the partial dividend by triple the square of the first root figure, and arbitrarily ckoose the last figure of the root, according to principles already explained. Perfect Cubes of More Than Three Periods. We will now extract the cube root of some numbers of more than three periods, and show that the new method applies to them, with a very slight amount of additional work. Let it be required to extract the cube root of the following numbers, viz.: 8024024008, 10460353203, 98867482624, 1 226 1 5327232, 1 54480441 6. Taking the first of the above numbers, we proceed as with powers of three periods, thus : 8,024,024,008 I 2002. Ans, Divisor 12.00 24.024 Explajtation.—Ymdimg the first root figure, deducting its cube, and at- taching the next period for a dividend, we find that the partial divisor, 12, is not contained in the par- tial dividend, 24, of the second period, and we attach the next period. The root thus far found is 20, and triple its square is 1200, the next partial divisor, which is again not contained in the brought-down dividend shorn of the two right-hand figures, and the root now found is 200, and we choose the last figure. The ■ 12.6. 1 10460353203 8 24.60 1261 2187. Ans. 212X3=1323 11993.53 Explanation. — We divide, as usual, the first partial dividend by the triple squai-e of the first root figure, which is 12, and obtain i for the second root figure. We finish the divisor, 12, by adding to it, successively, advanced one place to the right for each addition, the triple product of the two root figures, and the square of the last One. This finished divisor we multiply by the last root figure, and take the product from the brought-down dividend. We need only a second partial divisor, the triple square of 21, to find the third root figure, and we choose the last figure of the root, according to the character of the terminal figure of the power. 4624. The ^98867482624 = 64 Ans. Divisor 48.36 72 348.67 33336 Fin. div. 5556 15314.82 Using the same finished divisor as an approximate divisor, we find the next root figure to be 2, and then we select the last. Observe that 36 is put in the niche of the other two parts of the divisor, to save space. Obs. — To insure accuracy in finding the third root figure, it is generally best to take for a divisor the triple square of the first two root figures. Even then there is an immense saving of work, time and space over the old methods. By the modes of solution practiced heretofore, as treated in the books, the work of the above example is absolutely overwhelming, covering, with the rule and the explanation, from three to five pages. Let the pupil, for the present, accept on trust such parts or features of the rule as he may not thoroughly understand. Further elucidation will be given in due time. MATHEMATICAL ROOTS UPROOTED. 1st Divisor 48.81 ^122615327232 = 4968. 64 586.15 53649 5961 49663.27 43218 2nd Divisor 49''X 3 = 7203 . , ,. . 6445 It is only necessary to frame a second partial divisor to obtain the third root figure, and then we arbitrarily choose the last figure of the root. What have we saved by this abbreviated method ? We have saved the completion of the second divisor, the formation of the third, the completion of the third divisor, and all the multiplications and subtractions connected with these last periods, the prolixity and difficulty of which rapidly increase, under old methods, as we approach the end. The 1^1544804416 = 1156. Ans. 1st divisor 3 I Finished divisor 331 2d divisor 363 544 331 2138.04 .1815 There is no necessity for multiplying the second partial divisor, 363, by the third root figure, 5. We do so here, to show that the remainder of the dividend divided, is less than the divisor. The ^12,521,107,822,861 =23221. Ans. 1st divisor 12 Fin. divisor 2nd divisor 1389 45.21 4167 Explanation. — We complete the first partial divisor, and take its product, with the corresponding root figure, from the l^rought- I3174 down dividend. The triple of the square of the root now found is a partial divisor by which all the other root figures may be 367 found, except the last, and that is simply chosen. Having framed 317 the partial divisor, 1587, we ascertain how often it is contained in the corresponding partial dividend (always excepting the two 50 right-hand figures), and multiply the divisor by the quotient fig- ire, and subtract the result from the portion of the dividend divided, as in ordinary division. Having ound the third figure of the root, we use for a dividend the remainder of the last partial dividend, and lor a divisor we use the previous divisor shorn of its right-hand figure. And, in multiplying this divisor by the quotient figure, we reckon in the number of units that would be to carry from the figure cut off. If there were more periods than five, the process might be continued, by dropping figures, successively, from the right of the divisor. But the last figure of the root is always chosen. And, now, in what is said above, respecting the finding of some of the root figures by ordinary divison, lies the germ of the method herein treated, for the approximate extraction of the cube root of surds, to be explained in due time. Take one more example involving five periods. Let it be required to extract the cube root of the number 599)183,710,672,625 1 84305. Ans. 512 1st divisor 192.16 20176 Bsor 842x3 = 2116.8 I oot fieure. which, fror 871.83 80704 Elucidation.— E-aixTiCt in the usual way to find the first two figures of the root. Then, as will be seen by in- 64797.10 specting the work, there is a necessity for finishing only 63504 one partial divisor, and afterward forming another one from the first two root figures. This second partial divisor 1293 we use to find two more root figures, and then choose the figure, which, from the character of the terminal figure of the power, must be 5. Ods.—ln the above solution, there is an immense saving of labor, time, and space. The student cannot reahze the wide difference, if he has not gone through the labyrinths of the old methods, and the mazes of the old rule, in solving such problems. The ru?e and the explanation, under the old sys- tem would cover several pages of this work. By the new method here taught, we save the completion ot the second partial divisor, and the formation and completion of two other divisors, which it is the most desirable to obviate, because they become very large toward the end of the extraction, requiring much time and care in the work. But, for the encouragement of the student, it may be stated that few ^authors give numbers involving more than four periods; and it is also a rare thing in applied mathe- MATHEMATICAL ROOTS UPROOTED. matics to find questions involving the roots beyond the fourth or fifth decimal, which are exceedingly easy by the method here taught, but long and tedious by the old method. ADDITIONAL EXAMPLES FOR THE STUDENT. Find the cube root of these numbers, viz.: 1. 2176782336. 2. 87824421 125. 3. 43132764843. Solve, also, the following : 7. ^£TO5 8. 9- 132963364864. 225199.600704. 754863.574332608. f 17327 4H f2^% Note. — All perfect cubes of only two periods are to be solved at sight reduced to improper fractions, or to mixed decimals. Find the value of this express ion, viz.; f T6"6 -- f 6i- (4 X f Tsl^) = Solution: 162 -f 4 — (4 X .8) = 256 -r 4 — ( 3-2 ) = 64 — ( 3.2 ) = 60.8. Atu. Find the value of these, viz,: Mixed numbers should be f54.872 ^ 21.952 ^ 2326. 203125 ^64.964808 3- f2357947T^ff Answer to last : J^f . Let the student find it. Note. — If the terms of a fraction are not perfect powers of the indicated root, let them be reduced to such powers, where possible, before the extraction begins. The foregoing examples must suffice for illustration of the best method of extracting the root of exact cubes. The roots of higher powers will be discussed in connection with logarithms. CUBIC SURDS. We will now present a brief, easy, and practical method of treating imperfect powers of the third degree. In advance, we state that that method is, of course, one of approximation. Here it is impos- sible to choose the final figure, since there is no definite final root figure in surds. Let It be required to find the cube root, correct to four decimals, of the fraction %. = .500000000 I .7937+ ist divisor Finished divisor 2d divisor 147.81 189 1 667 1 1872.3 343 1570.00 150039 69610.00 This answer is true to the fourth place, inclusive, 56169 as verified by logarithms. We extract in the usual way until we obtain half the number of root figures 13441 desired. We then form the second partial divisor, 13 106 and with it obtain the other two root figures by a mode of contracted division, thus: Having found 335 the third figure of the root and taken its product with the divisor from the partial dividend, use, instead of attaching other periods, the same partial dividend, and drop one figure from the right of the divisor last found, and ascertain how often it will go into the remainder of dividend accruing from the previous division, reckoning in the number that would be to carry from the figure dropped. MATHEMATICAL ROOTS UPROOTED. 