Ver we < : SAN Sas SN THE UNIVERSITY OF ILLINOIS LIBRARY |) From the collection of » Julius Doerner, Chicago Purchased, 1918, SB+O-84 513.2 Gols | MATHEMATICS UBRARY ¢' e 1 i fo: VV. GOODHUPF, CHICAGO. Entered according to Act of Congress, in the year 1885, By 8S. W. GOODHUE, in the Office of the Librarian of Congress, at Washington, D. C. CHICAGO: Printed by H. H. HOFFMANN & Co. MATHEMATICS LIGRARL INTRODUCTION. 1. Addition, Subtraction, Multiplication and Division are commonly termed the four fundamental principles of Arithmetic. Operations in Addition, Subtraction and Division require a certain amount of mental effort which cannot be materially dimin- ished by any known labor-saving appliances. The expert arithmetician may shorten the time, but he cannot escape the labor incident to these processes. 2. But Multiplication is an abbreviated form of Addition, the distinction being that by one process we laboriously ascertain the sum of several unlike numbers, while by the other we more easily compute the sum of many like numbers. As in some branches of industry, one man aided by machinery may accomplish the work of 10 men, or may abridge the time, as well as improve the quality of his own labor in like proportion; so in Multiplication, a skillful use of mental appliances may effect a corres- ponding economy of time and effort. 01425 4 INTRODUCTION. 3. These points may be illustrated by two examples; the first showing the time-saving value ~ of expertness; the second, the labor-saving efficacy of abbreviation. I. A tailor sews upon 13 garments, the number of buttons expressed by the mar- ginal figures: How many buttons in all ? Solomon Slowbones points with his finger and drawls with his mouth, thus: “s-e-v-e-n a-n-d o-n-e a-r-e e-i-g-h-t, a-n-d f-o-u-r a-r-e t-w-e-l-v-e,” etc., while Ed. Expert, BD DO RH KR DDR BH BH — Hm Co bo bo ® WW Ol WwW > bo ee eee ae ee TU pi ~] holds his tongue, groups several figures mentally, and thinks nimbly, 12, 22, 38, 48, ete., or taking both columns at once, 38, 78, 112, 148, 171, 208. II. A tailor sews upon 13 coats, 16 buttons each: How many buttons in all? This, as a problem in Addition, requires the writing of the number 16 thirteen times; but by the abbreviated process of Multiplication as commonly practiced, 16 is written once with 13 under it—4 figures instead of 26—and then brain and hand work together, as shown on the following page. INTRODUCTION. 5 1 6 1 Brain work 3x6=18 Hand work 8 exe to 3+1—4 £6 6 6 ag Poe | 1 8 8 6+4=10 0 I+1= 2 2 Ed. Expert, abbreviating the abbreviation, with less brain work better directed and with no hand work at all, thinks, as the lightning flashes, . 19 tens and 6 threes, 208. Has he multiplied 16 by 13? By no means. He is exempted from so hard a task by his famil- iarity with the relations of numbers. He has simply apprehended 1910 as a relative approximation to 16 13, and 63 as the difference to be added. 4. With almost equal facility, in each case substituting for a difficult computation, AN EASY COMPUTATION AND A DIFFERENCE, he solves the following problems: 6 INTRODUCTION. 1. 17x19 8x 40 +3 323 R27 x27 17x 40 +49 729 3. 29x84 50x50 —64 2436 1 §382x86 32x 100 —8xX56 2752 " 13286 ¥% x 8600(2867) —115 2752 5. 36X38 40x40+-8 = 9407 ee 6. 44x48 1546x100 oe 2113 7. 57X64 60X61 i 3648 8. 64x64 35+4x100 +196 4096 9. 73x89 62 x 100 +297 6497 10. 263X278 270270 4-214) 7S 5. For all factors under 100, and for multi- tudes of greater ones, the GEOMETRICAL BIRD METHOD of multiplication, as developed in this work, fur- nishes the average accountant, if not the average school-boy, a system of simple and intelligible data for various mental formulas, easy, rapid and accu- rate; an insight into which will release him at once and forever from the slow, laborious and often in- correct drudgery of the ordinary written process. 6. By the geometrical diagrams with which the work is illustrated, integrity of process, and accu- INTRODUCTION. ~ 7 racy of result, are infallibly demonstrated,—the mind being, as it were, electrified by a broad scien- tific comprehension of the nature and scope of the problem, as a wHoxz, instead of being confused and lost in the multiplicity of its parts. 7. For example, with so simple a problem as 13X16, the ordinary cipherer is seldom quite sure of his correctness. His computation has many parts, as a tub has many staves. The tub, as a whole, is an incomprehensibility. He can only apprehend one staveatatime. So, fearing a hidden leak, he carefully examines stave by stave, and joint by joint, and finds only a negative assurance of soundness in the lack of a positive discovery of defect. He has no comprehensive idea of what the product of 13X16 ought to be. 8. Let him now inspect this 3 diagram, representing the space occupied by 13 rows of 16 but- 19 tons each, separated by crossed _ threads into 4 parts, viz: 10 tens 3 tens + and 6 threes. 6 tens 8 INTRODUCTION. Without ciphering, he may now perceive just what the product of 13x16 must be. His tub is glass, and through its transparency gleams its infallibility. 9. The general divisions of the subject are as follows :— I. The Multiplication Table, considered, 1st, in its superficial aspect, as a collection of 144 simple products to be memorized in childhood, and 2nd, in its deeper significance, as an agency in the devel- opment of countless larger products, by virtue of the geometrical position, or the mechanical manipu- lation of the figures of which it is composed. II. Multiplication, as a language, its letters, syllables and product-words. Slow and laborious writing and spelling backwards, carrying and blunder- ing, compared with quick, easy and demonstrably accurate reading forwards. Il. 99x99, or mental combination of several memorized partial products as syllables of a product- word—symbolized by the body, head and wings of of the Geometrical Bird. IV. Written Multiplication, or manual combi- nations of partial products, intelligibly arranged in INTRODUCTION. 9 the Diamond or Pyramid form. In this system, partial products being written entire, there is NOTHING TO CARRY, (except in the final addition of partial products), and consequently, a most prolific source of confu- sion, and error is effectually avoided. 10 THE MULTIPLICATION TABLE. THE MULTIPLICATION TABLE. 