1st divisor Find the ^.27 correct to four places. 108.16 72, "16 I22&.» .270000000 I .6463-f- 216 540.00 2d divisor I228t* ' Observe that 16 is written in the niche of the other two parts of the divisor, to save space. Explanation. — 108 is the partial divisor. (How obtained?) This divisor gives the next root figure. 1 1 536 is the finished divisor. (Hov/ obtained ?) Ob- serve that the two added parts, 72 and 16, are each advanced one place to the right. 12288 is the second partial, or approximate, divisor. 1. How is it obtained ? 2. In the product of the root figure, 3, with the approximate divisor, 1228, account for the figure 6 in the result. Obs. — Outside of mathematical astronomy, where, in a few instances, great precision is required, it is scarcely necessary to approximate a root beyond four decimal figures. The ^.640000000 &c. = .8617+ 7^6q.oo 73728 4832 3686 1 146 1st divisor 192.36 144 20676 512 1280.00 124056 2d divisor 2218.8 Observe that 36 is written in the niche of the other two parts of the divisor, to save space. 20676 is the only finished divisor it is necessary to make. 1720 After finding the third root figure, we add no other periods to the partial dividend, but use the same dividend, and use, as a divisor, the previous one shorn of its final figure. If we wish to find other root figures, we drop other figures from the divisor, one by one, and continiie to divide as in common division. But it is well to bear in mind that if v/e wish to obtain accurately a definite number of decimals in the root, we must extract in the ordinary way, until we have obtained one-half of them, and then make a partial divisor from the root thus far found, and use that as an approximate divisor, to find the other root figures. But this is a very great saving, as it is the final periods that are to be dreaded in extraction. Do not fail to be familiar with the cubes i, 8, 27, 64, 125, 216, 343 , 512, 729. Otherwise you cannot select at sight the first root figure. ~- ' ~" -8735+ 1st divisor Thef^/^ = 192.49 168 divisor 20929 2270.7 .666,666,666, &c. 512 1546.66 146503 81636.66 68i2i 135 1 5 See that 49 is written in the niche again, i'nstead 1 1353 of being written thus: 192 168 which would occupy 2162 49. too much space. Obs. — In completing any divisor, we must advance each part one place to the right. The f .097672831790877 = .46053+ 1st divisor 48.36 I 64 »72 — I 336.72 5556 33336 2d divisor 63480.0 3368.31 7.90 3174000 I 943 I 7 190440 3877 This is accurate as far as extracted ; and, ye , there is a necessity to frame only a second approxi- mate divisor. Referring to the third root figure, we see that the divisor is not contained in the partial divi- 10 MATHEMATICAL ROOTS UPROOTED. dend shorn of the last two figures, and we put a nought in the quotient, and two noughts to the right of the divisor, because the squaring of the root thus far found, namely 460, would give two noughts in the resulting partial divisor, 634800. Attaching another period, we use this divisor to find the remain- ing root figures. Required the indicated roots of the following surds, accurate to four deci mals, vi z.; I. f.171467 3- ^2.42999 1^19-44 4- f^.571428 5- 6. 7- f 5 f 4>^ f 42f 8. 9- 10. f 22.4 f 3 1^41-502 f II 1^9^ f 9 f 7 f48x 4=^-182 ^'300484 ^129.009 ^.6748 f 6 .2482 These examples must suffice for illustration of the brevity secured by this mode of approximating the cube root of surds, by which at least three-fourths of the work necessary under other methods is saved ; while, in perfect cubes, nine-tenths of the usual work is obviated. HIGHER ROOTS. To "roots of all powers," so called, but a small space can be allotted in this brief work. Some authors, with what seems to be a strange love of novelty, rather than a desire for utility, have made quite an array of numbers requiring the 5th, 6th, 7th, 8th, 9th, lOth, I2th, 15th, l8th, 20th; and the 25th root, to be extracted. Now, it is needless to say that no such roots occur in nature, or in the course of applied mathematics. It is rare, indeed, in applied mathematics, that a number or quantity occurs requiring a root beyond the ihh-d or fourth. Then why should such numbers encumber that most practical of all the mathematical branches — arithmetic ? One of the many authors on arithmet- ical science, whose works are in extensive use in this country, requires the 20th root of 617, the 15th root of 15, and the 25th root of 100. Wherefore? we ask ; what the need ? When will the necessity for their use arise? Such novelties are incubuses on the science of numbers, and ought to be relegated to the closets of defunct mathematics. But, if mathematicians must put such impractical problems in their books and have them solved, let them be solved by shorter and better methods than those pre- sented in their works. That briefer and better method is by m.eans of logarithms. Especially is this true for roots whose indices are not factorable into the square and cube roots. Indeed, even in this case, the logarithmic method would be far preferable, and, if once adopted, would supersede all other methods for the higher roots. For, although the 8th root can be taken by three successive extractions of the square root, the 9th root by two successive extractions of the cube root, and the 6th root by the cube root of the square root, or the square root of the cube root, still these roots can be much more easily and quickly taken by logarithms. We simply take, from a table of logarithms, the log. of the number whose root is sought ; divide this log. by the index of the root, and find, in the same table, the number corresponding to l^^^uotient, and it will be the required root. Ex. — Find the cttb^oot of 1.577635 — . The log. of this number is .19800+ ; divide it by 9, the index, and the result is .02200-{-, and the number in the table corresponding to these figures is 1. 05 19, the required root. The same result is also easily obtained by the abbreviated method for cube root, thus: 1st root. 2d root. 1-577635= 1.1641+ I I.05-I-. Ans. Divisor 3.3.1 i i Divisor 363.36 iq8 Divisor 383^6^ 403. 6. J 5-77 331 2466.35 229896 16739 16147 3.00 1. 641. 00 This is the answer to the example in the 1500 , work from which it has been drawn ; but it will be seen that the logarithmic method 141 gives the answer more accurately. We have simply taken the cube root of the cube root by our method. 592 404 MATHEMATICAL ROOTS UPROOTED. Find the 7th root of 308. The log. of 308, page 6 of the logarithmic table, is 2.48855 ; divide by the index of the root, 2.48855 4- 7 = .35550-]- ; and the number corresponding to this logarithmic quo- tient is 2.26729 + , the 7th root of 308. All that is necessary in order to extract, by logarithms, atty root, is to look in a table of logarithms for the log. of the number to be extracted ; divide this by the index of the root, and find, in the same table, the number corresponding to the quotient, and it will l)e the root sought. This method is vastly shorter, and only requires a little knowledge of logarithms, and a little facility in their use, to enable one to evolve, with despatch, all roots. But such roots belong rather to higher mathematics. We would advise that, by all means, all roots above the third be taken by means of logarithms. It is but an hour's work to teach, to anyone that can multiply and divide, the use of the table. The after work is simply routine, and much valuable time is saved. As a matter of curiosity, we give below the 7th root of the same number, as presented by its author in one of the books of the day. That is, the 7th root of 308. OPERATION, f 338"= 2.59 + ■1/308=2.04-1- 2.59-1-2.04 = 4.63 4.63 -=- 2 = 2.31 -|- assumed root. 2.316 = 151.93 308 -M51. 93 = 2-0272 -h 2.31X6-1-2.0272=15.8872 15.8872 -f 7 = 2.2696 1st approximation. 2.2696® = 136.6748 308-^136.6748 = 2.253452-}- 2 2606 V 6 -^ 2 2^'i±<2 —It; 87io-;2 ^'^ ^^^'^ reached the second approximation, Con- 2.2090 X -h 2.253452 -15.671052 g^j.^^^ thought ! And these are only indicated results, 15.871052^7 = 2.267293 2d approx. none of the multiplications, divisions, etc., being car- ried out. Now, if this belongs anywhere, and there is doubt of its having a place in applied mathe- matics, it belongs to higher mathematics. Such skirmishing in figures is calculated to keep one hum- ble, by giving him a modest estimate of his attainments in evolution. EVOLUTION BY LOGARITHMS. Required the 25th root of 100. The log. of 100 = 2.000000. 2.000000 -f- 25 = .080000, and the number corresponding is 1.202266 + , which is the root sought. The solution of the same example, as given by a standard author, is as follows : Now, as the The y/ 100 =10 25th root must / 100 = y 1 =3.1 622 be less than the y 1 00 = / 3.i622^ 1.7782 24th root, let us |^'ioo= ^1.7782=1.2115 take 1.2 = the assumed root. 1.224 = 79.49684-]- 100 -^ 79,49684= 1.25792 -f 1.2 X 24 -f 1.25792 = 30.05792 30.05792 -T- 25 = 1. 2023168 1st approx. 1.2023x682* =83.2677184 100-^83.2677184= 1.2009492-f- 1.2023168 X 24-j- 1,2009492^=30.0565524 30.0565524 -^ 25 =1.202262-]- 2d approx. We breathe a sigh of relief. Of course, not much space can be devoted, ii^ this small work, to such solutions. It is only to show the difficulty of the subject under the old methods, in contrast with the brevity and facility of the new, that we allow a little space for some solutions under existing methods. Find, by the logarithmic method, the 6th root of 25632972850442049. The log. of this large num- ber is 16.408800. Dividing by 6, we get 2.734800. The number found in the table, corresponding to this quotient, is 543, the sixth root of the above number. The foregoing number, treated by the new metho d, gives, for the square root, 160103007, and the ^160103007 = 543 125 Explanation. — With the approximate divisor, 75, find the second figure of the root, 4, and choose the last. Why is it 3 ? Thus, the 75 351-03 work of the heretofore difficult cube root is almost annihilated. 12 MATHEMATICAL ROOTS UPROOTED. What is the 20th root of 617 ? The log, of 617 is 2.790285. Dividing by 20, we have .139514, and the number corresponding is 1,378841, the ans. Find the 5th root of 5, The log. of 5 is .69897. Divide by 5, and get .13979, and the number cor- responding is 1.37973, the 5th root of 5. The 5th root of 120 is: Log. 120 = 2^piS= .41583 -(- ; and the number corresponding is 2.605174:5 the root wanted. ^ ^ Let the student find, by logarithms (see explanation of use of the table, pp. -62 and^, etc.,) the roots of the following numbers, viz.: 1. The 8th root of 109951 1627776, 2. The I2th root of 16.3939. 3. The i8th root of 104.9617, 4, The 7th root of 1.95678. 5, The loth root of 743044. 6, The 3rd root of 4330747. ' Find, also, by logarithms, or by the abbreviated method, at your option, the cube root of the fol- lowing numbers, viz.: 7. 7023 1 089 1 84307 2. I 9. 10964743589696. 8. 744935304423023. I 10, 1881365963625. Required the solution of these examples : '^"^' 1. If a ball 10 inches in diameter weighs 125 lbs,, what is the diameter of a ball that weighs 216 lbs.? Solution.— f \2i^ : ^216 :: 10 : ans= ^f|| X lO == 12, 5:6:: 10: ans, Ans. 12 inches. 2. How many balls % inch in diameter will be required to make a ball i inch in diameter ? Ans. 64 balls. 3. Suppose the diameter of the earth to be 7912 miles, and that it takes 1404928 bodies of the size of the earth to make one as large as the sun, what is the diameter of the sun ? ^i3o|2lsx 7912= 112 X 7912 = 886144 miles. Ans. 4. A bin is 8 feet long, 4 feet wide, and 2 feet deep ; what is the linear edge of a cubical box that will hold the same quantity of grain ? t^S X 4 X 2 = f8x8 = 2X2 = 4 feet. Ans. Let the curt processes be used. Extract the factors of products in preference to taking the roots of the products. 5. If a stack of hay 24 feet high weighs 27 tons, what is the hight of a stack weighing 8 tons? if Ty X 24 = 2^ X 24= 16 feet. Ans. 6. If a bell 4 inches high, 3 inches in diameter, and "% of an inch thick, weighs i pound, what are the dimensions of a similar bell weighing 27 pounds ? Ans. 12 inches high, 9 inches diameter, and ^ of an inch thick. 7. If a loaf of sugar 10 inches high weighs 8 pounds, what is the hight of a similar loaf weighing I pound ? fyiY, 10 = j^ X 10= 5 inches. Ans. 8. There is a bin 32 ft. long, 16 ft. wide, and 8 ft. deep; what must the side of a cubic bin be that shall contain the same quantity ? ^32 X 16X8= 1^64X64 = 4X4= 16 ft. Ans. 9. What must be the side of a cubic bin that shall hold 350 bushels of grain ? SOLUTION. 2150,4X350 = 752640 I 90.96+ =7 ft. 6,96 in. Divisor 24300.81 729 Ans. 2430 236,400.00 22089429 Fin. div. 245438. 1550571 We use the finished divisor, shorn of the final fig- , . 1472629 ure, as an approximate divisor to obtain the fourth figure of the root. For practical purposes, the above 77942 is a close approximation. 10. If a sphere of gold i inch in diameter is worth $100, what is the diameter of a sphere that is worth $6400 ? iff^Q = ^64 X I = 4 inches. Ans. 11. The cubic metre is 61026.048 cubic inches ; what is the linear metre ? Ans. 39.37 inches. Find it by approximation. We give one more illustration, each, of the new method of taking the cube root of perfect and im- perfect third powers : MATHEMATICAL ROOTS UPROOTED. 1st divisor 2d divisor The ^146113369163 = 5267. Ans. 75 30-4 7804 125 211.13 15608 55053-69 There is no necessity of the last multiplication, 6 times 48672 the second divisor. Satisfy yourself that the remainder will be less than the divisor, and then choose the last root figure, thus 6381 saving a vast amount of work. Why is the final root figure 7? What is the value of 1.05I to 6 decimals? The log of 1.05 is .021189. Divide this by 3, and multiply the result by 5, and we get .035315 ; the number corresponding is i. 084715 + , the answer. To solve the above example in the old way, will re- quire about 30 minutes; by the logarithmic method, 2 or 3 minutes. The ^1.1810108914205625, by the approximate method, accurate to 6 decimals, is 1.0570234;. Find it. ^ ^^ , p ^ ^ ,-^^ T v^ A i V 5 ; .:, ; ]. OSI^O ^ :i , ^> / s^ % -■ ^ > ^ :"■ We have presented an unusually large number of solutions, in order that the cube root method herein set forth may be clearly apprehended by all. For, knowing its advantage in brevity and sim- plicity, and, consequently, its economy of time and space, we are thoroughly convinced that, if once adopted, it will be abandoned for no other. Let it be remembered, that if, in approximating the cube root of a number, it is desirable to extend the answer to a given number of decimal figures, one-half of all the root figures must be obtained by extraction in the usual way. The other half may be obtained by contracted division. For instance, in example 9 of the 12th page, 90.9 is obtained by extraction, and 625 is obtained by contracted division. If it be asked how we determine when a number is a per- fect cube, and when it is a cubic surd, we answer that, in actual business, in applied mathematics, this fact is always known when the problem occurs in the course of our work. Perfect cubes are in the minority in the course of mathematics. The small number of problems given under the head of evolu- tion by logarithms, will be sufficient to illustrate the subject. The student may work any, or all, of the others by logarithms, if he chooses. For this purpose, a table of logarithms is appended, calculated as accurately as possible to five decimal places. The table is extensive enough to enable us to find the roots of all numbers correct to five decimals. An explanation of the use of the table is also appended. Should anyone desire further aid in the matter of a knowledge and ready application of logarithms in the extraction of roots, the author will take pleasure in rendering all the assistance in his power. In conclusion, if any have their pet theories, methods, or processes, in cube root, to which they cleave with such a blind adherence that they cannot, or will not, see merit in anything else, this book is not made for them. If any are moved by prejudice,or jealousy, or envy at my good fortune, or by a spirit of criticism, or are unduly inflated with the importance of their own knowledge of the subject, through the belief that nothing new can be presented in evolution, the book is not written for any of these classes. All true science consists, not in the discovery of any new truth, but in the right application of ex- isting truth, so as to render it subservient, in the highest degree, to the interests and to the pleasure of mankind. If *' brevity is the soul of wit," it is no less the key to successful business. The large cur- tailment of the amount of work done in book-keeping in the last few years, is only in harmony with the spirit of the age, and, reinforces the sentiment of "short profits and quick sales." So must our methods and processes in education be constantly improved and refined, so as to be the most highly contributory to the important interests of business and of society. 