10. A small child is taught to say, and to re- member the saying, 3 times 5 are 15. A parrot learns the same lesson, and then brute and human are equal. Hach repeats a form of sound without sense, of words without meaning, and one knows as much of arithmetic as the other. 11. The child takes another lesson—an object lesson, of fingers on three hands, or of buttons, 3 on each of five gloves; and by actual count of the entire 15, there comes to his mind a comprehension of the meaning of the repeated words, a demonstra- tion of the verity of the remembered theorem. Relations are now changed. The boy has knowl- edge, while the bird has only learning. 12. A few years later, the boy in partnership with a pencil, having learned the Multiplication Table and the Rule, is enabled, by writing 14 figures and making as many minor computations, to cipher out the product of 99x99. He could no THE MULTIPLICATION TABLE. La more do this without the pencil than the pencil could do it without him. Now, the parrot has learning, and the boy more learning, but he lacks knowledge. If the correctness of his cipher- ing is questioned, he dare not affirm it, for he can- not demonstrate it; but doubtfully reviews the work in a blind hunt for errors. He knows not how many 99x99 ought to be. 13. He takes a larger lesson in buttons, 100 rows of 100 buttons each, arranged as a geometrical square. One row being taken from a side and one from an end, 99x99 buttons remain. How many have been removed ? Not 200, for the corner but- ton was common to both rows. Then 200 minus 1 being taken from 100 hundred, there remain 98 hundred plus 1, or 9801. No pencil is needed to make this computation; no external proof to verify it. Independent, transparent, self-evident, the tub stands squarely on its own bottom. No questioning can now shake the boy’s confidence. He knows what 99x99 must be; and again the child with knowledge is superior to the parrot with learning. 14. There be multitudes of homo-parrots—school- boys, teachers, men of letters, and men of business 12 THE MULTIPLICATION TABLE. —with abundant learning of the multiplication table and the rule; but with scant knowledge of the underlying principles of Multiplication: They need to count buttons and dissect geometrical birds. 15. A child’s fingers are too few for larger demonstrations; and buttons in quantities are not conveniently portable—besides being noisy and troublesome” when dropping and _.rolling on the school-room floor—and so it was a happy thought of old Pythagoras, as tradition affirms, the con- struction of a geometrical table, the little squares of which, though convincing not the finger-tips like buttons, yet standing for buttons to the eye, may be counted, and thus convey to the child-mind a practical demonstration of each proposition, from 1x 1=—1 up to the grand climax 12x 12—144, 16. The beginning of knowledge, as distinet from learning, comes only from such actual count- ing of tangible objects representing small products. Later lessons may be based on immaterial buttons in imaginary rows.* * “Seemed washing his hands with invisible soap, In imperceptible water.”—Hood. THE MULTIPLICATION TABLE. 13 17. Our historic world was thousands of years old before it saw the first Multiplication Table; and even now there exist many tribes of men who can- not count their own fingers and toes. A man, wondrous learned and wise above his fellows was he, whether Pythagoras or another, who invented the geometrical table. But imitators now need not his full measure of wisdom. Indeed the 144 squares of the table may be correctly filled by a person utterly ignorant of Multiplication, or even of Addi- tion. He has only to number consecutively the 144 quarter-inch marks on a yard-stick, and then transfer to the first line of the table the first 12 numbers; to the second line, the first 12 which occur at half-inch intervals, and so on to the twelfth line, the appropriate numbers for which are found three inches apart. 18. Can these dry bones live? Can the soul of numbers vivity a dull geometrical. clod thus ignorantly conceived and mechanically developed ? Is there in this assemblage of figures any hidden thing beyond the 144 products which childhood cons by rote and crones in sing-song ? Investiga- tion proves that by virtue of their geometrical posi- 14 THE MULTIPLICATION TABLE. tion alone, these soulless characters have marvel- lous potency in the solution of stupendous problems, invisible if not incomprehensible to the childish student.* 19. Here, for example, is a miniature table in which the ordinary observer sees nothing beyond 2x2—4. If we add together the upper numbers, 1 and 2, and the side numbers 1 and 2—the number 1 doing double duty—we have 3x3, the product of which equals the sum of all the num- bers in the table. If, instead of add- ing the factor figures, we consider them as expressing the number 12, then the product of 12x12 appears by turning the point a to the left and adding per- pendicularly. Ifthe point a be turned to the right and the reversed figures be read as 21X21, the product 441, appears by adding as before. If the point a be turned upward or downward, and the © factors be read as 12X21, the product 252 is simi- larly obtained. 20. In the next table may be found the pro- * **There are more things in heaven and earth and the multipli- cation table, Horatio, than are dreampt of in your philosophy.”— Shakespeare & Co. THE MULTIPLICATION TABLE. 15 duct of 1+2x1+2-+3, or 3x6=the sum of all the numbers in the table; also with « at the Left, 12x 123=—1476 Bottom, 12 321=3852 Right, 21 x 321=6741 Top, 21 x 123=2583 If the side factors 1 and 2 be taken as factors only, and not included in the addition of products, we have with a at the Left, 12 23=276 Bottom, 12« 32—=384 Right, 21 x 32=672 Top, 21 « 23=483 21. In this table may be found the products of 4x 4=16 13x 13=169 13x 31=403 31x 31=961 With a cipher in each vacant square may be developed the products of 103 x 103=10609 301 x 103=31003 301 301=90601 16 THE MULTIPLICATION TABLE. 22. In this section representing the four central squares of the table, under and at the right of the factors 6 and 7, the product of 6+7x6+7, or 13X13 equals 3 the sum of the four numbers, | 6 169. Addition with the point 4 a3 a turned to the left developes ) 9 the product of- 67 x 67 = 4489. “ A change in the position of a a ‘ two figures, developes in the = E second diagram the product of - f 607 x 607 = 368449. 2,;4 9 23. This section represents the four squares at the right lower corner of the table, under and at the right of the factors 11 ; os Bl 10 and 12. The sum of the 1 1 four numbers 529 is the 9 3 product of 11+12x11-+4 1 - 12 or 23X23. The pro- duct of 1112 x 1112 — 1236544 is obtained by addition with the point a turned to the left. The factor numbers 11 ll and 12 overlapped as in the margin 12 form by addition the number 122. 122 THE MULTIPLICATION TABLE. 17 The numbers in the “ l second diagram similarly 9 overlapped, develop by ad- dition, the point a being pel hee 3 4 turned to the left, the pro- | duct of 122x 122—14884. 2 24, Finally, the whole Multiplication Table may be regarded as a single problem, one factor being the sum of the numbers in the upper line, the other, of those in the left column, while the pro- duct is the sum of the entire 144 numbers. , This problem may seem at first sight a formidable one; but the elements of its solution are wonderfully simple. The upper line of the table may be seen to consist of six pairs of numbers, viz: 112, 211, 3 10, 49,58and6 7; the sum of each pair being 13, the average of each number 63, and the sum total 78. In the second line these proportions are doubled, and so on by a regular progression to the twelfth line. Conditions precisely similar appear if the table is scanned in columns. 25. Itfollows that the square of 64 or 42+ must bethe average number for the entire table, and there- fore the product of 421 by 144 must be the sum 18 THE MULTIPLICATION TABLE. total: also, that the same aggregate must result from squaring the number 78. By methods here- after explained, the square of 78 is mentally de- duced from the square of 75 or of 80—an easy computation and a difference—thus, 5634+450= 6084, or 6404—320—6084. The problem 424 x 144 divested of its fractional complication by substitut- ing the equivalent factors 169 x 36 is solved at sight, thus: 4854+1230—6084. 26. As the number 169, the square of 13, repre- sents the average aggregate of four squares, the question naturally arises, are there in the table any four numbers of exactly that sum? Itis a curious fact that the entire table is made up of 36 such sroups of four—one odd number and three even ones in each group—all symmetrically arranged around a common center. The figures comprising each group are equi-distant from the center of the table, and also equi-distant from each other; and stand at the angles of a perfect geometrical square —unless the table itself varies from that form. The sum of a side of one of these squares is invari- ably a multiple of 13; and the product of the largest by the smallest number is in every case equal to that of the other two. THE MULTIPLICATION TABLE. 19 These groups of 169 may be readily located by reference to the table (28), thus: 1 12 2 24 Si Saplé 4 48 12 144 Blot 10 120 9F108 By analysis, the fourth group is seen to consist of 4-1s and 4 12s wersesetand 9. 1¢ 13 13 13 13 MULTIPLICATION TABLE. 1 2! 4] 6] 8/10/12/14]16]18] 20] 29) 24 3] 6] g/12/15/18/21|24|27] 30) 33) 36 ~4{ 8{12] 16/20 | 24/28 | 32 40; 44} 48! 610] 15 | 20 | 95| 30/35 140145] 50) 55] 60 6/12] 18 | 24| 30 | 36/42/48] 54] 60) 66] 72 714] 21 | 28 |35 | 42] 49 | 56 | 63 | 70] 77] 84 8 {16 | 24/32 | 40 | 48 | 56 | 64] 72] 80] 88] 96 ee | | | | | es | | io) — So bo we (SX) 3) eS es) pha ce On (=P) (one) as 90 |{0Q | 110/120 re) 12 | 24 | 36 | 48 108 120] 132/144 20 THE MULTIPLICATION TABLE. 27. The same principle prevails in any square multiplication table, whatever its magnitude, the sum of the group of four numbers being in any case equal to the next square number outside the table. In an odd-numbered table, as 5X5 or 9X9, the central figure, 9 or 25, represents the average num- ber, and is surrounded, the first by 6 groups of 36 each, the second by 20 groups of 100 each. 28. In the accompanying table of 12x12 squares, heavy lines at 9x9 indicate the extent of a table of unit factors. Factors too large to be multi- plied mentally are necessarily subjected to a written process, by which several partial products are obtained and added together. For the numbers representing these partial products, the operator depends almost invariably, not on original compu- tations but on the remembered lessons of this lesser table. 29. We may imagine a person unable, from some mental defect, to memorize these lessons and yet able by using this table for reference to obtain and set in order for addition the partial products of factors of any magnitude by means of an improvised geometrical table like the one here represented. In THE MULTIPLICATION TABLE. 21 this table the product of 684 x 397 is obtained by placing one factor at the top, one at the left, the 9 partial pro- ducts within the 9 squares, and their sum 971,548, extended around the bottom and right of the table, the point a being uppermost during the addition. 30. By the ordinary process, instead of 9, then are developed 3 partial products—which may be traced in the 3 lines of squares in this table, viz: 7X 684 4788 90 x 684 6156 300 x 684 2052 271548 31. By either method, only the significant. figures of the partial products are written, the final ciphers being omitted. The full expression of the partial products in the table requires that the ciphers in each square be placed at the right of the significant figures. 22 THE MULTIPLICATION TABLE. Thus there appear: 1 product of units or ones, 28 Dae Po tens 56, 36, 92 ote hundreds, 42, 72,* 12 126 EAS ry ten hundreds, 54, 24 . 78 iss? hundred-hundreds 18 271548 32. An observable feature in the Multiplication Table (28), is the diagonal line of square numbers, 1, 4, 9, 16, etc. A square number being the pro- duct of equal factors, has but one place in the table, but other products are duplicated by the inversion of their factors, the No. 132, for example, appearing both as the product of 11x12 and of 12x11. Hence the two sections of the table as divided by this line, are exact duplicates. From a table minus one section, a boy minus one eye might learn the 144 products; but both sections as well as both . eyes are usually considered desirable. 33. Diagonal lines parallel to that of square numbers, consist of numbers diminished by square numbers. Thus in 5 lines opposite 36 are 35 32 27 20 I1 the differ’ce from 36 being 1 4 Ee * 72is a product of ten-tens, identical with hundreds. THE MULTIPLICATION TABLE. 