14 MATHEMATICAL ROOTS UPROOTED. Simple Interest. Owing td the universal application and great practicality of this subject, we have thought proper' to give it a place in this work, and a treatment that, for brevity, utility, and simplicity, is in keeping with the constant drafts made upon it by all classes of men — those of inferior, as well as these of superior, attainments. The subject is what the name signifies, but is made rather complex by some authors and teachers, owing to the multiplicity of rules and tedious methods of treatment used by them. In simple interest, there is scarcely a necessity for more than one uniform rule, whatever be the rate or the time. Let us take some examples, by way of illustration : What is the interest of $450.87 for i yr. 7 m. and 9 da., at 6 per cent.? Operation. — 450.87 .0965 .005 225435 450.87 19.3 >&6. 270522 X — X — = 405783 I ^•i, I 9 days is .3 of a month, and the process Ans. 43.508,955 is simply one of cancellation. Find the interest of $125.16 for i yr. li m. 25 da., at 6 per cent.? Process. — 1.4298 10.43 10.43 '■^ 23.83+.06 42894 -X — X — == 57192 I -13. I 14298 One year and Ii months are 23 months, and An%. 14.91 25 d. are .83+ of a mo. So that the time is 23-834- mos., or I2ths of years, at the given rate per year. Let the cancellations, and all the multi- plications possible, be done mentally. Find the interest of $1500.60 for 2 yr. 4 mo., at 6^ per cent.? 125.05 1-75 125.05 >5aQ.6o 28 .06 >( 62525 X-X = 87535 I ^ta., I 12505 We take 6 times 28, plus the % of 28, Ans. $218.8375 mentally, which gives 1.75. Find the amount of $3050 for 4 yr. 8 m., at 51^ per cent.? i.245>^ 3050 14 .0175 3050 ^ ^e5^ 62250 X-X = 3735 I -« I -3- Ans. $3797.250 In making multiplications mentally, after the cancellations have been made, let the smallest numbers be so multiplied. Thus 14 times .0175, ^"^ then add i to the result, to get the amount of $1 for the lime, at the given rate. This result is then multiplied by the principal. MATHEMATICAL ROOTS UPROOTED. 15 Required the interest of $250 for i yr. 10 m. 15 da., 6 per cent. 125 .01 .225 --^^ 22.5 rD6- 125 I "T^ I II25 •'^ 2700 Ans. $28,125 If the amount had been required, we should have proceeded thus : .005 1. 1125 250 22.5 ^36 250 I -t^ I 556250 4ns, 22250 $278.1250 After the mental multiplication of the time and rate, add one to the result, be- fore the final multiplication by the prin- ciple. Find the interest of $51.10 for 10 m.'and 3 da., 4 per cent. .01 51.10 51.10 lo.i Tc^. .101 I r^ I 511 3 5" 3|5-i6ii Ans, 1.72 It may also be done thus : ===== 17-033+ -01 17.033 -3tTT& 10. 1 Aft^ .101 X X — = I ^s^ I 17033 ^ 17033 As before $1.720333 What is the interest of $175.40 for 15 m. 8 da., 10 per cent.? , 1.272+ 175.40 175.40 >5,.i66+ .10 .1272 X X — = I ^Ki I 3508 12278 21048 Ans. $22.3108 The multiplications are made mentally, except one. Required the amount of $1500 for 6 m. 24 da., JJ^ per cent. 7 -025 1.0425 =amt. of $1. 1500 1500 ~'&r5 >f5, I T-3 I :3 $1563.7500 Ans. Required the amount of 1.25 for I yr. 5 m. 10 da., 6}{ per cent. 8664 361 1.444+ 84.25 >>^+ .o6X_ I >2. I 1.09025 84.25 545125 218050 436100 872200 Ans. . $91.85 l6 MATHEMATICAL ROOTS UPROOTED. After first multiplication, add I to the result, before the formal multiplication. Find the interest of $112.50 for 3 m. I da. <)% per cent.? .2527+ 2274 112.50 112.50 ^TS954- .09^ 126 .02400 I 1^ I 4500 2250 Ans. $2.70 In multiplying the time and rate together, allow for the number of units that would be carried from 9 times 7. What is the interest of $408 for 20 da., 6 per cent.? .333 .01 408 408 ":64i44- 7o§-. .00333 1224 1224 1224 1. 35864=$!. 36, Ans. Always divide the days by 3, since every three days is tV of a month, thus reducing the days to decimals of a month. More than 5 mills should be called a cent. Less than 5 mills should be disregarded. Required the interest of $50 for i yr., 3 m., 27 da., 3 >^ per cent. 25 5-3 .05 1-25 -5€- -f^rf- .-K-^ 5.3 I -t5- -3- 375 -6- 625 ^ ^ 3 16^6^ Ans. $2.2o8>^ Find the interest of $384.50 for 2 yr., 8 m., 4 da., 8 per cent. 10.71 .02 384.50 384.50 -:5^~iad- -"^^ -2142 I -«- I 7690 3- 15380 3845 7690 5$82.35990=$82.36. Ans. Solve this by other methods and see if you get a different answer. Required the amount of $275 in 4 m., 25 da., 7 per cent. .4027+ 1. 02818 275 >>S5i+- -07 275 — X X — = I -tt- I 514090 719726 205636 $282.7495o=-$282.75. Ans. ^^ ^ ^ , .v 1 * After multiplying .4027 by .07, we add i to the result, for the amount of $1, before we make the last multiplication. Multiply .402 by .07, but carry from 7 times 7. Interest of $318.29 for 9 m., 10 da., "]}( per cent.? •777+ 5443 318.29 318.29 -9:395;^ .07X 194 .05637 X X = — I -in- I 222803 95487 190974 159145 Ans. 17.94.20 MATHEMATICAL ROOTS UPROOTED. 17 Required the interest of $4684.68 for 11 da., 12;^ per cent. 390-39 .04582 390.39 .183+ ,+§S:;:cS. -rS^-l- .25 — -— X X = 78078 I -K- -2- 3I23I2 I95I95 I56I56 $17.8876 = $17.89. Arts. In making the mental multiplication by 25, allow for the number that would be to carry, had the decimal .183+ been extended one figure further. Find the interest of $127.36 for I yr. 6 m. 21 da., 4^ per cent. 5-306+ 5.306+ . 1.683^ -is?.^^- 18.7 .09 15918 X X = 42448 I -r^ -2- 31836 5306 Ans. $8.93. $8.929998 Find the amount of $723.60 for 2 yr. 3 m. 18 da., 5^ per cent, 2-3 723.60 ^Jfr^. .23 .529 X — X— = — = .13225 I -f~ 4 4 Add I 1. 13225 723.6 679350 339675 22645b 792575 After cancelling and multiplying the expres- sions of time and rate, we add i to the product, $819,296 Ans. to get the amt. of $1. This saves time and one operation in the work. Now, if the work is short with these peculiar and mixed rates, it is much mo-e so with all ordinary rates. We will take only two or three illustrations : Find the interest of $780.26 for 90 da., without grace, at 1%. per cent, per month. Operation, — 195.065 195.065 .15 ;So.x6 — 3^ .15 — X— X— == 975325 I -rt- I 195065 _4_ Ans. $29.26975 What is the interest of $845 for i yn 10 m. 6 da., at I per cent a month? 845 .01 .222 845 2^.2 7t±- X X = 1690 I -1-3- I 1690 1690 Multiplying by .01 simply throws the point Ans. $187,590 two placesfurther to the left on the multiplicand. Find the interest of 1040 for i yr. 9 m. 9 da. at 8 per cent. .142 7.1 .02 1040 1040 -2*t5 "naS^ X X- 5680 I nr I 142 -3- Ans. $147,680 This method is equally expeditious in reck- oning up notes whereon partial payments have been made. Indeed, there is no department of interest i8 MATHEMATICAL ROOTS UPROOTED. where it may not be used with greater facility, and with much less work, than any other process. We have chosen to call it the Cancellation Method. The advantage lies in its simplicity, brevity, and uni- formity of treatment, there being but one process for all problems, whatever the rate, time or other conditions. And surely this, of itself, is a great saving, to both teacher and pupil, of much labor and taxing of memory, under the numerous methods and rules of interest laid down in the books. The plan here presented is strictly mathematical, depending on the principle, that the annual rate on a dol- lar, multiplied by the number of dollars in the principal, is equal to the interest. The time is reduced to months and decimals of a month; and, then, the expression for months is divided by 12, thus ex- pressing the time in years. Each example given in the foregoing pages, is simply a grouping of the sum at interest, the years, and the rate. Cancellation naturally follows. We might have reduced the time to days, dividing the number ot days by 360, thus making it express years. For instance, 2 yr. 4 m. 20 da. is 720 da. -f- 120 da. -|-20 da. = 860 da., or ||g years. But this is considerably longer, requir- ing more work every way. The briefer the method of reckoning interest, the less liability to mistakes. The one herein set forth,' takes the happy mean in all particulars. EXAMPLES FOR THE STUDENT. 