23 - THE TABLE OF SQUARE NUMBERS. 34. A method of Multiplication without multi- plying is based on the use of a reference table of square numbers, written or printed, combined with a process of Addition, Subtraction, and Division. 35. Consecutive square numbers are identical with the sums of consecutive odd numbers. Hence a person ignorant of Multiplication may construct this table by a series of additions, thus: Numbers, | yi 5 4. 5 6 Squares, 1+3—4+5—9+7=—16+9=25+11=36 As the work advances, its correctness may be tested at every 10th number. Thus 4x4 being 16, 40x40 must be 1600. To change 40x40 buttons to 4141 buttons, 2 sides and a corner must be covered or 2X40+1, making the square of 41=— 1681. Then to change the square of 41 to that of 42, the next odd number 83, must be added making 1764. 36. Square numbers, except the number 1, are either even, and multiples of 4, or odd, multiples of 4 plus 1. The first 10 square numbers will be found on the following page. 24 THE TABLE OF SQUARE NUMBERS. 1 4,..9 16 25.36 49° G25 sie The terminal figures of these 10 are repeated in every succeeding 10. Thus if a number ends with 0) its square ends with 00 00 0000 1 or 9 1 remove from 0 1 2 or 8 2 * 4 DOr. 3 . 9 4 or 6 4 a 6 5 5 A 25 No square number ends with 2, 8, 3 or 7, or with an odd number of ciphers, or with 5 other than 25. These characteristics may be verified by the accom- panying table, which contains the squares of all numbers up to 99. 25 THE TABLE OF SQUARE NUMBERS. TABLE OF SQUARE NUMBERS, F8IG | LPOG VGLO | 19E9 VHSE | LéELE FOLGE | 1096 POLL | T89T FCOL | 196 V8P LFF VFI LéT iy { G I v9FS | 18Eé8 _————— O0OF9 OO6F 0096 0066 0091 0018 06 08 OL 09 OG OF 0€ 26 THE TABLE OF SQUARE NUMBERS. 37. The method of obtaining products by refer- ence to this table is based on the fact that the square of the sum of 2 factors includes the square of their difference, plus 4 times their product. The process is as follows: I. Find the square of the sum of the factors. II. Find the square of their difference. 111. Subtract one from the other. IV. Divide the remainder by 4 Example I. 9x3. 9+3=12 Square of 12, 144 9—3=—6 Square of 6, 36 Remainder, 108 4 of remainder, 27 38. Demonstration. If 4 cards of buttons, 3x9 on each card, be placed together as repre- | a ae sented in the accompany- ing diagram, they form, 6 1. An outer square, measured on each side by 12 buttons. 2. An inner vacant square, measured on each side by 6 buttons. THE TABLE OF QUARTER-SQUARES. 27 Example I. 57 x 38. 57+38—95. Square of 95, 9025 57—38=19. Square of 19, 361 Remainder, 8664 3 of remainder, 2166 TABLE OF QUARTER-SQUARES. 39. The necessity for a final division by 4, in each operation, has been obviated by dividing the entire table by 4, or in other words, constructing a table of quarter-squares. 40. The quarter-square of any number is equal to the square of half that number. Thus the quarter square of 12 is the square of 6, or 36; of 13, the square of 64 or 424. In this system, how- ever, the fraction + pertaining to the quarter-squares of odd numbers is dropped without affecting the result. The proximate quarter-square is the pro- duct of the two numbers proximate to the half, as of 13, 6X 7=42. 41. This table, like the table of square num- bers, may be constructed by successive additions ; the consecutive quarter-squares of even numbers being identical with the sums of consecutive odd numbers, and vice versa, per example on next page. 28 THE TABLE OF QUARTER-SQUARES. ven Nos. , 4 Bose 10 ‘12 4 Squares, 143= 4+5=9+7=1649=25+11=36 Odd Nos, 3 5 i 9 11 13 a Syrs., 2+4—6+6=12+8=20+10=—30+12=42 42. By reference to a table of quarter-squares, products are obtained as follows: x, 1. 9X 3. 9+3=12. Quarter-square, 36 9—3—6. 7 9 Remainder, 27 Ex. IT. 57 X 38. 57+38=—95. Quarter-square, 2256 57—38=—19. 2 90 Remainder, 2166 43. This method has the following earnest com- mendation, in “Sane’s Hicguer Aritumetic.” Kd. and Lond, 1857: “M. Anton Voisin published in 1817 a table of quarter-squares of all numbers up to 20000, which enables us to find the product of any two numbers up to 1000. Mr. Laundy has lately published a table of quarter-squares up to 100000. This work should be in the hands of every professional cal- culator.”’ Popular judgment, however, has consigned the system to oblivion, from which it is here resurrected in compliment to its ingeniousness rather than to its usefulness. MULTIPLICATION AS A LANGUAGE. 29 MULTIPLICATION AS A LANGUAGE. 44, Multiplication may be considered as a lan- suage, whereof the numerals are letters; the par- tial products, syllables, and the sum of the sylla- bles, product-words. 45. The alphabet of this language consists of the 10 numerals, 0 1 2 3 + 5 6 7 8 4 commonly termed the cipher and the 9 digits. The digits are also called significant figures. 46. A syllable is the product of two numbers, as 9X9, or any lesser factors. No larger syllables than those contained in the lesser Multiplication table, (28) occur in the ordinary written process ; but in mental computation, any product may be treated as a syllable when recalled by an act of memory, or when instantly computable, as of 12x 12, 15x18, 25x 25, 75 x 64. 47. Letters or syllables followed by ciphers assume magnified proportions, but are not thereby rendered more difficult of computation, Thus in the 30 SPELLING BACKWARDS. formula 80x 1200=96000, the 3 ciphers from the 2 factors mechanically supplement the simple product of 8x12, just as 3 dots are made to complete the writing of the word Division. 48. In any product word the number of sylla- bles equals the product of the number of digits in one factor by the number in the other, thus: 2, 3, or 4 digits in each factor produce respectively 4, 9, or 16 syllables; while 2, 3, or 4 digits multiplied by 1 digit produce 2, 3, or 4 syllables. es ae SPELLING BACKWARDS. 49. The word ‘‘ Multiplication”’ requires for its writing, some 50 distinct movements of pen or pencil, for its spelling, 14 letters; for its pro- nouncing, 5 syllables; but for its silent reading, only the passing glance of a practiced eye. 50. The various methods of Multiplication, as applied to factors under 100; for example, 46 x 48, exhibit contrasts equally striking. The ordinary computer, ignorant of syllables, takes the slowest CARRYING AND BLUNDERING. 31 and most toilsonie method, wasting time and energy on 50 manual movements in 14 figures and 2 lines, meanwhile making 10 computations—thus laboriously writing backwards with many letters a word which may be instantly read forwards with 2 syllables, thus 44x50+4x2. When all is done it is only half done. There are 24 chances of error in his 14 figures and 10 computations, and he seldom ventures to omit reviewing the entire work in search of the possible mistake. ep eres iSeeate CARRYING AND BLUNDERING. 51. The written process prescribed in the “Rule’’ of the school-books, although theoretically sound and philosophically correct, is practically the occas- sion of infinite blundering. While sufficiently simple to be apprehended and practiced by children 7 years of age, it is yet so complex as hardly to be comprehended by the same children when that age is doubled. “ Likewise ye Monkie, albeit he turneth about y. Crank of ye Hande-Organ, hath natheless Small Knowledge of ye Musickal Product thereof.’’— Antique Author. 32 CARRYING AND BLUNDERING. 52. The hidden spring of this defect is the ignoring and the unconscious mutilation of the natural syllables of a product. A child, unable to read the word, Multiplication, may yet succeed with - the syllables, Mul-ti-pli-ca-tion, and again be sorely puzzled by the no-syllables, Mu-lt-ip-li-cat-ion. So a child may comprehend the 9 syllables of the fac- tors 684 « 397, either within or without the lines of a geometrical table (29, 117); and may further apprehend the fact that each syllable is unques- tionably right and in its right place; and thata correct addition only is required to insure a correct product. | 53. Turning from these 9 simple syllables, all verified by the multiplication table, to the three complex partial products (30); each consisting of syllables butchered as soon as born—the tens chopped off from one and stuck on to another—what can the child make of them? They may be right, if the ciphering is right ; as the tune may be ground out if the crank is properly turned; but neither child nor monkey can distinguish the right from the wrong. 54. The school-boy, by the time he reaches the ‘‘first class,” is so thoroughly satwrated with the CARRYING AND BLUNDERING. 33 Addition and Multiplication tables, that he cannot miss the resulting syllable of 8-+-8 or 4x4. He fails, not in setting down the 6, but in carrying the 1. 55. The device of carryine, the child’s riddle, and the man’s life-long stumbling-block, is the source of nine-tenths of the every-day blundering in Multiplication. 56. If the world were restricted to one tool, it should be the jack-knife, which fits the universal pocket of boy or man, and is pre-eminently the implement of all work; if to one method of Multi- plication, the old one, with its little and big blades, the ‘‘table’’ and the ‘‘ rule,”’ has been fitted to the universal head, and like the ox-team, is to be com- mended for the slow and heavy labor it has accom- plished in the past. 57. But, the jack-knife being too delicate for the ship-carpenter and too clumsy for the surgeon, the broad-ax and the lancet have been evolved; and the ox-express is displaced by steam and electricity. So for the practical computer the Multiplication table is too short, the old written process too long and laborious. The child’s table, 12x12, should be supplemented by the man’s table, 99x99, not 34 LIGHTNING CALCULATORS. written, not memorized, but instantly computable; and the written process—never applied to factors under !00--shortened, simplified and freed: from the errors incident to carrying, should partake more of the electric alertness of the brain, less of the dull plodding of the hand. LIGHTNING CALCULATORS. 58. Partly in this direction, and partly oppo- site thereto, is the method of the ‘lightning caleu- lator’’; combining on the one hand more head- work with less hand-work, and on the other greater speed with less simplicity, By this method the product is written directly in one line, without the intervention of partial products. Ex. 684 x 397. 684 397 Written, 271548 Performed—the numbers carried being expressed in Italics : — 7x4 28 7X8 and 9x4 +2 94 7x6 and9x8and3x4 +49 135 9x6 and 3x8 +13 91 3X6 +9 27 LIGHTNING CALCULATORS. 35 59. The student may be assisted in compre- hending this process by practicing the following ex- amples,in which the digits correspond to the several orders of units numbered from right to left. i Dees 321 X 321. Ex. IT. 4321 x 4321. Ex. WI. 54321 54321. Ex. I is written, _ 321 321 103041 Performed: 1x1 1 (1 product of units. 1x2 and 2x1. 4 (2 products of tens. 1X3 and 2X2 and 3X1 IO (3 “ hundreds. 2x3 and 3x2 +] 13 (a? thousds. brag +i 10 (1 “ 10 thousds, 60. This operation may be made partly mechan- ical, by writing one of the factors with the digits transposed, thus: 123 on a moveable slip of paper and passing it from right to left under. the other factor, taking the products, or sums of pro- ducts, of digits perpendicular to each other in each of 5 positions, as follows: Position : 5 4. 3 2 1 Nature of Products. 1o-thousands. thousands. hundreds. tens. units. 321 321 321 A Ee WA | 123 123 123 123 123 ee ee mews arc (10.4 36 LIGHTNING CALCULATORS. 61. Arithmeticians generally are aware of the existence of this method; but comparatively few conquer the first discouragements attending its practice. Lightning calculators, like poets, are born, not made; and their exceptional powers can- not be transferred to their less gifted admirers. With the operator of only average capacity, this method increases the lability to error, and dimin- ishes the facility of its detection. ‘¢And yet shew I unto you a more excellent way.’—I Cor. XII, 31. THE GEOMETRICAL BIRD. 37 THE GEOMETRICAL BIRD. ee 62. In the miniature Multiplication Table (19), the product of 1212 is developed by adding the four numbers in a certain direction. The accompanying diagram instead of vias consisting of four squares of uniform boy 4 size, represents the space occupied by 12 rows of 12 buttons each, 10 2 separated by crossed threads into four sections, correspond- ing to arithmetical syllables. The relative proportion of these syllables suggests the similitude of four parts of a bird, viz: 1 ten-ten, the body, 100 buttons. 