2. 3- 4- 5- 6. 7- 8. 9- 10. Let it be required to find the interest on the following, viz.: $300 for 2 yr. 7 m. 24 da., at 6 per cent. $700 for I yr. 9 m. 12 da., at 6 per cent. $400 for 2 yr. 6 m. 15 da., at 6 per cent. $350 for 3 yr. 8 m. 24 da., at 6 per cent. $450.87 for I yr. 7 m. 9 da., at 7 per cent. $375.50 for 2 yr. i m. 8 da., at 7 per cent. $125.16 for I yr. ii m, 25 da., at 7 per cent. $658.25 for I yr. 2 m. 13 da., at 7 per cent. 1 $187.44 for I yr. 10 m. 24 da., at 7^,7 per cent. [-Find the amount. $444.84 for I yr. i m. 16 da., at 5 per cent. j Also, reckon up the following promissory notes, on which indorsements have been made, viz.: $167.42. Selma, Apr. 15, 1882. 1. For value received, I promise to pay Judge Fowler, or order, one hundred sixty-seven and -^ dollars, in 6 months from date with interest. Tom Scroggins. Indorsements : May 21, 1883, $42.18; July 17, 1884, $6.25; Sept. 9, 1884, $48.16; Jan. 27, 1885, $27.47. What was due Apr. 15, 1886? ^ns. $72,277. $472.76. Selma, June 4, 1884. 2. For value received of Arrents & Longacre, I promise to pay them, or their order, four hundred seventy-two and ^^% dollars, in 6 months from date, with interest at 7 per cent, afterward. Jno. Grubs. Indorsements : Apr. 10, 1885, $125,843 ; Nov. 28, 1885, $133,724; Apr. 15, 1886, $223,081. What will due Nov. 13, 1886? ylns. $24.95. Let these be done strictly by the abbreviated process — it saves half the usual work. These exam- ples must suffice for our book. The student will find abundant material for practice from other sources. We will present the solution of the last promisory note, to show the plan by the cancellation process. W^e first write the dates in succession, thus ; 1886, II, 13 =6 m. 28 da. 1886, 4, 15 = 4 m. 17 da. 1885, II, 28 = 7 m. 18 da. 4, 10 = 4 m. 6 da. JV. B. — The four cancellations and multiplications" following, present 12, 4= the entire work, and not simply the indicated vioxY'. 1885, 472.76 1.0245 236380 189104 94552 47276 $484,342 Payment — 125.843 $358-499 .63+ 358.50 .i^ .07 X X 358-50 1.0443 10755 14340 14340 3585 $374,381 Payment— 133.724 $240,657 MATHEMATICAL ROOTS UPROOTED. 19 1.026635 240.657 •3805 240.657 4.566 U X X— I -^-^- I 07 7186445 5133175 6159810 4106540 2053270 $247.06689 Payment— ^223 .08 1 $23.99 •574- 23-99 -^^^^f -07 I -irt- I •57 X. 07+ I = 1.0399 $24,947 = $24.95. Ans. ramt. of $1. In partial payments, most authors, in illustrating their methods, give simply a brief of results, as if they would make the work appear short. After cancellation and multiplication of the expressions lor time and rate, let i be added to the result, for the amt. of $1, before the mechanical multiplication is made. See above. Observe the manner of writing the dates, all at once, and all in one group, from the latest to the earliest, and then the consecutive subtraction of them, thus giving the several periods of time at once, before the cancellation processes are commenced. This is a great saving of time, and promotes simplicity. And here it may be stated that, with respect to the subject of interest, the author does not so much claim to have discovered new truth, as he does a new and right application of it. Much of scientific truth is as good as lost, through the circuitous, obscure processes under which it is presented. Required the interest on the following, viz.: 1. $1284.60 for 5 m. 12 da., at ^ per cent, a month. 2. $621.09 for 7 m. 16 da., at ^ per cent, a month. 3. $818.26 for 9 m. 3 da., at \ per cent, a month. 4. $220.38 for 2 m. 21 da., at lOj^ per cent, per annum. 5. $62.96 for I yr. 8 m. 23 da., at 11 per cent per annum. 62.96 1-73, .1903 62.96 —20^76-+- -X X = 1888 56664 6296 $11.98. Ans. to last. $614.42. For value received, I promise to pay D on demand, with interest at 6^ per cent. Indorsements : May 15, 1886, $169.30; June 10, 1886, $88.40; Sept. I on this note Nov. 20, 1886? Selma, Cal., May i, 1886. Wagner, or order, six hundred, fourteen and -^ dollars, John Davis. 1886, $325.80. How much will be due Write the dates in a group, as above ; begin with the date of giving the note, and subtract each from the next succeeding, as 5 m. i da. from 5 m. 15 da., always making the subtractions mentally, and writing the payments opposite the intervening times. Find the amount of the original principal for the first term of time, and of each succeeding principal for its term of time, subtracting each payment, in order, from the corresponding amount, till you come to the maturity of the note. Then find the amount of the last principal for the corresponding term of time, and it will be the balance. Solution of the last example : 614.42 1.00262 amt. of $1. 1886, II, 20 = 2 m. 2 da. PAYMENTS 1886, 9, 18 = 3 m. 8 da. 1886, 6, 10 = m. 25 da. $325.80. IS86, S, 15 = om. 14 da. 88.40. 1886, 5, I 169.30. .0291 -.1165- .09 614.42 ~:^f6B- .2f- X X — I -12- -4- 122884 368652 122884 61442 $616.03 169.30 $446.73 2d prin. 20 MATHEMATICAL ROOTS UPROOTED, 1.00468 amt. of $1. 44673 20S .02^ 357384 446.73 .-853. -r^T^ 268038 X X 178692 I -«- S^ 44673 $448.82 88.40 $360.42 3d prin. 1.01836 amt. of $1. 360.42 .204 .09 203672 360.42 - 3-^6^ : >^ 407344 X X — = 611016 I -13— _4_ 305508 After multiplying .204 $367,037 325.80 by .09, prefix I to the result. $41.24 4tli prin. I.01162 amt. of $1. 41.24 404648 202324 101 162 404648 $41,719. Ans. $41.72 Ba/. The above is the entire work. It is comparatively short, and is quickly done. Let the student use the stereotyped methods in use, and he will at once see a vast difference in favor of this curt cancella- tion process, uniform for a/l rates, times, and conditions, and equally easy for all questions in interest. For these and other reasons, this method, once adopted, will be used in preference to any other. In reckoning up the balance due on promissory notes whereon partial payments have been made, let the cancellations be so managed that the uncancelled factors may, as far as possible, be multiplied iogQihev mentally ; or, at least, maybe reduced to one formal multiplication. The advantage of the cancellation process may be seen in the following problem : Find the amount of $235.18 for 2 yr. 8 m. and 12 da., at 5>^ per cent. $235.18 .9 I. 144 •w>8. 235.18 :yM. -16 94072 X X = 94072 I Hri- -5-. 258698 $269.04592 = $269,046. Ans. Observe that, after multiplying together the expressions of time and rate, .g and .16, we add I to the result, which makes the amt. of $1. This multiplied into the principal gives the amt. of the debt. Hence, in the cancellations, it is not proper to cut down the principal, when the amount is to be found. If only the interest is required, factors may be stricken from any of the three parts. Those who have not tried this method cannot realize how easily these factors (principal, time in years, and rate) can be thrown together, and cut down to the answer, with a very small amount of fig- uring, whatever be the nature of the parts. The years and months are reduced to months, simply by inspection, without a mental effort ; the days are reduced to the decimal of a month, by dividing them by 3 ; and under this mixed expression we place 12, thus expressing the whole time in years. We have little patience with sticklers for analysis in everything. It is very essential in some depart- ments of arithmetical science, but is utterly useless in many of its most practical subjects. As an inci- dental feature, and as conducing to thoroughness in the fundamental principles of a mathematical edu- cation, analysis, in a few subjects in arithmetic, should be thoroughly taught to the young, but beyond this it is useless. In interest it is of no value. The banker, the lawyer, the real estete man, and all other practical people, have no use for analysis in any of the several departments of interest, but must have the most direct, curt processes for all problems arising under this broad department in the science MATHEMATICAL ROOTS UPROOTED. 