2 tens, a wing, PAB ied 2 tens, a wing, open 2 twos, the head, Ao 63. Dispensing with the formal lines of a table, a ready method is here suggested of obtaining the products of factors from 11 to 19, by either a written or a mental process, as shown at the top of next page: 38 THE GEOMETRICAL BIRD. Written process: Mental process: 1 EEX 13S salero 11+3 tens, 143 3 1x3 units 143 2 ISSAG AAG ae 13+6 tens, 208 6 3x6 units, 208 7 1 Xa 17+7 tens, 989 ff 7X7 units, 289 5 15x19 1456 15+9 tens, 28h 9 5x9 units, ; 285 64, The same method may be applied to larger — factors, but not with any practical advantage, thus: 4 4 4 4 4 3) 4 24 34 3 44 24X27 28 34x37 328 34x47 3h or 4.28 BY 37 3 47 7 7 7 er 7 7 648 1258 1598 THE GEOMETRICAL BIRD. 39 65. ae es 400 x 400 NM > = 400 40 4 ‘¢ Hach one had six wings; with twain he covered his face, and with twain he covered his feet, and with twain he did fly.” —Isa. VI, 2-3. Diagrams on the same scale representing the squares of 4444 and of 44444 would measure on each side, the first 22 inches. the second, 18 feet. 119. Product syllables arrranged for addition must each be represented by 2 figures, a cipher 70 THE PYRAMID FORM. being prefixed when the syllable is less than 10, except that at the extreme left the cipher may be omitted. Example 31632. a Es 3X3 09 SC Os ulin oO 1803 DS OKy LO, Oe 020618 32 12 62 32 9013609 1X3, 6561)3 506 030618 6x3, 3X1 1803 3X3 09 10004569 120.. The number of figures in any product is either equal to the number in both factors, asin the squares of 32, 317 and 3163, or one figure less, as in the squares of 31, 316 and 3162. THE PYRAMID FORM. 121. The diamond form may be condensed, and the partial products arranged in pyramid form, by coupling syllables of like order. THE PYRAMID FORM. Le Example 684 x 397. 6 8 4 397 6xX7+3x4 54 6x9+3x8, 8x7+9x4 7892 6x3 8x9 4X7 187228 271548 Coupled syllables may be written separately, except at the extreme left, when their sum exceeds 99, as in the following example: Ex. 897X789. 789 897 7X7+8x9 121 7X9+8Xx 8534 12739 meres; 9X7 567263 707733 122. In squaring a number it should be remem- bered that one factor must be doubled in computing wing products. Ex. 438. 4 38 8x8 64 3x3, 6X8 2448 42 32 So 160964 191844 THE INVERTED PYRAMID FORM. THE INVERTED PYRAMID FORM. 123. Practically, this form is preferable to the preceding ones, as the products of digits in perpen- dicular lines are taken first, and thereby the posi- tions of the several orders of units are determined at the outset. Ex. I 721834 Yaa a 8 3 4 7x8 2x3 Tose 4 560604 7X38 2) 94 35] 3711 7x4+8x1 36 601314 Ex. Il. 4372 8694. 8694 ; 4372 8x4 6x3 9x7 4X2 32186308 8x 3+4x6, 6xX713x4, 9Xx217x4 486946 8x714x9, 6X2+43x«4 92294 8x2+4x«4 32 38010168 124. As each digit in a factor is represented by 2 figures in its product syllable, so each cipher must be represented by 2 ciphers. THE INVERTED PYRAMID FORM. 73 Hx. I. 2703x8976. 2703 Ex. IT. 3024 2850. S-¥eieG 302 4 . 12 28 5 0 1442 — 184900 06001000 16630018 241620 560021 1932 0027 08 24 8618400 24262128 125. In examples lke the following, the entire product may be written in one line, or may be com- puted mentally. Ex. 307 x 906 7009 x 4008 307 i 0;,.059 9 0 6 4008 278142 28092072 126. With factors of an unequal number of digits the product forms are incomplete. Ex. 6789 x 24. 6789 24 6789 24 24 1228 $s * 1432 01636 * 1636 * 1450 ke 1228 es 24 * 162936 162936 74 RAPID MULTIPLICATION. RAPID MULTIPLICATION 127. Expert mental computers may abridge the written formula, and perform more rapid work by dividing the factors partially or wholly into sec- tions or periods of 2 figures each, thus making the product syllables larger in amount and less in number. The product of a period of 2 figures must occupy 4 places, ciphers being prefixed if occasion requires. Ex. 1312x914. Lge ooied 139, 12X14 1170168 13x 14+12x9 (”) 290 1199168 269 X 248. 269 248 26x24, 9x8 62472 25x17—1 ~~ (°) 424 66712 45672, 4567 452 672 20254489 90X67 (°°) §030 20857489 MULTIPLICATION. BY SUBTRACTION. 75 7128x7246. peed) ities 7-9 486 “x7, 12x24, 8x6, 49028848 area (2) 12% 22 (°) 952264 Tx 14-(°°) 98 51649488 MULTIPLICATION BY SUBTRACTION. 128. The product of any number multiplied by nines may be determined by annexing a cipher for each nine and then subtracting the original number. Ex. I. 76499. 76400 764 75636 DemonstTraTIon. 764 ones from 764 hundreds leave 764 ninety-nines. ieee e900 C099, 9999000 or 9990000 9999 wou 9989001 9989001 76 MULTIPLICATION BY SUBTRACTION. 129. With factors of a like number of figures, the process—either written or mental—may be abridged by annexing to a number J less than the multiplicand the successive remainders obtained by subtracting each figure of that number from nine, commencing at the left. Examples: | 68x99 6732 695 999 694805 7890 x 9999 78892110 7891x9999 78902109 The first example is thus performed: 67 with (6 from 9) 3 and (7 from 9) 2 annexed. 130. Demonstration. 68 rows of 99 buttons each, changed by taking one row from the side and adding one to the end became 67 rows of 100 each. The side row, 99, being 32 in excess of the end row, 67, the total number must be 6732. Again, the inner factors, 68 x 99, exceed the outer factors, 67: °>e aie by the product of differences (from 68)l1 xX 32. See Relative Factors. BREVITY. ak lA ae 131. Philological “Josh Billings” playfully pities punctilious pundits, who with infinite labor acquire just education enough to spell a word in one partic- ular way; but not enough to indulge themselves in 66 that variety of forms wherewith he, “ good easy man,’ may give his word a new dress—usually a very short one—for every day in the week. 132. The adept in the various short and easy methods developed in this work, may well be moved to a more hearty compassion for the unprogressive cipherer who knows only one process of multiplica- tion, and that a laborious, blind, and often inaccur- ate one. Whether ‘‘fonetik orthografy’”’ shall ever overcome popular opposition or not, no reasonable objection can be made to shorter, surer, and more intelligble methods of spelling products. The brevity of free and easy spelling is illustrated in the treatment of the following 10 problems, each of which is solved in 2 or more different ways. The . operations extend only to the development of partial products from which the full products may usually be read from left to right. 78 BREVITY. Problem. First Process. Second Process. l. 1962. 36186 40016 228 1s LR TO OE: 17412 1) 218 526 40% 872 3. 825x426. 320650 (8) 3550 308 (—19%) 3550 4. 9864297. (25) C50). pn | at x28} 76 gov 5728 ee x64 1728 De SLIT Sas 13606 « 103 17716 67 —>x 20 344 6) 1400 aoe 13524 x 40 18120 | 369 5% . 906 26428 X77); 4482100 x750 4821000 4996 < 200eteu 8. 46.52 1642.25 1 2325 a. 520 =e 162.75 9. . 32502. 9062500 4 10833333* B. 150 — le ee 1GS6 C87 Be 121645625 fc 1484375 9c 176 —59% 7421875 1200 : 1650 } Third Process. A. 46X47 and .25 2162.25 B. 3233 and 2500 10562500 Cc. 118-+19.5 and 752 137505625 18 19.5 351 * This product must be expressed by 8 figures, the last 4 being 2500 (119, 36.) ERRORS. 79 ERRORS: 133. The way to avoid errors in computation is _to avoid them.* There is no other way. ‘The "process of multiplication being learned in childhood, if in after life all its conditions be strictly observed, error is impossible, and each and every attempted pro- duct must infallibly be right. Failing this, the defect is not in knowledge nor in memory, but in attention and precision. When Grandame ‘‘ drops a stitch,’ it is not because knitting is too intricate or too laborious. Nothing can be easier than for her to make a correct stitch; but in the monotonous making of a thousand, she becomes tired or sleepy or inattentive. Her boys and girls all inherit in a greater or lesser degree, this trick of inattention; and consequently as errors in computation are likely to continue for some time to come, methods must be practiced for their detection and correction. _ * Horace Greely said: “The way to resume specie payments, is to resume.” 80 PROOFS WHICH DO NOT PROVE. PROOFS WHICH DO NOT PROVE, 134. Under the fallacious title of PROOFS, some arithmeticians have prescribed certain forms, which like the forms of criminal jurisprudence, sometimes result ina conviction of the wrong, never in a vindi- cation of the right. When a prisoner is not proven to be guilty, his innocence may be legally assumed, although that also is unproven, and so when acom- putation is not convicted of error, the evidence of its correctness is only negative. 135. These methods of ‘‘ proof’’ are 30. 1. Inverting the factors. 2. Dividing the product by a factor. 3. Casting out the nines. By inverting the factors the multiplier in the first operation becomes the multiplicand in the second; new partial products are introduced, and the agree- ment of the two products establishes a preswmption of correctness, strong or weak, according to the proficiency of the operator. 136. Dividing the product by one factor gives the other factor as a quotient; provided, that both PROOFS WHICH DO NOT PROVE. 81 operations are correct, which is not proven, and also provided that if both are incorrect, one error counterbalances another. 137. Ben. Blunderhead tries his hand at these two methods of proof, with the following result— his ciphering being omitted : Examples: Proof. 78 X 43=3344 43 x 78=3254 68 x 92—6246 6246+68= 99 As the two products of the first example do not agree, there must be an error either in the compu- tation or the proof, or both. He has doubled his work to no good purpose, and now his only remedy is to review carefully what he has performed carelessly. In the second example he has written 14 figures to obtain a false product, and 17 more to prove it by a false quotient. Ben’s chief recommendation for a clerkship is the fact that he can prove, in the last half of each day, the computations made in the first half. 82 CASTING OUT THE NINES. CASTING OUT THE NINES. 138. The question, how the correctness of a computation could be demonstrated by casting out the nines, has been to the partially informed, a mystery deep and awful as that of witchcraft, or of the casting out of devils. When the demonstration proves to be no demonstration, then the mystery vanishes. 139. Hvery ten equals 9 and 1; every hundred, 11 nines and 1; every thousand, 111 nines and 1. Hence the number, 7468 contains In 7000, 777 nines and 7, In 400, 44 nines and 4, In 60, 6 nines and 6, In 8, é 8. The 4 remainders are duplicates of the original digits, and in their sum, 25, are 2 additional nines and a final remainder of 7. This final remainder is the essential object of search, the number of nines being immaterial; and as the digits of any number may CASTING OUT THE NINES. 83 be considered as so many remainders in excess of an unknown number of nines, it is easier to cast the nines from them than to divide the whole number _by 9. The casting consists in dropping 9 at every opportunity from the sum of the digits, as in the number 7468, 7+4=—2+6—8 x8=7, or taking 8, 6, and 4 as 2 nines, the number 7 stands alone as a remainder. 140. An operation in multiplication is tested by casting the nines from each factor, multiplying the remainders together and casting the nines from their product. Then casting the nines from the main product, the two remainders should be equal. Their inequality demonstrates the existence of error. Then equality demonstrates—nothing! The product may be right: it may be wrong: but according to the theory of the test, it is presumably correct. 141. This device serves as a detective for errors not measurable by the number 9; but any falsity, however glaring, which involves a multiple of 9— including all transpositions of figures—may pass the test unchallenged, as appears by the example shown on the following page. 84 CASTING OUT THE NINES. B56 was. 423 4 32122728 403657 6038 60 36823098 1 Error, 36822098 2 Errors, 37822098 Tranposition, 36832098 9 instead of 0, 36823998 0 omitted, 3682398 Assumed product, 12 Remainder 3 6é 4 12 Remamder, 3 Remainder 3, Presumably correct. 2, Error demonstrated. 3, Presumably correct. If we know that 12 is not the true product, we know it by insight. Itis not disproved by casting out the nines. Language should not be perverted by applying the term proof to such a piece of no- evidence. DEMONSTRATION. 