21 of numbers. Teachers should not lose sight of the fact that, in our practical civilization, we are, in many things, reducing more and more to practicality. Hence, as business increases and ramifies into various new departments, thus multiplying our cares and duties, we must seek to do our work with the utmost despatch consistent with accuracy. The cancellation process in interest meets the demands of business, on the subject to which it belongs. It supersedes the necessity and the utility of any 6 per cent, method, or 12 per cent, method, or i per cent, method, or 10 percent method, or any other specific method. It secures uniformity, simplicity, brevity, accuracy, FINIS. 22 LOGARITHMS. Logarithms. A Logarithm is simply an exponent of a power. The logarithm of a number is the exponent of the power to which the base of the number must be raised, to produce the number. Thus, in the following equations : 5" = 1 and lo<* = I 51 = 5 loi = 10 52 = 25 lo* = 100 53 = 125 10* = loog, O, I, 2, 3 are the logarithms of the respective numbers to which they stand opposite in the several equa- tions. 5 is the base in the one set, and ten in the other. In any one of the above quotations, the value of the second member depends on the numerical value of the base and the exponent attached. In a sys- tem of logarithms, any number above I may be taken as a base, and, by suitably varying the exponent, the base being unaltered, all possible numbers may be represented. For example, lo^ -82000 represents the number 6607, and 3.82000 is the log. of this number. io3'7'*225 represents the number 5524, and 3.74225 is the log. of this niimber. It means that 10 must be raised to the power denoted by 3.74225 to produce 5524. In all practical mathematics, 10 is the base. The system is called the Common, or Brigg's system, and, in it, all numbers, integral or fractional, are regarded as some power of 10. 10° is no power of 10, and is equal to 10 divided by 10, or to I. That is, the log. of i is o. All numbers be- tween I and 10 have, for their logarithm, a decimal fraction ; all numbers between 10 and 100 have, for their logarithm, i -[-a decimal ; and all numbers between 100 and 1000 have, for their logarithm, 2 -(- a decimal ; and so on. See, page I of the table of logarithms, in column headed N, that numbers be- tween I and 10, 10 and loo, and 100 and 1000, respectively, fulfill the above conditions. The log. of 7, for example, is .84510, and that of 25 is 1-39794, and these logs, are simply exponents. io''-845io __ 7^ and ioi-^9794 =25, signifying that 10 must be raised to these powers, respectively, to produce 7 and 25. To find the logarithms of numbers over 100, and under looo. — Look opposite the number, in column headed O, and find the logarithm. The log. of 398, page 7 of the table, is 2.59988. .-To find the logarithms of numbers of four figures. — Look under caption iV'for the first three 'fig- ures of the number, and at the top of the page for the fourth figure ; and opposite the one part of the number and under the other, find the logarithm. Thus, the log. of 6982 is 3.84398. The decimal part of any logarithm is called the mantissa, and the integral part, the characteristic. In the log. of 1840, which is 3.26482, 3 is the characteristic, and .26482 is the mantissa. The characteristic of all numbers between I and 10 is o. To find the logarithms of numbers of more than four figures. — Find, for example, the logarithm of 248963. On page 4 we find, as previously directed, the «/rt«/wa*corresponding to the first four figures, 2489, to be .39602 ; and, to this partial mantissa, there must be an addition for the remaining part of the number, 63. And since this addition affects only the decimal part, or mantissa, and not the char- acteristic, 63, the remaining part of the number must be regarded as a decimal. This decimal, .63, we multiply by the tabular difference, opposite the mantissa, in column D, which is 17+, or 17.5, and get .63 X 17.5 = 11.025, giving II to be added to the final figures of the partial mantissa, .39602, already taken out, making .39613 ; and the characteristic is 5, being always one less than the number of figures in the integral part of the number whose logarithm is sought. Thus, the log. 248963 = 5.39613. Required the logarithm of 142967542. The mantissa of the first four figures, 1429, page 2 of the table, is .15503, and the tabular difference is 30+, or 30.5. This multiplied into .67542, the remainder of the number treated as a decimal, gives 20.6, or 21, to be added to the terminal figures of the partial mantissa already taken out, making .15524, and the characteristic is 8. Thus, the log. 142967542 = 8.15524. In making additions to the mantissa, more than 5 decimal units should be reckoned i , less than 5 should be disregarded. For the same figures, in the same order, the mantissa is the same, whatever the local value or the fig- ures. Thus, the mantissa of the logs, of these numbers, viz.: 8328, 832.8, 83.28, 8.328, .8328, .08328. ,008328, etc., is .92054, the same for each of the numbers. The characteristic of the first is 3 ; of the second, 2 ; of the third, i ; of the fourth, o; of the fifth, — i ; of the sixth, —2 ; of the seventh, — 3. The LOGARITHMS. 23 characteristic of a decimal is always negative, and numerically one more than the number_of noughts prefixed to the decimal. The negative sign is usually vsrritten over the characteristic, thus 2, 6 9 8 9 7, in the log. of .05. Find the logarithm of .6423. On page 12 we find the mantissa to be .80774, and the characteristic is T, making T. 8 7 7 4, for the log. of .6423. To find the number corresponding to a given logarithm. — What is the number whose log. Is 1. 681^4? Looking on page I, we find this log. opposite 48. Hence, 48 is the number whose log. is I. 68 I 24. Find the number having for its logarithm 2 , 3 6 3 5_. Looking on page 4, we find opposite 230 and under 7, the number 230.7, the answer required^. Find the number having for its logarithm 2,64367. Looking for the nearest mantissa to the given one, we find it, page 8, opposite 440 and under 2, to be .64365. This mantissa we subtract from the given one, and divide the difference, 2, by the tabular diff^ence, 10, and get .2. Appending this to the 4402, already taken out, we get 44022. And now, as the characteristic is — 2, we prefix one o to the last result, and get .044022 for the required number. What is the number having for its logarithm .29824? The nearest mantissa is .29820, page 3, op- posite 198 and under 7 ; ,29824— ,29820 = 4; 4-4-22, the tab. difif., gives 1.8, or 2 nearly. Appending 2 to 1987, we have 19872 — . And, since there is no characteristic, the integral part is I. Hence, .29824 = log. I. 19872. ; Find the cube root of .4986. The log. of .4986 (p. 9) is T.6 9 775, This we divide by 3, the in- dex of the required root. But since the characteristic is negative, while the mantissa is always positive, we cannot directly divide the logarithm by the index 3. But T,6 9 7 7 S == ^-f 2^.69775, in which the characteristic is exactly divisible by the index 3. Dividing, we get T,?99 2 5, We now find the num- ber corresponding to this logarithm. The nearest mantissa is .89922.' Subtracting it from the given .89925, we get 3, which, divided by the corresponding tabular diffe^nce, §-f ' gives 5. The number corresponding to the mantissa .89925 is 79295. To this prefix^'wic. o, to correspond to the negative characteristic, and the cube root of .4986 is .-&7929§^ ^^ ,^ y- ^7 <^J^ When the characteristic is negative, and not divisible by me4nclex^of any root, add to it the smallest nega- tive number that will render it divisible, and then prefix the same number, with a plus sign, to the mantissa. What is the 5th root of 512.8? The log. of 512.8 = 2.70995 ; 2,70995 -f- 5 "= -54199 ; the nearest mantissa, opposite 348 and 3, is. 54195; .54199 — .54195=4; and 4 -r 12^-, the tab, diff,, gives 3 to to be apended to 3.483, making 3.4833, for the 5th root of 512.8. k 5,if2^«02. T THE COMMOH OR BRIGGS LOGARITHMS —OF THE— . iT-A.rrxTie-A.ij 3sr"U-3iv^BEies FROM 1 TO 10000. I— 100. H loS If log ir log N log N log I 0. 00 000 21 I 32 222 41 I. 61 278 61 I 78533 81 I. 90 849 2 0. 30 103 22 I 34 242 42 62325 62 I 79 239 82 I. 91 381 3 0. 47 712 23 I 36 173 43 63347 63 I 79 934 83 I. 91 908 4 0. 60 206 ■ 24 I 38021 44 64345 64 I 80618 84 I. 92 428 5 0. 69 897 25 I 39 794 45 65 321 65 I 81 291 85 I. 92 942 6 0. 