85 DEMONSTRATION, 142. The positive evidence that an assumed product is correct, as distinguished from the nega- tive evidence that no error is detected, has its primi- tive foundations in the actual counting of tan- gible objects representing small products. Every civilized child sooner or later demonstrates by his fingers that 2 fives are ten. The Multiplication Table is a systematic collection of small products which have been demonstrated in the counting of their squares by thousands of children. Memor- izing this table in childhood obviates the necessity of innumerable demonstrations by actual counting, otherwise required in the daily affairs of life. 143. From the known in the Multiplication Table, the child may push his demonstrations out- ward to the unknown, thus: If 12 times 9 are 108, 36 times nine are 3 hundreds and 3 eights; or if 12 times 12 are 144, 24 times 12 must be 288. The demonstrations of the problems 13 x 16=208 ( 8) 99 x 99=9801(13) 86 DEMONSTRATION. are brought within the mental grasp of childhood. In fact, the Geometrical Bird method, for all num- bers under 100, is from first to last to the adult mind, if not to the child mind, a self-demonstrating method. 144. We have witnessed the unsuccessful wrestling of Ben. Blunderhead with the problem, 68x92. (137) Tid. Expert solves it thus: 80% 8012513 or (0X) 902% 22-) O26 or 60X 100+ 8x32 His mind grasps the problem as a whole: the operation is self-demonstrating; and he knows that error therein is impossible. But Ben. fails to comprehend the process, and so to him there is no demonstration; yet there is even for him a partial demonstration. The product is the sum of the accompanying 4 partial products, which his memory may draw from the multiplication table. Thus far, he may comprehend that the 72 work is correct beyond a peradventure. So 54/6 renouncing all ‘‘ ways that are dark, and 12 9 tricks that are vain,’ such as mutilating partial products, and carrying from one to another, DEMONSTRATION. 87 or inverting the factors, or dividing the product, or casting out the nines, if he can now trust himself to read the product-word from the product syllables, or to add these numbers correctly, his task will be accomplished, the only element of uncertainty being in this addition. 145. To larger minds than Ben’s, larger prob- lems, as 684 x 397 may not be demonstrable as a whole, but the nine partial products (117), or the 6 partial products (121), may be examined and cross-examined one by one from right to left, or from top to bottom, and thus verified as absolutely right. Then if correctly added, their sum cannot be other than the true product. 146. Notso with the 3 partial products (30), obtained by mutilation of syllables and carrying. They cannot besoimplicitly trusted. They are right if they are not wrong, but begging the question is not sound logic. Universal experience shows that this method is everywhere and always a fruitful source of error. MULTIPLICATION. 88 CO ri C2 Ominrand DW l= Ors NM OO HH DrOlowiNa aA {Yen} DWP O1O HNO ww 10 SH eS DOI TNO SH <1 69 6 faa Drowdrsio sH OD 1IDSH CO RAM Dr OnmNwolo H om 1d CO OMm DMP OI AND SH SH 10 69 TD 6 10 41 GR OD SH ID Heo Dre Oily iNow cH 1 Drornc sas Ot IANO A TRANSPARENT PROBLEM. APPENDIX, 89 APPENDIX. The Combination Table. A case has been supposed (29), of a person unable, from some mental defect, to memorize the multiplication table, and yet able by reference thereto to obtain and set in order for addition the partial products of factors of any magnitude, by means of an improvised geometrical table. Proceeding on this principle, a combination of unit tables may be so arrangedas to indicate by means purely mechanical, the partial products of any and all factors, leaving the operator only the task of adding them. | The accompanying table of four sections, one representing units, two tens, one hundreds, consists of unit tables so inverted and arranged that factor digits, in distinctive type, run on internal lines, one horizontal and the other perpendicular, in order that any required partial products may be inclosed within a hollow square or rectangle. This table is sufficient for all factors up to 99. Factors of 3, 4, or 5 figures each, would respectively require tables of 9, 16 or 25 sections. 90 THE COMBINATION TABLE. THE COMBINATION TABLE. 99 x 99 TENS. UNITS. 81 | 72 | 63 | 54 | 45 | 36 72 | 64 | 56 | 48 | 40 | 32 63 | 56 | 49 | 42 | 35 | 28 21 | 28 | 35 | 42 | 49 54 | 48 | 42 | 36 | 30 | 24 | 18 | is | 24 | 30 | 36 | 42 | 48 | 54 45 | 40 | 35 | 30 | 25 | 20 | 45 | 15 | 20 | 25 | 30 | 35 | 40 | 45 36 | 32 | 28 | 24 | 20 | 16 12 | 16 | 20 | 24 | 28 | 32 | 36 a7 | 24 | 21 | 18 | 15 | 12 | 09 09 | 12 | 15 | 18 | 21 | 24 | 27 18 | 16 | 14 | 12 | 10 | 08 | 06 | | 06 | 08 | 10 | 12 | 14 | 16 | 18 09) 08) 0'7| 06) 05) 04 03 08) 04) 05) 06} 07; O8 O9 03 | 04 | 05 | 06 | 07 | 08 | 09 06 | 08 | 10 | 12 | 14 | 16 | 18 09 | 12 | 15 | 18 | 21 | 24 | 27 12 | 16 | 20 | 24 | 28 | 32 | 36 15 | 20 | 25 | 30 | 35 | 40 | 45 18 | 24 | 30 | 36 | 42 | 48 | 54 21 | 28 | 35 | 42 | 49 | 56 | 63 24 | 32 | 40 | 48 | 56 | 64 | 72 54 | 63 | 72 | 81 27 | 24 | 21 | 18 | 15.| 12 | 09 36 | 32 | 28 | 24 | 20 | 16 | 12 ) 45 | 40 | 35 | 30 | 25 | 20 54 | 48 | 42 | 36 | 30 | 24 | 18 63 | 56 | 49 | 42 | 35 | 28 | 21 72 | 64 | 56 | 48 | 40 | 32 | 24 81 | 72 | 63 | 54 | 45 | 36 | 27 i HUNDREDS. TENS, The method of operation is as follows: the factors being, for example, 74x63. Two squares of card-board or other material are so adjusted as to inclose on the factor lines. THE COMBINATION TABLE. 91 07 tens and 04 units horizontally, 06 tens and 03 units perpendicularly, as is here represented in diagram. ae 21 03 12 “02 07 06 05 04 08 02 O1f 01 02 03 04 49 06 o4 The point a being uppermost, the par- 21 2 tial products may be read, one in each 4212 corner of the rectangle, and the product is 24 perceived to be 4662. In a table of nine sections, the partial products of two 3-digit factors may be indicated by stretching threads squarely across the table so as to intersect the tens figure of each factor, and then adjusting the two squares to the factor figures expressing units and hundreds. The nine partial products then appear, one in each corner and one at each 92 THE COMBINATION TABLE. intersection of the threads with each other, and with the sides of the inclosure. In the accompany- ing diagram are thus shown the nine partial pro- ducts of 361 x 427, the crossed threads being repre- sented by dotted lines. THE COMBINATION TABLE. 93 The point a being uppermost, the par- 21 : 7 0642 Ep rerodacts appear as here arranged for 121907 addition. 9402 04 In this diagram may also be seen an orderly arrangement of the partial products of many lesser factors, formed by dropping one or more digits from the original numbers. o wn c uu = = —] csc = «© oc be oc) = wn HU THE GEOMETRICAL BIRD