77 815 26 I 41 497 46 66 276 66 I 81 954 86 I. 93 450 7 0. 84 510 27 I 43 136 47 67 210 67 I 82 607 87 I- 93 952 8 0. 90 309 28 I 44 716 48 68 124 68 I ^3 251 88 I. 94 448 9 0. 95 424 29 I 46 240 49 69 020 69 I S3 885 89 I- 94 939 lO I. 00 000 30 I 47 712 50 69897 70 I 84 510 90 I. 95 424 II I. 04 139 31 I 49 136 51 70757 71 I 05 126 91 I. 95 904 12 I. 07 918 32 I 50515 52 71 600 72 I 85 733 92 I. 96 379 13 I. II 394 33 I 51 851 53 72428 73 I 86 332 93 I. 96 848 14 I. 14 613 34 I 53 148 54 73 239 74 I 86923 94 I. 97313 15 I. 17 609 35 I 54407 55 74036 75 I 87506 95 I. 97 772 16 I. 20 412 36 I 55630 56 74819 76 I 88 081 96 I. 98 227 17 I. 23 045 37 I 56 820 57 75587 77 I 88 649 97 I. 98 677 18 I. 25 527 38 I 57978 58 76 343 78 I 89 209 98 I. 99 123 19 I. 27 875 39 I 59 106 59 77085 79 I 89763 99 I. 99 564 20 I. 30 103 40 I 60 206 60 I- 77815 80 I 90309 100 2. 00 000 N log N log H log H log V log N 1 % 3 4 5 6 7 8 9 1) 100 00 000 00043 po 087 00130 00 173 00 217 00 260 00 303 00 346 00 389 43 lOI 00432 00475 00 518 00 561 00 604 00 647 00 689 00 732 00 775 00 817 43 I02 00860 00903 00945 00988 01 030 01 072 01 115 01 isf 61 199 01 242 42 103 01 284 01 326 01 368 01 410 01 452 01 494 01536 01 578 01 620 01 662 42 104 01 703 01 745 01787 01 828 01 870 01 912 01953 01 995 02 036 02 078 42 105 02 119 02 160 02 202 02 243 02 284 02325 02 366 02 407 02 449 02 490 41 106 02531 02572 02 612 02653 02 694 02735 02776 02 816 02 857 02 898 41 107 02938 02979 03019 03 060 03 100 03 141 03 181 03 222 03 262 03302 40 108 03342 03383 03423 03 463 03503 03543 03583 03 623 03 663 03703 40 109 03743 03782 03 822 03 862 03902 03941 03 981 04021 04060 04 100 40 110 04139 04179 04 218 04258 04297 04336 04376 04415 04454 04493 39 III 04532 04571 04 610 04 650 04 689 04727 04766 04 805 04 844 04 883 39 112 04 922 04961 04999 05 038 05 077 05 115 05 154 05 192 05 231 05 269 39 113 05 308 05 346 05 385 05423 05 461 05500 05538 05 576 05 614 05652 38 114 05 690 05729 05767 05805 05 843 05 881 05 918 05 956 05 994 06 032 38 115 06 070 06 108 06 145 06 183 06 221 06 258 06 296 06333 06371 06408 38 116 06 446 06 483 06 521 06558 06595 06 633 06 670 06 707 06 744 06 781 37 117 06819 06856 06 893 06 930 06 967 07004 07041 07 078 07 ii5 07 151 37 118 07 188 07 225 07 262 07 298 07335 07372 07 408 07 445 07 482 07518 37 119 07555 07 591 07 628 07 664 07 700 07 737 07773 07 809 07 846 07882 36 120 07 918 07 954 07 990 08 027 08063 08 099 08135 08 171 08 207 08 243 36 121 08 279 08 314 08350 08386 08 422 08458 08493 08 529 08 56S 08 600 36 122 08 636 08 672 08 707 08743 08778 08814 08849 08 884 08 920 08955 35+ 123 08 991 09 026 09 061 09 096 09132 09 167 09 202 09237 09 272 09307 35 124 09342 09377 09 412 09447 09 482 09517 09552 09 587 09 621 09 656 35 125 09 691 09 726 09 760 09795 09830 09 864 09899 09 934 09 968 10003 35 126 10037 10 072 10 106 10 140 10 175 10 209 10243 10 278 10 312 10 346 34 127 128 10 380 10 721 10 415 10755 10449 10 789 10483 10517 10857 10551 10 890 10 585 10924 i4o 619 10 653 10958 10992 10687 II 025 34 34 10823 129 II 059 1 1 093 II 126 II 160 II 193 II 227 11 261 II 294 II 327 II 361, 33+ 130 II 394 II 423 II 461 1 1 494 II 528 II 561 II 594 II 628 II 661 11 694 33 131 11727 11 760 II 793 II 826 II 860 II 893 II 926 II 959 II 992 12 024 33 132 12057 12 090 12 123 12 156 12 189 12 222 12 254 12 287 12 320 12352 33 ^33 12385 12 418 12450 12483 12516 12548 12 581 12 613 12 646 12 678 32+ 134 12 710 12743 12775 12808 12 840 12872 12905 12937 12969 13 001 32 135 13033 13066 13098 13 130 13 162 13 194 13226 13 258' 13 290 13322 32 ^3^ 13354 13386 13 418 13450 13 481 13 513 13545 13577 13609 13 640 32 137 13672 13 704 13735 13767 13799 13830 13862 13893 13925 1395^ 31+ ^38 13988 14 019 14 051 14082 14114 14 145 14 176 14 208 14 239 14270 31 139 14301 14333 14364 14395 14426 14457 14489 14520 14 551 14582 31 uo 14613 14644 14675 14706 14737 14768 14799 14829 14860 14 891 31 141 14922 14953 14983 15 014 15 045 15076 15 106 15 137 15 168 15 198 31 142 15229 15259 15 290 15320 15 351 15 381 15 412 15 442 15 473 15503 30+ 1 143 15534 15564 15594 15625 15655 15685 15 715 15 746 15 776 15 806 30 144 15836 15866 15897 15927 15957 15987 16 017 16047 16077 16 107 30 145 16 137 16 167 16 197 16227 16 256 16286 16316 16346 16376 16 406 30 146 16435 16465 16495 16524 16554 16584 16613 16 643 16 673 x6 702 30 147 16732 16 761 16 791 16820 16850 16 879 16 909 16 938 16 967 16997 29+ 148 17 026 17056 17085 17 114 17 143 17 173 17 202 17 231 17 260 17289 29 149 17319 17348 17377 17406 17435 17464 17493 17522 17551 17580 29 150 17609 17638 17667 17696 17725 17754 17782 17 811 17 840 17 869 29 N 1 2 3 4 5 6 7 8 9 N 12 3 4 5 6 7 8 9 D 150 17609 17638 17667 17696 17.725 17754 17782 17811 17840 17869 29 151 17898 17926 17955 17984 18 013 X804X 18070 18 099 18 127 18x56 29 152 18 184 18213 18241 18270 18298 X8327 18355 18384 184x2 18 44X 28+ 153 18469 18498 18526 18554 18583 18 6x1 18639 18667 18696 18724 28 154 18752 18780 18808 18837 18865 18893 18 921 18949 18977 19005 28 155 19033 19 061 19089 19 117 19 145 19173 19 201 19229 19257 19285 28 156 19 312 19340 19368 19396 19424 19 451 19479 19507 19535 19562 27 157 19590 19 618 19645 19673 19700 19728 19756 19783 X981X X9 838 28 158 19866 19893 19 921 19948 19976 20003 20030 20058 20085 20x12 27 159 20 140 20 167 20 194 20 222 20 249 20276 20303 20330 20358 20385 27 160 20412 20439 20466 20493 20520 20548 20575 zo 602 20629 20656 27 161 20683 20710 20737 20763 20790 208x7 20844 20871 20898 20925 27 162 20952 20978 21005 21032 21059 21 085 2X XX2 21 139 21 X65 21 I92 27 ^63 2x2x9 2x245 21272 2x299 21325 21352 2x378 2x405 21431 21458 27 164 21484 21 511 21537 21564 21590 21 6x7 21 643 21 669 21 696 21 722 26 165 21 748 21 775 21 801 21 827 2X 854 21 880 21 906 21 932 21 958 2X 985 26 166 220x1 22037 22063 22089 22 115 22 141 22 167 22 194 22 220 22 246 26 167 22 272 22 298 22 324 22 350 22 376 22401 22427 22453 22479 22505 26 168 22531 22557 22583 22608 22634 22 660 22 686 22 712 22 737 22 763 26 169 22 789 22 814 22 840 22 866 22 89I 229x7 22943 22968 22994 23019 26 170 23045 23070 23096 23 121 23147 23 172 23 198 23 223 23 249 23 274 25 171 23300 23325 23350 23376 234OX 23426 23452 23477 23502 23528 25 172 23553 23578 23603 23629 23654 23679 23704 23729 23754 23779 25 173 23 8o5 23 830 23 855 23 830 23 905 23930 23955 23980 24005 24030 25 174 24055 24080 24105 2413,0 24x55 24180 24204 24229 24254 24279 25 175 24304 24329 24353 24378 24403 24428 24452 24477 24502 24527 25 176 24551 24576 2460X 24625 24650 24674 24699 24724 24748 24773 25 177 24797 24822 24846 24871 24895 24920 24944 24969 24993 25018 24+ 178 25 042 25 066 25 09X 25 115 25 139 25 164 25 188 25 2X2 25 237 25 261 24 179 25285 25310 25334 25358 25382 25406 25431 25455 25479 25503 24 180 25527 25551 25575 25600 25624 25 648 25 672 25 696 25 720 25 744 24 181 25 768 25 792 25 8x6 25 840 25 864 25888 25912 25935 25959 25983 24 182 26007 26031 26055 26079 26x02 26 126 26 150 26 174 26 198 26 221 24 183 26245 26269 26293 26316 26340 26364 26387 26411 26435 26458 24 184 26482 26505 26529 26553 26576 26 600 26 623 26 647 26 670 26 694 23+ 185 26717 26741 26764 26788 26811 26834 26858 26881 26905 26928 23 186 26951 26975 26998 27021 27045 27 068 27 091 27 114 27 138 27 x6i 23 187 27 184 27 207 27 231 27 254 27 277 27300 27323 27346 27370 27393 23 188 27416 27439 27462 27485 27508 27531 27554 27577 27600 27623 23 189 27646 27669 27692 27715 27738 27761 27784 27807 27830 27852 23 190 27875 27898 27 921 27944 27967 27989 28012 28035 28058 28081 23 191 28 103 28 126 28 149 28 171 28 194 282x7 28240 28262 28285 28307 23 192 28330 28353 28375 28398 28421 28443 28466 28488 28 511 28533 23 193 28556 28578 28601 28623 28646 286-68 28691 28713 28735 28758 22+ 194 28 780 28 803 28 825 28 847 28 870 28892 28914 28937 28959 28981 22 195 29 003 29 026 29 048 29 070 29 092 29 115 29137 29x59 29181 29203 22 196 29226 29248 29270 29292 29314 29336 29358 29380 29403 29425 22 197 29447 29469 29491 29513 29535 29557 29579 2960X 29623 29645 22 198 29 667 29 688 29 7x0 29 732 29 754 29776 29798 29 S20 29842 29863 22 199 29885 29907 29929 29951 29973 29994 300x6 30038 30060 30081 22 200 30 103 30 12S 30 146 30 168 30 190 30 211 30233 30255 30276 30298 22 N 12 3 4 5 6 7 8 9' N 12 3 4 5 6 7 8 9 1) 200 30 T03 30 125 30 146 30 168 30 190 30 211 30233 30255 30276 30298 22 20I 30320 30341 30^63 30384 30406 30428 30449 30471 30492 30514 22 202 30535 30557 30578 30600 30621 30643 30664 30685 30707 30728 21+ 203 30750 30771 30792 30814 30835 30856 30878 30899 30920 30942 21 204 30 963 30 984 31 006 31 027 31 048 31 069 31 091 31 112 31 133 31 154 21 205 31 175 31 197 31 218 31 239 31 260 31 281 31 302 31 323 31 345 31 3^^ 21 206 31 387 31 408 31 429 31 450 31 471 31492 31 513 31534 31555 31576 21 207 31 597 31 618 31 639 31 660 31 681 31 702 31 723 31 744 31 765 31 785 21 208 31 806 31 827 31 848 31 869 31 890 31 911 31931 31952 31973 31994 21 209 32015 32035 32056 32077 32098 32 1X8 32 139 32 160 32 181 32 20I 21 210 32 222 32 243 32 263 32 284 32 305 32325 32346 32366 32387 32408 21 211 32 428 32 449 32 469 32 490 32 510 32531 32552 32572 32593 32613 20+ 212 32 634 32 654 32 675 32 695 32 715 32736 32 756 32777 32797 32818 20 213 32 838 32 858 32 879 32 899 32 919 32 940 32 960 32 980 33 001 33 021 20 214 33 041 33 062 33 082 33 102 33 122 33 143 33 ^^3 33 ^^3 33 203 33 224 20 215 33 244 33 264 33 284 33 304 33 325 33 345 33 365 33 385 33 405 33 425 20 216 33 445 33 465 33 486 33 506 33 526 33 546 33 566 33 586 33 606 33 626 20 217 33 646 33 666 33 686 33 706 33 726 33 746 33 766 33 786 33 806 33 826 20 218 33 846 33 866 33 885 33 905 33 925 33 945 33 9^5 33 985 34 oo5 34 025 20 219 34044 34064 34084 34104 34124 34143 34163 34183 34203 34223 20 220 34242 34262 34282 34301 34321 34341 34361 34380 34400 34420 20 221 34 439 34 459 34 479 34 498 34 5^8 34 537 34557 34577 34596 34616 20 222 34635 34655 34674 34694 34713 34 733 34 753 34 772 34 792 34 811 19+ 223 34830 34850 34869 34889 34908 34928 34 947 34967 34986 35005 19 224 35 025 35 044 35 064 35 083 35 102 35 122 35 141 35 160 35 180 35 199 19 225 35218 35238 35257 35276 35295 35 315 35 334 35 353 35 372 35 392 19 226 35 411 35 430 35 449 35 468 35 488 35 507 35 526 35 545 35 564 35 583 19 227 35 603 35 622 35 641 35 660 35 679 35 698 35 717 35 736 35 755 35 774 19 228 35 793 35813 35832 35851 35870 35 889 35 908 35 927 35 946 35 965 19 229 35984 36003 36021 36040 36059 36078 36097 36 116 36135 36154 19 230 36173 36192 36 211 36229 36248 36267 36286 3630^ 36324 36342 19 231 36361 36380 36399 36418 36436 36455 36474 36493 36 511 36530 19 232 36549 36568 36,586 36605 36624 36642 36661 36680 36 6g8 36717 19 233 36 T36 36 754 36 773 36 791 36 810 36829 36847 36866 36884 36903 19 234 36922 36940 36959 36977 36996 37014 37033 37051 37070 37088 18+ 235 37 107 37 125 37 144 37 162 37 181 37 199 37 218 37 236 37 254 37 273 18 236 37291 37310 37328 37346 37365 37383 37401 37420 37438 37 457 18 237 37475 37 493 37 511 37 53o 37 548 37566 37585 37603 37621 37639 18 238 37 658 37 676 37 694 37 712 37 731 37749,37767 37785 37803 37822 18 239 37 840 37 858 37 876 37 894 37 912 37 931 37 949 37 967 37 985 38 003 18 240 38021 38039 38057 38075 38093 38 112 38 130 38 148 38 166 38 184 18 241 38 202 38 220 38 238 38 256 38 274 38292 38310 38328 38346 38364 18 242 38382 38399 38417 38435 38453 38471 38489 38507 38525 38543 18 243 38561 38578 38596 38614 38632 38 650 38 668 38 686 38 703 38 721 18 244 38739 38757 38775 38792 38810 38828 38846 38863 38881 38899 18 245 38917 38934 38952 38970 38987 39005 39023 39041 39058 39076 18 246 39094 39 III 39 129 39 146 39 164 39182 39199 39217 39235 39252 18 247 39270 39287 39305 39322 39340 39358 39 375 39 393 39 4io 39428 18 248 39445 39463 39480 39498 39515 39 533 39550 39568 39585 39602 17+ 249 39620 39637 39655 39672 39690 39 707 39 724 39 742 39 759 39 777 17 250 39 794 39 811 39829 39846 39863 39.881 39898 39915 39 933 39950 17 N 12 3 4 5 6 7 8 9 250 251 252 253 254 255 256 257 258 259 260 26l 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 39 794 39967 40 140 40312 40483 40654 40 824 40993 41 162 41330 41 497 41 664 41 830 41 996 42 160 42325 42488 42 651 42813 42975 43 136 43 297 43 457 43616 43 775 43 933 44091 44248 44404 44560 44716 44871 45025 45 179 45332 45 484 45 637 45788 45 939 46 090 46 240 46389 46538 46687 46 835 46 982 47 129 47 276 47422 47567 47 712 398II 39985 40157 40329 40 500 40 671 40 841 41 010 41 179 41347 4^ 514 41 681 41 847 42 012 42 177 42341 42504 42 667 42 830 42991 43 152 43313 43 473 43 632 43 791 43949 44 107 44 264 44420 44576 44731 44886 45 040 45 194 45 347 45 500 45652 45803 45 954 46 io5 46255 46 404 46553 46 702 46850 46997 47 144 47 290 47436 47582 47727 39 829 39 846 39 863 40 002 40 019 40 037 40 175 40 192 40 209 40346 40364 40381 40518 40535 40552 40 688 40 705 40 722 40 858 40 875 40 892 41 027 41 044 41 061 41 196 41 212 41 229 41 Z^2, 41 380 41 397 41 531 41 547 41 564 41 697 41 714 41 731 41 863 41 880 41 896 42 029 42 045 42 062 42 193 42 210 42 226 42357 42374 42390 42521 42 537 42 553 42 684 42*700 42 716 42 846 42 862 42 878 43 008 43 024 43 040 43 169 43 185 43 201 43329 43345 43361 43 489 43505 43 521 43 648 43 664 43 680 43 807 43 823 43 838 43 96S 43 981 43 996 44122 44138 44154 44279 44295 44 311 44436 44451 44467 44592 44607 44623 44 747 44762 44778 44902 44917 44932 45 056 45071 45 086 45 209 45 225 45 240 45 362 45 378 45 393 45515 45530 45 545 45 667 45 682 45 697 45 818 45 834 45 849 45 969 45 984 46 000 46 120 46 135 46 i5o 46 270 46 285 46 300 46 419 46 434 46 449 46 568 46 583 46 598 46 716 46 731 46 746 46 864 46 879 46 894 47 012 47 026 47 041 47 159 47 173 47 188 47305 47319 47 334 47451 47465 47480 47596 47 611 47 625 47 741 47 756 47 770 39881 39898 39915 40 054 40 071 40 088 40 226 40 243 40 261 40398 40415 40432 40569 40586 40603 40 739 40 756 40 773 40 909 40 926 40 943 41 078 41 095 41 III 41 246 41 263 41 280 41 414 41 430 41 447 41 581 41 597 41 614 41 747 41 764 41 780 41 913 41 929 41 946 42 078 42 095 42 III 42 243 42 259 42 275 42 406 42 423 42 439 42 570 42 586 42 602 42 732 42 749 42 765 42 894 42 911 42 927 43 056 43 072 43 088 43217 43233 43 249 43 377 43 393 43 409 43 537 43 553 43 569 43696 43 712 43 727 43 854 43 870 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4 981 99 167 99 171 99 176 99 180 99 185 99 189 99 193 99 198 99 202 99 207 4 982 99 211 99 216 99 220 99 224 99 229 99 233 99 238 99 242 99 247 99 251 4 983 99 255 99 260 99 264 99 269 99 273 99277 99282 99286 99291 99295 4 984 99300 99304 99308 99313 99317 99322 99326 99330 99335 99ZZ9 4 98s 99 344 99348 99352 99357 99361 99366 99370 99 374 99 379 99383 4 986 99388 99392 99396 99401 99405 99410 99414 99419 99423 99427 4 987 99 432 99 436 99 441 99 445 99 449 99 454 99458 99463 99467 99471 4 988 99 476 99 480 99 484 99 489 99 493 99498 99502 99506 99 511 99515 4 989 99520 99524 99528 99533 99.537 99542 99546 99550 99555 99 559 4 990 99564 99568 99572 99577 99581 , 99 585 99 590 99 594 99 599 99 603 4 991 99607 99612 99616 99621 99625 99 629 99 634 99 638 99 642 99 647 4 992 99651 99656 99660 99664 99669 99673 99677 99682 99686 99691 4 993 99 695 99 699 99 704 99 708 99 712 99717 99721 99726 99730 99 734 4 994 99 739 99 743 99 747 99 752 99 756 99 760 99 765 99 769 99 774 99 778 ■ 4 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Page 4. II. 276 3. 2.6888+ 4. 1. 10065+ I. 34 12. 408 4. .8298+ 5- 3-8645+ 2. .45 Page 8. 5- 1.7099+ 6. 163 3. 36 I. 1296 6. 1.6509+ 7. 88888 4- 74 2. 4445 7. 3-4956+ 8. 90647 5- 73 3. 3507 8. 2.8189+ 9- 22216 6. 78 4- 5104 9- 1.2599+ 10. 12345 7. .014 5- 60.84 10. 1.4422+ Page 18. 8. .029 9. .043 10. .085 6. 7. 8. 91.052 II. 12. 13. 3.4622+ 1.5874+ 2.2239+ I. 2. 3- 4. 5. 6. 7. 8. 9. 10. $47.70 74.90 61 11. 1.30 12. 9.80 Page 6. 1. 439 2. 141 3- 147 4. 726 5- 3-43 6- 115 9- 10. 12. 2. 3. 4. 5. 1-3 ^% M 3.8 2.8 I33-I 13-25 9V, 14. 15. 16. 17. 18. 19. 20. 21. 2. 1 1 79+ 2.08008+ 1. 91 29+ 7.6288+ 66.9792+ 5.0528+ .7528- 1.8171+ Page 12. 78.40 50.72 55.31 17.40 713.66 213-44 469.92 Page 19. 7. 238 6. 4.02 I. $46.25 8. 20.50 Page 10. I. 32 2. 35.085 9. 9.04 I. •5555+ 2. 1.2624+ 3- 62.05 to. 8.25 2. 1.3444+ 3. 1.2950+ 4. 5-21 ^ -vi 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewals only: Tel. No. 642-3405 Renewals may be made 4 days priod to date due. Renewed books are subject to immediate recall. Ill LD- SENT ON ILL AUG 2 3 1995 U. C. BERKELEY Jl ,xT^?o^i^n^9^;^'"!i°^o UniSS of cSrnia (N8837sl0)476 — A-32 Berkeley LIBRARY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. THIS BOOK IS DUE BEFORE CLOSING TIME ON LAST DATE STAMPED BELOW '■IBRARY mi OCT 18 U D LP |ia'64-iOPM ni l ttRY USE 0CT2i, ' 64 f" R efe? t > L.D ocDgr-.avsf.iM LD 62A-50m-2,'64 (E3494sl0)9412A General Library University of California Berkeley